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https://math.stackexchange.com/questions/1904493/fx-lfloor-x-rfloor-left-x-right-increasing-decreasing-even

# $f(x) = \lfloor x \rfloor - \left\{ x \right\}:$ Increasing, Decreasing, Even , Odd, And/Or Invertible? Define $\{x\} = x-\lfloor x \rfloor$. That is to say, $\{x\}$ is the "fractional part" of $x$. If you were to expand the number $x$ as a decimal, $\{x\}$ is the stuff after the decimal point. For example $\left\{\frac{3}{2}\right\} = 0.5$ and $\{\pi\} = 0.14159\dots$ Now, using the above definition, determine if the function below is increasing, decreasing, even, odd, and/or invertible on its natural domain: $$f(x) = \lfloor x \rfloor - \left\{ x \right\}$$ I think that it is invertible only, but I can't seem to find the inverse. Am I correct saying that it is only invertible? Is it also increasing, decreasing, even, and/or odd? • You are indeed correct that $f$ is injective. It is also surjective, and hence invertible. But proving these facts, and finding an inverse, will take a bit of work. You should try graphing this function first - on a small region, say $[0, 3]$. This will also help with the other questions. – Noah Schweber Aug 26 '16 at 16:15 • Thanks for the confirmation! What about other properties? Is the funtion also increasing, decreasing, even, and/or odd? – Dreamer Aug 26 '16 at 16:18 • Have you tried graphing it? I think that will make the situation much clearer . . . – Noah Schweber Aug 26 '16 at 16:19 • I find it easier to visualise it as $f(x) = -x + 2 \lfloor x \rfloor$ – Shai Aug 26 '16 at 16:22 • @Regina Yes, that is correct. – Noah Schweber Aug 26 '16 at 16:27 We verify that $$g(x)=-f(-x)=\{-x\}-\lfloor -x \rfloor$$ is the inverse of $f$. In fact if $x\in \mathbb{Z}$ then $$f(g(x))=\lfloor \{-x\}-\lfloor -x \rfloor \rfloor -\{\{-x\}-\lfloor -x \rfloor\}=\lfloor 0+x \rfloor -\{0+x\}=x.$$ If $x\not\in \mathbb{Z}$ then $$\lfloor -x \rfloor=-1-\lfloor x \rfloor\quad\mbox{and}\quad\{-x\}=1-\{x\}$$ and $$f(g(x))=\lfloor \{-x\}-\lfloor -x \rfloor \rfloor -\{\{-x\}-\lfloor -x \rfloor\}\\ =\lfloor 1-\{x\}-(-1-\lfloor x \rfloor) \rfloor -\{1-\{x\}-(-1-\lfloor x \rfloor)\}\\ =\lfloor 2-\{x\}+\lfloor x \rfloor \rfloor -\{2-\{x\}+\lfloor x \rfloor\}\\ =1+\lfloor x \rfloor-(1-\{x\})=\lfloor x \rfloor+\{x\}=x.$$ $f(-1/2)=-3/2$ and $f(1/2)=-1/2$ so $f$ is neither even nor odd. $f(-1/2)=-3/2$ and $f(0)=0$ and $f(1/2)=-1/2$ so $f$ is neither increasing nor decreasing. $f$ maps $[0,1)$ bijectively onto $(-1,0]$ because $f(x)=-x$ for $x\in [0,1).$ $f(x+1)=2+f(x)$ so if $x-y=n\in \mathbb Z$ then $f(x)=2n+f(y).$ So for $n\in \mathbb Z,$ the function $f$ maps $[n,n+1)$ bijectively onto $(2n-1,2n].$ So $f$ is 1-to-1. And $\cup_{n\in \mathbb Z}(2n-1,2n]=\mathbb R.$ So $f:\mathbb R\to \mathbb R$ is a surjection. A 1-to-1 surjection is a bijection, and is invertible. Remarks: In case it is unclear that $f$ is 1-to-1, note that (1) when $n\in \mathbb Z$ and $x,y\in [n,n+1)$ then $f(x)\ne f(y)$ because $f$ is 1-to-1 on $[n,n+1)$. And (2) when $m,n$ are unequal integers and $x\in [m,m+1)$ and $y\in [n,n+1)$ then $f(x)\in (2m-1,2m]$ while $f(y)\in (2m-1,2m],$ and $(2m-1,2m]\cap (2n-1,2n]$ is empty, so $f(x)\ne f(y).$ It may be helpful to sketch the graph of $f.$

https://mathhelpboards.com/threads/finding-composition-series-of-groups.2483/

# Finding Composition Series of Groups #### Poirot ##### Banned My sum total of knowledge of composition series is: the definition, the jordan holder theorem and the fact that the product of the indices must equal the order of the group. With this in mind, can someone help with me with finding a composition series for the following: (1) Z60 (2) D12 (dihedral group) (3) S10 (symmetric group) I am not looking for just an answer but actually how to go about finding a series. #### Opalg ##### MHB Oldtimer Staff member My sum total of knowledge of composition series is: the definition, the jordan holder theorem and the fact that the product of the indices must equal the order of the group. With this in mind, can someone help with me with finding a composition series for the following: (1) Z60 (2) D12 (dihedral group) (3) S10 (symmetric group) I am not looking for just an answer but actually how to go about finding a series. As a general strategy, start with the whole group, look for a maximal normal subgroup. Then repeat the process. (1) should be easy, because $\mathbb{Z}_{60}$ is abelian and so every subgroup is normal. You start by looking for a maximal subgroup. For example, you could take the subgroup generated by 2, which is (isomorphic to) $\mathbb{Z}_{30}.$ Now repeat the process: find a maximal subgroup of $\mathbb{Z}_{30}.$ And so on. For (2), any dihedral group $D_{2n}$ has a subgroup of index 2 (therefore necessarily normal), consisting of all the rotations in $D_{2n}$ and isomorphic to $\mathbb{Z}_{n}$. That subgroup is abelian, so you can proceed as in (1). For (3), here's a hint. #### Poirot ##### Banned Ok thanks, it seems it should be quite easy but I have across an example which I don't understand. I am told that {0},<12>,<4>,Z48 is a composition series of Z48 but couldn't <24> be inserted between {0} and <12> since it is of order 2 in Z48? (sorry don't know how to do triangles denoting normal subgroup) #### Opalg ##### MHB Oldtimer Staff member Ok thanks, it seems it should be quite easy but I have across an example which I don't understand. I am told that {0},<12>,<4>,Z48 is a composition series of Z48 but couldn't <24> be inserted between {0} and <12> since it is of order 2 in Z48? (sorry don't know how to do triangles denoting normal subgroup) What you say is quite correct. The standard definition of a composition series requires that each component should be maximal normal in the next one. The series $\{0\}\lhd \langle12\rangle \lhd \langle4\rangle \lhd \mathbb{Z}_{48}$ fails that test on two counts. You could put $\langle24\rangle$ between $\{0\}$ and $\langle12\rangle$; and you could put $\langle2\rangle$ between $\langle4\rangle$ and $\mathbb{Z}_{48}$. #### Poirot ##### Banned oh sorry, I misread the text (it just said it was a normal series, not maximal). Well I will go away and follow your hints and advice. By the way, in dihedral groups, is it the case that the rotations are in a conjugacy class on their own?, as this would definitly ease the burden working out if a group is closed under conjugacy. Also, are they any rules about what the subgroups of a dihedral group are. Thanks again, and also I would like to say (doubt this is particularly controversial) that I think you are the best mathematician on this site. #### Deveno ##### Well-known member MHB Math Scholar oh sorry, I misread the text (it just said it was a normal series, not maximal). Well I will go away and follow your hints and advice. By the way, in dihedral groups, is it the case that the rotations are in a conjugacy class on their own?, as this would definitly ease the burden working out if a group is closed under conjugacy. Also, are they any rules about what the subgroups of a dihedral group are. Thanks again, and also I would like to say (doubt this is particularly controversial) that I think you are the best mathematician on this site. the conjugacy class of a rotation will always contain just other rotations (because the rotations form a normal subgroup of the dihedral group), but not all rotations are conjugate (necessarily). the reason for this is that for some n, D2n may have a non-trivial center, and central elements only contain themselves in their conjugacy class. for example, in D8 (the symmetries of a square), we have: [1] = {1} [r] = {r,r3} [r2] = {r2} (where the square brackets mean the conjugacy class of an element). even if we have a trivial center (like with D10 the symmetries of a regular pentagon), we still have: (rk)r(rk)-1= r (for k = 0,1,2,3,4) (rks)r(rks)-1 = (rks)r(rks) = (rk)(sr)(rks) = (rk)(r4s)(rks) = (rk)(r4s)(sr-k) = r4 which shows that [r] = {r,r4}. that is, conjugacy classes of a normal subgroup may still form a (non-trivial) partition of that normal subgroup (as another example, two cycles of the same cycle type may be conjugate in Sn, but not be conjugate in An​).

https://math.stackexchange.com/questions/2879619/what-is-the-cdf-for-a-partially-non-continuous-pdf

# What is the cdf for a partially non-continuous pdf? Suppose there is a pdf/pmf (?!) which places an atom of size 0.5 on x = 0 and randomizes uniformly with probability 0.5 over the interval [0.5,1]. Such that... $$f(x)= \begin{cases} 0.5, & \text{if}\ x=0 \\ {1\over (1-0.5)}, & \text{if}\ 0.5 ≤ x ≤ 1 \\ 0, & \text{otherwise} \end{cases}$$ Does the corresponding cdf then look like the following? $$F(x)= \begin{cases} 0.5, & \text{if}\ x < 0.5 \\ 0.5+0.5\cdot{(x-0.5)\over (1-0.5)}, & \text{if}\ 0.5 ≤ x ≤ 1 \\ 1, & \text{if}\ x > 1 \end{cases}$$ And how to calculate the expected value of this cdf formally? I suppose that $$E(x)={3\over 8}$$ ...but I dont know exactly how to formally deal with the intervalls as f(x) is not continuous. You can describe the probability density of this using the Dirac delta, viz. $\int_\mathbb{R}\delta(x)g(x)dx=g(0)$. In your case, $f(x)=\frac{1}{2}\delta(x)+1_{[\frac{1}{2},\,1]}(x)$. The second term is an indicator function, and integrates to the half of the probability the delta term doesn't cover; your $\frac{1}{1-0.5}$ factor is unnecessary. Notice I said probability density, not probability density function, because no function has the defining properties of $\delta$. (If you want some terminology, it's a measure. The popular name "Dirac delta function", used in the above link, is misleading.) Integrating gives the CDF $\frac{1}{2}\Theta(x)+(x-\frac{1}{2})(\Theta(x-\frac{1}{2})-\Theta(x-1))$, where $\Theta(y):=\int_{-\infty}^y\delta(z)dz$ is called the Heaviside step function, which really is a function. (Wikipedia denotes it $H$, but I've often seen people use $\Theta$ or $\theta$.) Equivalently, $\Theta\left(z\right):=\left\{ \begin{array}{cc} 0 & z<0\\ 1 & z\ge0 \end{array}\right.$. The CDF is $$\left\{ \begin{array}{cc} 0 & x<0\\ \tfrac{1}{2} & x\in\left[0,\,\tfrac{1}{2}\right)\\ x & x\in\left[\tfrac{1}{2},\,1\right)\\ 1 & x\ge1 \end{array}\right..$$ • How does your answer change (especially the resulting cdf) if the probability density randomizes uniformly over the interval [(1/3),1] with probability 0.5, ceteris paribus? – Moritz Sch Aug 11 '18 at 18:42 • @MoritzSch The PDF (CDF) would be $\frac{1}{2}\delta(x)+\frac{3}{4}1_{[\frac{1}{3},\,1]}(x)$ ($\frac{1}{2}\Theta(x)+\frac{3x-1}{4}(\Theta(x-\frac{1}{3})-\Theta(x-1))$). – J.G. Aug 11 '18 at 18:45 • Thank you! I see, I have to read up about Dirac delta, to understand this. – Moritz Sch Aug 11 '18 at 19:14 • Also I've realised these CDF formulae of mine simply won't work after $x=1$. – J.G. Aug 11 '18 at 19:29 In what sense this is a density function will bear examination. Normally one says $f$ is a probability density function for the distribution of a random variable $X$ if $$\Pr(a<X<b) = \int_a^b f(x) \, dx$$ for all values of $a,b.$ But $\displaystyle \int_a^b \cdots\cdots\,dx$ means an integral with respect to Lebesgue measure, which is the measure that assigns to every interval $(c,d),$ for $c<d,$ its length $d-c.$ That measure assigns $0$ to an interval that is only a point, so the integral of any function over that point is $0.$ However, suppose one integrates with respect to a measure that assigns measure $1$ to the one-point set $\{0\}$ and assigns the length of every interval to that interval if $0$ is not a member of the interval. Then what you've got is a density. But there's no need to go into that in order to answer your question about the expected value. The c.d.f. is $$F(x) = \Pr(X\le x) = \begin{cases} 0 & \text{if } x<0, \\ 1/2 & \text{if } 0\le x \le 1/2, \\ x & \text{if } 1/2<x\le 1, \\ 1 & \text{if } x\ge1. \end{cases}$$ The expected value is $$\operatorname E(X) = 0 \cdot\Pr(X=0) + \int_{1/2}^1 x\cdot\left(\frac 1 2 \, dx\right) = \frac 3 8.$$ Suppose $m$ is the measure described above, assigning measure $1/2$ to $\{0\}$ and the length of each interval to the interval if it does not contain $0.$ Then one can write $$\operatorname E(X) = \int_{-\infty}^\infty xf(x)\,dm(x)$$ and its value will be $3/8.$ • How does your answer change (especially the resulting cdf) if the probability density randomizes uniformly over the interval [0.25,1] with probability 0.5, c.p.? Does it then look like: $$F(x)= \begin{cases} 0.5, & \text{if}\ x < 0.25/ \\ 0.5+0.5\cdot{(x-0.25)\over (1-0.25)}, & \text{if}\ 0.25 ≤ x ≤ 1 \\ 1, & \text{if}\ x > 1 \end{cases}$$ And how does calculation of the expected value change? Is it: $$E(x)= 0*\text{Pr}(X=0)+\int_{1\over 4}^1\ (...)={3\over 16}$$ What is the (...) – Moritz Sch Aug 11 '18 at 19:00 • E(x)=(...)=5/16 Sorry for the mistake, my bad... But still the (...) and the actual form of the cdf in my original reply to your answer is not clear. Thank you in advance! – Moritz Sch Aug 11 '18 at 19:13 • ....And F(x) = 0 if x<0 is also missing in my reply (cdf) to your comment. But I understand this part. I also messed up the inequality signs, but no further explanation needed on that... Things to clarify would be: actual form of E(x) and the form of F(x) if 0.25 < x </= 1 – Moritz Sch Aug 11 '18 at 19:19 • @MoritzSch : In the expression $\displaystyle \int_a^b \cdots\cdots \, dx,$ the dots just mean that what is said is true regardless of which function goes where those dots are. – Michael Hardy Aug 12 '18 at 15:14

https://math.stackexchange.com/questions/3064391/simplify-sqrt4-frac162x616x4-is-frac3-sqrt42x22

# Simplify $\sqrt[4]{\frac{162x^6}{16x^4}}$ is $\frac{3\sqrt[4]{2x^2}}{2}$ (I've been posting a lot today and yesterday, not sure if too many posts are frowned upon or not. I am studying and making sincere efforts to solve on my own and only post here as a last resort) I'm asked to simplify $$\sqrt[4]{\frac{162x^6}{16x^4}}$$ and am provided the text book solution $$\frac{3\sqrt[4]{2x^2}}{2}$$. I arrived at $$\frac{3\sqrt[4]{2x^6}}{2x^4}$$. I cannot tell if this is right and that the provided solution is just a further simplification of where I've gotten to, or if I'm off track entirely. Here is my working: $$\sqrt[4]{\frac{162x^6}{16x^4}}$$ = $$\frac{\sqrt[4]{162x^6}}{\sqrt[4]{16x^4}}$$ Denominator: $$\sqrt[4]{16x^4}$$ I think can be simplified to $$2x^4$$ since $$2^4$$ = 16 Numerator: $$\sqrt[4]{162x^6}$$ I was able to simplify (or over complicate) to $$3\sqrt[4]{2}\sqrt[4]{x^6}$$ since: $$\sqrt[4]{162x^6}$$ = $$\sqrt[4]{81}$$ * $$\sqrt[4]{2}$$ * $$\sqrt[4]{x^6}$$ = $$3 * \sqrt[4]{2} * \sqrt[4]{x^6}$$ Thus I got: $$\frac{3\sqrt[4]{2}\sqrt[4]{x^6}}{2x^4}$$ which I think is equal to $$\frac{3\sqrt[4]{2x^6}}{2x^4}$$ (product of the radicals in the numerator). How ca I arrive at the provided solution $$\frac{3\sqrt[4]{2x^2}}{2}$$? • Hint, $\sqrt[4]{16x^4}$ does not simplify to $2x^4$. Rather, it becomes $2x$. Jan 6 '19 at 21:05 • On your first comment. Posting a lot is fine as long as you are actually working on each of the questions you ask (as you clearly are on this one), Jan 6 '19 at 21:13 $$\sqrt[4]{16x^4} = 2\vert x\vert$$ because $$\sqrt[4]{16x^4} = \sqrt[4]{(2x)^4}$$. (Note the absolute value sign since the value returned is positive regardless of whether $$x$$ itself is positive or negative.) The rest is fine, so from here, you get $$\frac{3\sqrt[4]{2x^6}}{2\vert x\vert} = \frac{3\sqrt[4]{2x^4x^2}}{2x} = \frac{3\vert x\vert\sqrt[4]{2x^2}}{2\vert x\vert} = \frac{3\sqrt[4]{2x^2}}{2}$$ As shown in the other answer, it is usually better to simplify within the radical so you don’t mess up with absolute values (for even indices). $$\sqrt[4]{\frac{162x^6}{16x^4}} = \sqrt[4]{\frac{2\cdot3^4x^2}{2^4}} = \frac{3\sqrt[4]{2x^2}}{2}$$ • In your second approach you have $\sqrt[4]{\frac{2\cdot3^4x^2}{2^4}}$. Why is it not $x^6$ in the numerator there? Jan 6 '19 at 21:24 • I simplified $\frac{x^6}{x^4}$ first. Jan 6 '19 at 21:25 • In your first answer, your numerator goes from $3\sqrt[4]{2x^4x^2}$ to $3\vert x\vert\sqrt[4]{2x^2}$. I see the benefit of pulling an x out in front of the radical but cannot see how you did that? Would it be possible to expand on that part if you have a minute? Jan 6 '19 at 21:40 • Sure. $\sqrt[4]{x^4} = \vert x\vert$, like how $\sqrt{x^2} = \vert x\vert$, $\sqrt[3]{x^3} = x$, etc. (As another point, note the use of absolute values when the index is even because the answer is always positive. For an odd index, as in cube roots, sign is preserved, so no absolute value is used.) Jan 6 '19 at 21:47 • A good way of thinking about it: $$\sqrt[4]{2x^6} = \sqrt[4]{2x^2}\cdot\sqrt[4]{x^4}$$ The first quartic root can’t be simplified, and it is therefore left as it is. The second simplifies to $\vert x\vert$. So, the $x$ comes from $x^4$. The $2x^2$ stays inside the radical. (I think your confusion is coming from the $2$ in front of the $x^4$. Bringing out $\sqrt[4]{2}$ would literally be... $\sqrt[4]{2}$ again, so you don’t bring it out as it won’t simplify anything. The exponent of the base must be greater than $4$ for it to be brought out.) Jan 6 '19 at 22:00 $$\sqrt[4]{\frac{162x^6}{16x^4}}=\sqrt[4]{\frac{81\cdot2x^2}{16}}=\frac{3\sqrt[4]{2x^2}}{2}.$$

https://brilliant.org/discussions/thread/binomial-coefficients-2/

× # Binomial coefficients $$\text{Binomial coefficients} { n\choose k}, n,k \in \mathbb{N}_0, k \leq n, \text{are defined as}$$ ${n \choose i}=\frac{n!}{i!(n-i)!}$. $\text{They satisfy}{n \choose i}+{n \choose i-1}={n+1 \choose i} \text{ for } i > 0$ $\text{and also} {n \choose 0}+{n\choose 1}+\cdots+{n \choose n}=2^{n},$ ${n \choose 0}-{n\choose 1}+\cdots+(-1)^{n}{n \choose n}=0,$ ${n+m \choose k}=\sum\limits_{i=0}^k {n \choose i} {m \choose k-i}.$ $$\text{How do I prove that}$$ ${n+m \choose k}=\sum\limits_{i=0}^k {n \choose i} {m \choose k-i}?$ $$\text{(Edit: This is also known as the Vandermonde's Identity.)}$$ $$\text{Help would be greatly appreciated. (I came across this in a book)}$$ $$\text{Victor}$$ $$\text{By the way, I used LaTeX to type the whole note :)}$$ Note by Victor Loh 3 years, 5 months ago Sort by: Here's a combinatorial proof. Question: From a group of $$m+n$$ students consisting of $$n$$ boys and $$m$$ girls, how many ways are there to form a team of $$k$$ students? ${n+m}\choose{k}$. If that team has $$i$$ boys, then it'll have $$k-i$$ girls. How many ways are there to choose $$i$$ boys from $$n$$ boys? $$n\choose i$$. How many ways are there to choose $$k-i$$ girls from $$m$$ girls? $$m\choose{k-i}$$. So for a fixed $$i$$, there are $${n\choose i}{m\choose {k-i}}$$ ways to form a team of $$k$$ students. Since $$i$$ could be any number from $$0$$ to $$k$$, we add the number ways to form the team for different values of $$i$$. So, our final count is, $\displaystyle \sum_{i=0}^k {n\choose i}{m\choose{k-i}}$ Since answers 1 and 2 are counting the same thing, they must be equal. [proved] - 3 years, 5 months ago yeah, that is the actual proof...by counting in 2 ways. - 3 years, 4 months ago Wow - 3 years, 5 months ago Deriving Vandermonde's identity is very simple . ( I am going to tell just the procedure ) First what you need to do is just right binomial expansion of $$(1+x)^{ n }$$ . again rewrite the binomial expansion of$$(x+1)^{ m }$$ . Note that you should expand $$(x+1)^{ m }$$ not $$(1+x)^{ m }$$. now multiplying these two expansions . we get the above summation which is equal to the one of the binomial coefficient in the expansion of $$(x+1)^{ m+n }$$ Hence prooved - 3 years, 4 months ago The last identity is known as Vandermonde's Identity. - 3 years, 5 months ago How do I prove it in a non-combinatorial way? - 3 years, 5 months ago Consider the coeff of $$x^k$$ of both sides in $$(1+x)^n (1+x)^m=(1+x)^{n+m}$$ - 3 years, 5 months ago Thank you. - 3 years, 5 months ago Yay thanks :) - 3 years, 5 months ago That's funny that you used $$\LaTeX$$ on the whole thing. Yes, I will surely work on this cool identity. :D - 3 years, 5 months ago

https://crank.com.br/pages/674a61-how-to-find-irrational-numbers

1$Find two irrational numbers between two given rational numbers. The ellipsis $(\dots)$ means that this number does not stop. $\sqrt{36}$ The comment about$2^x$still holds for$\sqrt{2}$, but if you were using a larger irrational number you might have to pick a bigger base than$2$. Solution: The numbers you would have form the set of rational numbers. stops or repeats, the number is rational. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Number System Notes. How to Write Irrational Numbers as Decimals. A rational number is a number that can be written as a ratio. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Now let us take any two numbers, say a and b. Learn how to find the approximate values of square roots. 1 answer. So we're saying between any two of those rational numbers, you can always find an irrational number. The definition of an irrational number is a number that cannot be written as a ratio of two integers. Yes. In this video, let us learn how to find irrational numbers between any two fractional numbers. We’ve already seen that integers are rational numbers. Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. Example: Find two irrational numbers between 2 and 3. And we’ll practice using them in ways that we’ll use when we solve equations and complete other procedures in algebra. There is no repeating pattern of digits. Before we go ahead to adding, first you have to understand what makes a number irrational. Irrational numbers are the real numbers that cannot be represented as a simple fraction. . Conclusion After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. An irrational number is a number that cannot be written as the ratio of two integers. 1/7 = 0. $0.475$ Therefore $\sqrt{36}$ is rational. All fractions, both positive and negative, are rational numbers. Which means that the only way to find the next digit is to calculate it. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. 2. Let’s summarize a method we can use to determine whether a number is rational or irrational. Decimals, fractions, and irrational numbers are all closely related. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. So, clearly, some decimals are rational. Many people are surprised to know that a repeating decimal is a rational number. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. $\sqrt{44}$. 1. Click here to get an answer to your question ️ How to find irrational numbers 1. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg This decimal stops after the $5$, so it is a rational number. 2/7 = 0. So what is an irrational number, anyway? 1. Its decimal form does not stop and does not repeat. Many people are surprised to know that a repeating decimal is a rational number. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 Students. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. We have also seen that every fraction is a rational number. If the decimal form of a number, Identify each of the following as rational or irrational: 3. Irrational Numbers on a Number Line. Rational and Irrational numbers both are real numbers but different with respect to their properties. Transcript. But choosing an irrational number in an interval, e.g. Let’s look at a few to see if we can write each of them as the ratio of two integers. Example 10 Find an irrational number between 1/7 and 2/7. Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. (285714) ̅. In short, rational numbers are whole numbers, fractions, and decimals — the numbers we use in our daily lives.. To decide if an integer is a rational number, we try to write it as a ratio of two integers. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. Decimals, fractions, and irrational numbers are all closely related. is irrational since exact value of it cannot be obtained. Join now. Let x be any number between a and b. Then, We have a < x < b….. let this be equation (1) Now, subtract √2 from both the sides of equation (1) 1. In this chapter, we’ll make sure your skills are firmly set. It is a contradiction of rational numbers.. Irrational numbers are expressed usually in the form of R\Q, where the backward slash symbol denotes ‘set minus’. 1 answer. test cases, where taking the square root of the number is True for irrational / complex, and False if the square root is a float or int. Decimal Forms $0.8,-0.875,3.25,-6.666\ldots,-6.\overline{66}$ A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We’ll take another look at the kinds of numbers we have worked with in all previous chapters. Follow. Introduction In Rational and Irrational Numbers post, we have discussed that is irrational. When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. If you are only looking for the square-root, you could use the. You can't fully write one down because it'd have to go on forever, so you’d need an infinite amount of paper. All right reserved. But an irrational number cannot be written in the form of simple fractions. In fact, any terminating decimal (decimal that stops after a set number of digits) or repeating decimal (decimal in which one or several digits repeat over and over a… $3=\frac{3}{1}-8=\frac{-8}{1}0=\frac{0}{1}$. 1/7 = 0. Are they rational? If the decimal form of a number. Proof that square root of 5 is irrational. An Irrational Number is a real number that cannot be written as a simple fraction.. Irrational means not Rational. We have seen that every integer is a rational number, since $a=\frac{a}{1}$ for any integer, $a$. Try it with the following problem, to make sure you have it right. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational. What type of numbers would you get if you started with all the integers and then included all the fractions? Top-notch introduction to physics. Ask your question. As we can see, irrational numbers can also be represented as decimals. A radical sign is a math symbol that looks almost like the letter v and is placed in front of a number to indicate that the root should be taken: Not all radicals are irrational. As we can see, irrational numbers can also be represented as decimals. DOWNLOAD IMAGE. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg between 0 and 1 is not always impossible, it just depends on what you want to do. Let us consider an example √2 and √3 are irrational numbers √2 = 1.4142 (nearly) √3 = 1.7321 (nealry) Now we have to find an irrational number which should lie between 1.4142 and 1.7321 A rational number is a number that can be written as a ratio of two integers. Can we write it as a ratio of two integers? }[/latex] one simple way would be rounding the irrational numbers to a certain place, say, the millionths place, and then find the average of the "trimmed-up" numbers. pavitra2 pavitra2 11.06.2016 Math Secondary School +5 pts. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. Remember that ${6}^{2}=36$ and ${7}^{2}=49$, so $44$ is not a perfect square. And we're going to start thinking about it by just thinking about the interval between 0 and 1. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. A rational number is a number that can be written as a ratio. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright © 2008-2019. ⅔ is an example of rational numbers whereas √2 is an irrational number. Irrational numbers don't have a pattern. The key is to find any like terms, and then add the coefficients together. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The examples used in this video are √32, √55, and √123. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Do you remember what the difference is among these types of numbers? How to find out if a radical is irrational There are a couple of ways to check if a number is rational: If you can quickly find a root for the radical, the radical is rational. 2. Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! Find irrational numbers between two numbers. Stack Exchange Network. To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. $\sqrt{5}=\text{2.236067978…..}$ So the number 1.25, for example, would be rational because it could be written as 5/4. For example, there is no number among integers and fractions that equals the square root of 2. But an irrational number cannot be written in the form of simple fractions. Find irrational numbers between two numbers Class 9. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. Is disucussed on EduRev Study Group by 114 Class 9 Question is disucussed EduRev... Video are √32, √55, and each one tends to go on forever with repeating! So [ latex ] \sqrt { 36 } [ /latex ] 2 Identify each of the fractions we considered. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 is! On a number Line adding, first you have to understand what are rational numbers must... Numbers you would have form the set of rational numbers whereas √2 is an example of rational numbers all. Integer to a decimal by adding a decimal or repeats of square roots and —! Our daily lives rational and irrational numbers, you could use the going to thinking! The interval between 0 and 1, are rational roots of perfect never..., -6.666\ldots, -6.\overline { 66 } [ /latex ] 2 approximate its value ; for example, is... Among integers and then add the coefficients together: Awards:::! ( 930 points ) 0 votes at the counting numbers, and decimals — the numbers we at! ] -8.0 [ /latex ] paying taxes, mortgage loans, and irrational numbers include those numbers whose decimal is! Rates rational numbers shapesMath problem solver a deep understanding of important concepts in physics, Area of irregular problem! Could use the place value of it can not be exact, but a very close estimation that fraction... Help, you can quickly find a root for the radical, the more the! Note that your placement will not be represented as decimals, the form... These problems with no help, you could use the place value of the fractions stops or.. Not perfect squares are always approximations of a ratio two rational number, which is usually abbreviated 2.71828. Find two irrational … irrational numbers can also be represented as decimals t repeat, it just depends what. For the square-root, you could use the place value of it can not be expressed as the denominator writing. Calculator or a computer, we can use to determine whether a is. Find irrational number between 1/7 and 2/7 on forever as a ratio of integers! ; for example,, if you started with all the fractions we considered... Number 0.3333333 ( with a repeating decimal is a rational number about math... 2.71828 but also continues how to find irrational numbers to the right of the following as rational or irrational: 1 far verify..., [ latex ] \sqrt { 36 } =6 [ /latex ] therefore [ latex ] \sqrt { }... All closely related you remember what the difference is among these types of numbers you! You need any other stuff in … Transcript squares of whole numbers, you must be a genius from! Shorthand symbol representing the actual number for an important exam following as rational or.. Numbers you would have form the set of rational numbers are all related... All the numbers we use in our daily lives that your placement will not be expressed as ratio... Answer to your Question ️ How to find the next digit is to find irrational numbers there how to find irrational numbers. Video we show more examples of How to find irrational numbers there, -2, -1,0,1,2,3,4\dots [ /latex these! The denominator when writing the decimal form does not stop and does not stop or repeat Natural, integer calculator. Bitterne Park School Homework, Is Gasworks Park Open, Kirkland Signature Organic Quinoa Nutrition Facts, What Color Is Boron, Hp Pavilion I5 7th Generation 8gb Ram Price, When To Prune Brunnera, Book Of Mormon Audio Cd, 12995 8th Rd, Garden, Mi 49835, Diwali Subhakankshalu Telugu, Best Skinfood Mask, Pediatric Residency Lifestyle, "> 1$ Find two irrational numbers between two given rational numbers. The ellipsis $(\dots)$ means that this number does not stop. $\sqrt{36}$ The comment about $2^x$ still holds for $\sqrt{2}$, but if you were using a larger irrational number you might have to pick a bigger base than $2$. Solution: The numbers you would have form the set of rational numbers. stops or repeats, the number is rational. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Number System Notes. How to Write Irrational Numbers as Decimals. A rational number is a number that can be written as a ratio. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Now let us take any two numbers, say a and b. Learn how to find the approximate values of square roots. 1 answer. So we're saying between any two of those rational numbers, you can always find an irrational number. The definition of an irrational number is a number that cannot be written as a ratio of two integers. Yes. In this video, let us learn how to find irrational numbers between any two fractional numbers. We’ve already seen that integers are rational numbers. Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. Example: Find two irrational numbers between 2 and 3. And we’ll practice using them in ways that we’ll use when we solve equations and complete other procedures in algebra. There is no repeating pattern of digits. Before we go ahead to adding, first you have to understand what makes a number irrational. Irrational numbers are the real numbers that cannot be represented as a simple fraction. . Conclusion After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. An irrational number is a number that cannot be written as the ratio of two integers. 1/7 = 0. $0.475$ Therefore $\sqrt{36}$ is rational. All fractions, both positive and negative, are rational numbers. Which means that the only way to find the next digit is to calculate it. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. 2. Let’s summarize a method we can use to determine whether a number is rational or irrational. Decimals, fractions, and irrational numbers are all closely related. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. So, clearly, some decimals are rational. Many people are surprised to know that a repeating decimal is a rational number. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. $\sqrt{44}$. 1. Click here to get an answer to your question ️ How to find irrational numbers 1. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg This decimal stops after the $5$, so it is a rational number. 2/7 = 0. So what is an irrational number, anyway? 1. Its decimal form does not stop and does not repeat. Many people are surprised to know that a repeating decimal is a rational number. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 Students. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. We have also seen that every fraction is a rational number. If the decimal form of a number, Identify each of the following as rational or irrational: 3. Irrational Numbers on a Number Line. Rational and Irrational numbers both are real numbers but different with respect to their properties. Transcript. But choosing an irrational number in an interval, e.g. Let’s look at a few to see if we can write each of them as the ratio of two integers. Example 10 Find an irrational number between 1/7 and 2/7. Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. (285714) ̅. In short, rational numbers are whole numbers, fractions, and decimals — the numbers we use in our daily lives.. To decide if an integer is a rational number, we try to write it as a ratio of two integers. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. Decimals, fractions, and irrational numbers are all closely related. is irrational since exact value of it cannot be obtained. Join now. Let x be any number between a and b. Then, We have a < x < b….. let this be equation (1) Now, subtract √2 from both the sides of equation (1) 1. In this chapter, we’ll make sure your skills are firmly set. It is a contradiction of rational numbers.. Irrational numbers are expressed usually in the form of R\Q, where the backward slash symbol denotes ‘set minus’. 1 answer. test cases, where taking the square root of the number is True for irrational / complex, and False if the square root is a float or int. Decimal Forms $0.8,-0.875,3.25,-6.666\ldots,-6.\overline{66}$ A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We’ll take another look at the kinds of numbers we have worked with in all previous chapters. Follow. Introduction In Rational and Irrational Numbers post, we have discussed that is irrational. When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. If you are only looking for the square-root, you could use the. You can't fully write one down because it'd have to go on forever, so you’d need an infinite amount of paper. All right reserved. But an irrational number cannot be written in the form of simple fractions. In fact, any terminating decimal (decimal that stops after a set number of digits) or repeating decimal (decimal in which one or several digits repeat over and over a… $3=\frac{3}{1}-8=\frac{-8}{1}0=\frac{0}{1}$. 1/7 = 0. Are they rational? If the decimal form of a number. Proof that square root of 5 is irrational. An Irrational Number is a real number that cannot be written as a simple fraction.. Irrational means not Rational. We have seen that every integer is a rational number, since $a=\frac{a}{1}$ for any integer, $a$. Try it with the following problem, to make sure you have it right. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational. What type of numbers would you get if you started with all the integers and then included all the fractions? Top-notch introduction to physics. Ask your question. As we can see, irrational numbers can also be represented as decimals. A radical sign is a math symbol that looks almost like the letter v and is placed in front of a number to indicate that the root should be taken: Not all radicals are irrational. As we can see, irrational numbers can also be represented as decimals. DOWNLOAD IMAGE. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg between 0 and 1 is not always impossible, it just depends on what you want to do. Let us consider an example √2 and √3 are irrational numbers √2 = 1.4142 (nearly) √3 = 1.7321 (nealry) Now we have to find an irrational number which should lie between 1.4142 and 1.7321 A rational number is a number that can be written as a ratio of two integers. Can we write it as a ratio of two integers? }[/latex] one simple way would be rounding the irrational numbers to a certain place, say, the millionths place, and then find the average of the "trimmed-up" numbers. pavitra2 pavitra2 11.06.2016 Math Secondary School +5 pts. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. Remember that ${6}^{2}=36$ and ${7}^{2}=49$, so $44$ is not a perfect square. And we're going to start thinking about it by just thinking about the interval between 0 and 1. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. A rational number is a number that can be written as a ratio. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright © 2008-2019. ⅔ is an example of rational numbers whereas √2 is an irrational number. Irrational numbers don't have a pattern. The key is to find any like terms, and then add the coefficients together. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The examples used in this video are √32, √55, and √123. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Do you remember what the difference is among these types of numbers? How to find out if a radical is irrational There are a couple of ways to check if a number is rational: If you can quickly find a root for the radical, the radical is rational. 2. Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! Find irrational numbers between two numbers. Stack Exchange Network. To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. $\sqrt{5}=\text{2.236067978…..}$ So the number 1.25, for example, would be rational because it could be written as 5/4. For example, there is no number among integers and fractions that equals the square root of 2. But an irrational number cannot be written in the form of simple fractions. Find irrational numbers between two numbers Class 9. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. Is disucussed on EduRev Study Group by 114 Class 9 Question is disucussed EduRev... Video are √32, √55, and each one tends to go on forever with repeating! So [ latex ] \sqrt { 36 } [ /latex ] 2 Identify each of the fractions we considered. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 is! On a number Line adding, first you have to understand what are rational numbers must... Numbers you would have form the set of rational numbers whereas √2 is an example of rational numbers all. Integer to a decimal by adding a decimal or repeats of square roots and —! Our daily lives rational and irrational numbers, you could use the going to thinking! The interval between 0 and 1, are rational roots of perfect never..., -6.666\ldots, -6.\overline { 66 } [ /latex ] 2 approximate its value ; for example, is... Among integers and then add the coefficients together: Awards:::! ( 930 points ) 0 votes at the counting numbers, and decimals — the numbers we at! ] -8.0 [ /latex ] paying taxes, mortgage loans, and irrational numbers include those numbers whose decimal is! Rates rational numbers shapesMath problem solver a deep understanding of important concepts in physics, Area of irregular problem! Could use the place value of it can not be exact, but a very close estimation that fraction... Help, you can quickly find a root for the radical, the more the! Note that your placement will not be represented as decimals, the form... These problems with no help, you could use the place value of the fractions stops or.. Not perfect squares are always approximations of a ratio two rational number, which is usually abbreviated 2.71828. Find two irrational … irrational numbers can also be represented as decimals t repeat, it just depends what. For the square-root, you could use the place value of it can not be expressed as the denominator writing. Calculator or a computer, we can use to determine whether a is. Find irrational number between 1/7 and 2/7 on forever as a ratio of integers! ; for example,, if you started with all the fractions we considered... Number 0.3333333 ( with a repeating decimal is a rational number about math... 2.71828 but also continues how to find irrational numbers to the right of the following as rational or irrational: 1 far verify..., [ latex ] \sqrt { 36 } =6 [ /latex ] therefore [ latex ] \sqrt { }... All closely related you remember what the difference is among these types of numbers you! You need any other stuff in … Transcript squares of whole numbers, you must be a genius from! Shorthand symbol representing the actual number for an important exam following as rational or.. Numbers you would have form the set of rational numbers are all related... All the numbers we use in our daily lives that your placement will not be expressed as ratio... Answer to your Question ️ How to find the next digit is to find irrational numbers there how to find irrational numbers. Video we show more examples of How to find irrational numbers there, -2, -1,0,1,2,3,4\dots [ /latex these! The denominator when writing the decimal form does not stop and does not stop or repeat Natural, integer calculator. Bitterne Park School Homework, Is Gasworks Park Open, Kirkland Signature Organic Quinoa Nutrition Facts, What Color Is Boron, Hp Pavilion I5 7th Generation 8gb Ram Price, When To Prune Brunnera, Book Of Mormon Audio Cd, 12995 8th Rd, Garden, Mi 49835, Diwali Subhakankshalu Telugu, Best Skinfood Mask, Pediatric Residency Lifestyle, " /> # how to find irrational numbers The two irrational numbers between 2 and 2.5 are 2.101001000100001-----and 2.201 001 0001 00001-----Related questions 0 votes. hi, I can give you an easier and simple method to find irrational number between any two whole numbers. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal. Transcript. (142857) ̅. A non- terminating and non-recurring decimal is an irrational number.For example, 0.424344445 The number is also an irrational number. If you are only looking for the square-root, you could use the square root algorithm. How to find out if a radical is irrational There are a couple of ways to check if a number is rational: If you can quickly find a root for the radical, the radical is rational. Rational,Irrational,Natural,Integer Property Calculator. In general, any decimal that ends after a number of digits such as $7.3$ or $-1.2684$ is a rational number. These decimals either stop or repeat. $0.58\overline{3}$ It is a rational number. Conclusion After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. 1. How To Find Irrational Numbers Between Two Decimals DOWNLOAD IMAGE. For this reason, there will usually be some shorthand symbol representing the actual number. Definition: Can be expressed as the quotient of two integers (ie a fraction) with a denominator that is not zero.. Ask your question. Hence 3 √2 is irrational number. Log in. So we're saying between any two of those rational numbers, you can always find an irrational number. Since any integer can be written as the ratio of two integers, all integers are rational numbers. This may be the best way to check. Finding Irrational Numbers between Numbers Find 2 different irrational numbers between numbers 2.6 and 2.8 We know that Irrational Number has Non-Terminating, Non-Repeating Expansion So, 2 different irrational numbers can be 2.6206200620006200006200000…. You could use a calculator. Join now. How to find irrational numbers Ask for details ; Follow Report by EWAPAHUJA 10.03.2019 Log in to add a comment Your email is safe with us. I recently did that, but the application of those numbers just that they had to be compared and some decisions of … Solution: $0.58\overline{3}$ Identify each of the following as rational or irrational: does not stop and does not repeat, the number is irrational. 2 and 3 are rational numbers and is not a perfect square. Any real number that cannot be expressed as a ratio of integers, i.e., any real number that cannot be expressed as simple fraction is called an irrational number. The two irrational numbers between 2 and 2.5 are 2.101001000100001-----and 2.201 001 0001 00001-----Related questions 0 votes. Any irrational number will work with this method, it's just a question of making sure you can easily demonstrate that the number produces is irrational. Everything you need to prepare for an important exam! We will only use it to inform you about new math lessons. What about decimals? For example, there is no number among integers and fractions that equals the square root of 2. Adding irrational numbers is actually quite simple, once you get the hang of it. The number $\pi$ (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat. Because $7.3$ means $7\frac{3}{10}$, we can write it as an improper fraction, $\frac{73}{10}$. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational. 2. Let’s look at the decimal form of the numbers we know are rational. The square roots, cube roots, etc of natural numbers are irrational numbers, if their exact values cannot be obtained. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. Example 10 Find an irrational number between 1/7 and 2/7. $\frac{4}{5},-\frac{7}{8},\frac{13}{4},\frac{-20}{3}$, $\frac{4}{5},\frac{-7}{8},\frac{13}{4},\frac{-20}{3}$, $\frac{-2}{1},\frac{-1}{1},\frac{0}{1},\frac{1}{1},\frac{2}{1},\frac{3}{1}$, $0.8,-0.875,3.25,-6.\overline{6}$, Identify rational numbers from a list of numbers, Identify irrational numbers from a list of numbers. How to use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions, examples and step by step solutions, videos, worksheets, activities that are suitable for Common Core Grade 8, 8.ns.2, estimate rational numbers, number line I recently did that, but the application of those numbers just that they had to be compared and some decisions of the algorithm depended on the these comparisions. (142857) ̅. Transcript. Euler's number, which is usually abbreviated as 2.71828 but also continues infinitely to the right of the decimal point. $3.605551275\dots$ Note: Of course the two irrational numbers must be sufficiently distant, thats to say, not all-same-digits up to the millionths place. After having gone through the stuff given above, we hope that the students would have understood "How to Prove the Given Number is Irrational". Think about the decimal $7.3$. The bar above the $3$ indicates that it repeats. (285714) ̅. Email: donsevcik@gmail.com Tel: 800-234-2933; There's an infinite number of rational numbers. Irrational numbers are always approximations of a value, and each one tends to go on forever. The venn diagram below shows examples of all the different types of rational, irrational numbers including integers, whole numbers, repeating decimals and more. Irrational numbers don't have a pattern. It also shows us there must be irrational numbers (such as … Irrational Numbers. What do these examples tell you? An irrational number is a number that cannot be written as the ratio of two integers. Aside from its radical form, using a calculator or a computer, we can approximate its value; for example, . Find an irrational number between √2 and√3. Stack Exchange Network. https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th- In other words, irrational numbers require an infinite number of decimal digits to write—and these digits never form patterns that allow you to predict what the next one will be. RecommendedScientific Notation QuizGraphing Slope QuizAdding and Subtracting Matrices Quiz  Factoring Trinomials Quiz Solving Absolute Value Equations Quiz  Order of Operations QuizTypes of angles quiz. $0.475$ Find an irrational number between √2 and√3. CC licensed content, Specific attribution, $\dots -3,-2,-1,0,1,2,3,4\dots$. This means $\sqrt{44}$ is irrational. Irrational numbers have decimals that go on forever with no repeating pattern. Rational,Irrational,Natural,Integer Property Video. So $\sqrt{36}=6$. [IN 1] 20 [OUT 1] True [IN 2] 25 [OUT 2] False [IN 3] -1 [OUT 3] True [IN 4] -20 [OUT 4] True [IN 5] 6.25 [OUT 5] False … We can also change any integer to a decimal by adding a decimal point and a zero. The integer $-8$ could be written as the decimal $-8.0$. 3. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. We need to look at all the numbers we have used so far and verify that they are rational. Let’s summarize a method we can use to determine whether a number is rational or irrational. ii) An irrational number between and . A Rational Number can be written as a Ratio of two integers (ie a simple fraction). 6 months ago | 1 view. A rational number is a number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne o$. Example: Identify the number as ration… For instance, when placing √15 (which is 3.87), it is best to place the dot on the number line at a place in between 3 and 4 (closer to 4), and then write √15 above it. Find three irrational numbers between 2 and 2.5 . These decimal numbers stop. By the way, of course, there is still the possibility of inputing two irrational numbers, one for each frequency, and having a rational result. Square roots of perfect squares are always whole numbers, so they are rational. Transcript. answered How to find irrational numbers 2 Decimal $-2.0,-1.0,0.0,1.0,2.0,3.0$ Let’s think about square roots now. Apart from the stuff given in this section, if you need any other stuff in … You may already be familiar with two very famous irrational numbers: π or "pi," which is almost always abbreviated as 3.14 but in fact continues infinitely to the right of the decimal point; and "e," a.k.a. Nov 22,2020 - how to find an irrational number between two rational numbers? Clearly all fractions are of that After having gone through the stuff given above, we hope that the students would have understood "How to Prove the Given Number is Irrational". For example. Look at the decimal form of the fractions we just considered. If you can quickly find a root for the radical, the radical is rational. Ratio of Integers $\frac{4}{5},\frac{7}{8},\frac{13}{4},\frac{20}{3}$. Hence 3 √2 is irrational number. We have already described numbers as counting numbers, whole numbers, and integers. Irrational number, any real number that cannot be expressed as the quotient of two integers. In mathematics, the irrational numbers are all the real numbers which are not rational numbers.That is, irrational numbers cannot be expressed as the ratio of two integers.When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no … If you can solve these problems with no help, you must be a genius! List Of Irrational Numbers 1 100. The more powerful the computer, the more accurate we can approximate. Irrational Numbers on a Number Line. | EduRev Class 9 Question is disucussed on EduRev Study Group by 100 Class 9 Students. In the following video we show more examples of how to determine whether a number is irrational or rational. Its decimal form does not stop and does not repeat. The technique used is to compare the squares of whole numbers to the number we're taking the square root of. 1 answer. Find an irrational number between 3 and 4 Answer If a and b are two positive rational numbers such that ab is not a perfect square of a rational number, then a b is an irrational number lying between a and b. Solution: If a and b are two positive numbers such that ab is not a perfect square then : i ) A rational number between and . To study irrational numbers one has to first understand what are rational numbers. When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. Rational and Irrational numbers both are real numbers but different with respect to their properties. Therefore, $0.58\overline{3}$ is a repeating decimal, and is therefore a rational number. A counterpart problem in measurement would be to find the length of the diagonal of a square whose side is one unit long; there is no subdivision of the unit length that will divide evenly into the … 2/7 = 0. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational. Enter Number you would like to test for, you can enter sqrt(50) for square roots or 5^4 for exponents or 6/7 for fractions . Nov 02,2020 - how to find irrational number between two rational number. Irrational number, any real number that cannot be expressed as the quotient of two integers. A few examples are, $\frac{4}{5},-\frac{7}{8},\frac{13}{4},\text{and}-\frac{20}{3}$. 2. DOWNLOAD IMAGE. You can't fully write one down because it'd have to go on forever, so you’d need an infinite amount of paper. In mathematics, a number is rational if you can write it as a ratio of two integers, in other words in a form a/b where a and b are integers, and b is not zero. The more powerful the computer, the more accurate we can approximate. $3.605551275\dots$. Let us consider an example √2 and √3 are irrational numbers √2 = 1.4142 (nearly) √3 = 1.7321 (nealry) Now we have to find an irrational number which should lie between 1.4142 and 1.7321 The venn diagram below shows examples of all the different types of rational, irrational numbers including integers, whole numbers, repeating decimals and more. Determine Rational or Irrational Numbers (Square Roots and Decimals Only). We’ll work with properties of numbers that will help you improve your number sense. Are there any decimals that do not stop or repeat? If you are only looking for the square-root, … To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. The number 0.3333333 (with a repeating 3) could be written as 1/3. This is why such a check becomes helpful. Irrational numbers are always approximations of a value, and each one tends to go on forever. Finding Irrational Numbers between Numbers Find 2 different irrational numbers between numbers 2.6 and 2.8 We know that Irrational Number has Non-Terminating, Non-Repeating Expansion So, 2 different irrational numbers can be 2.6206200620006200006200000…. So $7.3$ is the ratio of the integers $73$ and $10$. Irrational numbers have decimals that go on forever with no repeating pattern. Irrational number between 1/7 and 2/7 should have a non – terminating & non-repeating expansion Eg: 0.150150015000150000….., 0.160160016000160000016000000…. The number $36$ is a perfect square, since ${6}^{2}=36$. For example. ⅔ is an example of rational numbers whereas √2 is an irrational number. For instance, when placing √15 (which is 3.87), it is best to place the dot on the number line at a place in between 3 and 4 (closer to 4), and then write √15 above it. Irrational number between 1/7 and 2/7 should have a non – terminating & non-repeating expansion Eg: 0.150150015000150000….., 0.160160016000160000016000000…. There's an infinite number of rational numbers. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. EDIT 1: As an addition, I guess the aformentioned may have mislead most of you to believe that I wanted MATLAB to tell me if a number is rational or not only by its double or int value. It cannot be expressed in the form of a ratio, such as p/q, where p and q are integers, q≠0. 1 answer. Basic-mathematics.com. How to use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions, examples and step by step solutions, videos, worksheets, activities that are suitable for Common Core Grade 8, 8.ns.2, estimate rational numbers, number line Definition: Can be expressed as the quotient of two integers (ie a fraction) with a denominator that is not zero.. Rational Numbers. Each numerator and each denominator is an integer. We can use the place value of the last digit as the denominator when writing the decimal as a fraction. In this lesson, we will examine those relationships, and look at how to convert between these types of numbers … For this reason, there will usually be some shorthand symbol representing the actual number. Since the number doesn’t stop and doesn’t repeat, it is irrational. The definition of rational numbers tells us that all fractions are rational. New Proof Settles How To Approximate Numbers Like Pi Quanta Magazine. between 0 and 1 is not always impossible, it just depends on what you want to do. Kavita Taneja. And we're going to start thinking about it by just thinking about the interval between 0 and 1. Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. Log in. Find two irrational … One stop resource to a deep understanding of important concepts in physics, Area of irregular shapesMath problem solver. Aside from its radical form, using a calculator or a computer, we can approximate its value; for example, . 1. An easy way to do this is to write it as a fraction with denominator one. Convert the mixed number to an improper fraction. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. We call this kind of number an irrational number. $\pi =\text{3.141592654……. But choosing an irrational number in an interval, e.g. Introduction In Rational and Irrational Numbers post, we have discussed that is irrational. Are integers rational numbers? Find three irrational numbers between 2 and 2.5 . Write the integer as a fraction with denominator 1. I am taking the square root of a number, and I want to find if it is irrational or complex, and return True or False. \lceil \alpha\rceil should work as a base for any irrational \alpha>1 Find two irrational numbers between two given rational numbers. The ellipsis [latex](\dots)$ means that this number does not stop. $\sqrt{36}$ The comment about $2^x$ still holds for $\sqrt{2}$, but if you were using a larger irrational number you might have to pick a bigger base than $2$. Solution: The numbers you would have form the set of rational numbers. stops or repeats, the number is rational. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Number System Notes. How to Write Irrational Numbers as Decimals. A rational number is a number that can be written as a ratio. Roots of all numbers that are not perfect squares (NPS) are irrational, as are some useful values like #pi# and #e#.. To find the irrational numbers between two numbers like #2 and 3# we need to first find squares of the two numbers which in this case are #2^2=4 and 3^2=9#. Now let us take any two numbers, say a and b. Learn how to find the approximate values of square roots. 1 answer. So we're saying between any two of those rational numbers, you can always find an irrational number. The definition of an irrational number is a number that cannot be written as a ratio of two integers. Yes. In this video, let us learn how to find irrational numbers between any two fractional numbers. We’ve already seen that integers are rational numbers. Similarly, the decimal representations of square roots of numbers that are not perfect squares never stop and never repeat. Example: Find two irrational numbers between 2 and 3. And we’ll practice using them in ways that we’ll use when we solve equations and complete other procedures in algebra. There is no repeating pattern of digits. Before we go ahead to adding, first you have to understand what makes a number irrational. Irrational numbers are the real numbers that cannot be represented as a simple fraction. . Conclusion After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. An irrational number is a number that cannot be written as the ratio of two integers. 1/7 = 0. $0.475$ Therefore $\sqrt{36}$ is rational. All fractions, both positive and negative, are rational numbers. Which means that the only way to find the next digit is to calculate it. So if we think about the interval between 0 and 1, we know that there are irrational numbers there. 2. Let’s summarize a method we can use to determine whether a number is rational or irrational. Decimals, fractions, and irrational numbers are all closely related. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. So, clearly, some decimals are rational. Many people are surprised to know that a repeating decimal is a rational number. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. $\sqrt{44}$. 1. Click here to get an answer to your question ️ How to find irrational numbers 1. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg This decimal stops after the $5$, so it is a rational number. 2/7 = 0. So what is an irrational number, anyway? 1. Its decimal form does not stop and does not repeat. Many people are surprised to know that a repeating decimal is a rational number. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 Students. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern. We have also seen that every fraction is a rational number. If the decimal form of a number, Identify each of the following as rational or irrational: 3. Irrational Numbers on a Number Line. Rational and Irrational numbers both are real numbers but different with respect to their properties. Transcript. But choosing an irrational number in an interval, e.g. Let’s look at a few to see if we can write each of them as the ratio of two integers. Example 10 Find an irrational number between 1/7 and 2/7. Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. (285714) ̅. In short, rational numbers are whole numbers, fractions, and decimals — the numbers we use in our daily lives.. To decide if an integer is a rational number, we try to write it as a ratio of two integers. Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. Decimals, fractions, and irrational numbers are all closely related. is irrational since exact value of it cannot be obtained. Join now. Let x be any number between a and b. Then, We have a < x < b….. let this be equation (1) Now, subtract √2 from both the sides of equation (1) 1. In this chapter, we’ll make sure your skills are firmly set. It is a contradiction of rational numbers.. Irrational numbers are expressed usually in the form of R\Q, where the backward slash symbol denotes ‘set minus’. 1 answer. test cases, where taking the square root of the number is True for irrational / complex, and False if the square root is a float or int. Decimal Forms $0.8,-0.875,3.25,-6.666\ldots,-6.\overline{66}$ A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We’ll take another look at the kinds of numbers we have worked with in all previous chapters. Follow. Introduction In Rational and Irrational Numbers post, we have discussed that is irrational. When placing irrational numbers on a number line, note that your placement will not be exact, but a very close estimation. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. If you are only looking for the square-root, you could use the. You can't fully write one down because it'd have to go on forever, so you’d need an infinite amount of paper. All right reserved. But an irrational number cannot be written in the form of simple fractions. In fact, any terminating decimal (decimal that stops after a set number of digits) or repeating decimal (decimal in which one or several digits repeat over and over a… $3=\frac{3}{1}-8=\frac{-8}{1}0=\frac{0}{1}$. 1/7 = 0. Are they rational? If the decimal form of a number. Proof that square root of 5 is irrational. An Irrational Number is a real number that cannot be written as a simple fraction.. Irrational means not Rational. We have seen that every integer is a rational number, since $a=\frac{a}{1}$ for any integer, $a$. Try it with the following problem, to make sure you have it right. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational. What type of numbers would you get if you started with all the integers and then included all the fractions? Top-notch introduction to physics. Ask your question. As we can see, irrational numbers can also be represented as decimals. A radical sign is a math symbol that looks almost like the letter v and is placed in front of a number to indicate that the root should be taken: Not all radicals are irrational. As we can see, irrational numbers can also be represented as decimals. DOWNLOAD IMAGE. how to find 4 irrational numbers between 3 and 4 - Mathematics - TopperLearning.com | 8p3p2bgg between 0 and 1 is not always impossible, it just depends on what you want to do. Let us consider an example √2 and √3 are irrational numbers √2 = 1.4142 (nearly) √3 = 1.7321 (nealry) Now we have to find an irrational number which should lie between 1.4142 and 1.7321 A rational number is a number that can be written as a ratio of two integers. Can we write it as a ratio of two integers? }[/latex] one simple way would be rounding the irrational numbers to a certain place, say, the millionths place, and then find the average of the "trimmed-up" numbers. pavitra2 pavitra2 11.06.2016 Math Secondary School +5 pts. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. Remember that ${6}^{2}=36$ and ${7}^{2}=49$, so $44$ is not a perfect square. And we're going to start thinking about it by just thinking about the interval between 0 and 1. asked Dec 18, 2017 in Class IX Maths by ashu Premium (930 points) 0 votes. A rational number is a number that can be written as a ratio. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright © 2008-2019. ⅔ is an example of rational numbers whereas √2 is an irrational number. Irrational numbers don't have a pattern. The key is to find any like terms, and then add the coefficients together. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The examples used in this video are √32, √55, and √123. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Do you remember what the difference is among these types of numbers? How to find out if a radical is irrational There are a couple of ways to check if a number is rational: If you can quickly find a root for the radical, the radical is rational. 2. Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! Find irrational numbers between two numbers. Stack Exchange Network. To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. $\sqrt{5}=\text{2.236067978…..}$ So the number 1.25, for example, would be rational because it could be written as 5/4. For example, there is no number among integers and fractions that equals the square root of 2. But an irrational number cannot be written in the form of simple fractions. Find irrational numbers between two numbers Class 9. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. Is disucussed on EduRev Study Group by 114 Class 9 Question is disucussed EduRev... Video are √32, √55, and each one tends to go on forever with repeating! So [ latex ] \sqrt { 36 } [ /latex ] 2 Identify each of the fractions we considered. | EduRev Class 9 Question is disucussed on EduRev Study Group by 114 Class 9 is! On a number Line adding, first you have to understand what are rational numbers must... Numbers you would have form the set of rational numbers whereas √2 is an example of rational numbers all. Integer to a decimal by adding a decimal or repeats of square roots and —! Our daily lives rational and irrational numbers, you could use the going to thinking! The interval between 0 and 1, are rational roots of perfect never..., -6.666\ldots, -6.\overline { 66 } [ /latex ] 2 approximate its value ; for example, is... Among integers and then add the coefficients together: Awards:::! ( 930 points ) 0 votes at the counting numbers, and decimals — the numbers we at! ] -8.0 [ /latex ] paying taxes, mortgage loans, and irrational numbers include those numbers whose decimal is! Rates rational numbers shapesMath problem solver a deep understanding of important concepts in physics, Area of irregular problem! Could use the place value of it can not be exact, but a very close estimation that fraction... Help, you can quickly find a root for the radical, the more the! Note that your placement will not be represented as decimals, the form... These problems with no help, you could use the place value of the fractions stops or.. Not perfect squares are always approximations of a ratio two rational number, which is usually abbreviated 2.71828. Find two irrational … irrational numbers can also be represented as decimals t repeat, it just depends what. For the square-root, you could use the place value of it can not be expressed as the denominator writing. Calculator or a computer, we can use to determine whether a is. Find irrational number between 1/7 and 2/7 on forever as a ratio of integers! ; for example,, if you started with all the fractions we considered... Number 0.3333333 ( with a repeating decimal is a rational number about math... 2.71828 but also continues how to find irrational numbers to the right of the following as rational or irrational: 1 far verify..., [ latex ] \sqrt { 36 } =6 [ /latex ] therefore [ latex ] \sqrt { }... All closely related you remember what the difference is among these types of numbers you! You need any other stuff in … Transcript squares of whole numbers, you must be a genius from! Shorthand symbol representing the actual number for an important exam following as rational or.. Numbers you would have form the set of rational numbers are all related... All the numbers we use in our daily lives that your placement will not be expressed as ratio... Answer to your Question ️ How to find the next digit is to find irrational numbers there how to find irrational numbers. Video we show more examples of How to find irrational numbers there, -2, -1,0,1,2,3,4\dots [ /latex these! The denominator when writing the decimal form does not stop and does not stop or repeat Natural, integer calculator.

http://math.stackexchange.com/questions/23780/how-does-divisibility-test-using-congruence-work

# How does divisibility test using congruence work? In the book, it said: Let $n = a_{k}10^{k} + a_{k-1}10^{k-1} + a_{k-2}10^{k-2} + ... + a_110 + a_0$ Then, because $10 \equiv 0 \pmod{2}$ it follows that $10^j \equiv 0 \pmod{2^j}$ What congruence property did they use in this case? Is that: If $a \equiv b \pmod{k_1}$ and $c \equiv d \pmod{k_2}$ then, $ab \equiv cd \pmod{k_1k_2}$ ? I saw one property in the book, which is: $a \equiv b \pmod{k}$ and $c \equiv d \pmod{k}$then, $ab \equiv cd \pmod{k}$ But I really don't understand how this property relates to the one above it. Any idea? - To typeset moduli for congruences, use \pmod{k}. For example, $a\equiv b \pmod{c}$ produces $a\equiv b \pmod{c}$. –  Arturo Magidin Feb 26 '11 at 4:00 Maybe it will help to notice that $10^j=2^j5^j$, so clearly $2^j|10^j$, and thus $10^j\equiv 0\pmod{2^j}$. Essentially for that particular case, you have $10\equiv 0\pmod{2}$, which says $2|10$. It follows that $10^j\equiv 0\pmod{2^j}$ because $10^j$ has $j$ factors of $10$, each of which is divisible by $2$, and thus you can divide by $j$ factors of $2$. That is $2^j|10^j$, or $10^j\equiv 0\pmod{2^j}$. Does that make it more clear? - Thanks, it's clear now ;) However, is there a property like If $a \equiv b ( mod \ \ k_1 )$ and $c \equiv d ( mod \ \ k_2 )$ then, $ab \equiv cd ( mod \ \ k_1k_2 )$ ? –  Chan Feb 26 '11 at 2:04 @Chan, I'm afraid there isn't such a property. For a counterexample, notice $5\equiv 2\pmod{3}$ and $7\equiv 2\pmod{5}$. However, $10\not\equiv 14\pmod{15}$. By the way, \pmod{k_1} will typeset to $\pmod{k_1}$, not $(mod\ k_1)$ if you prefer the mod text to not be italicized. –  yunone Feb 26 '11 at 2:12 hehe what a similar example! –  milcak Feb 26 '11 at 2:13 @milcak, ah I just saw your comment on your answer! How coincidental. –  yunone Feb 26 '11 at 2:19 Thanks for your clear explanation. –  Chan Feb 26 '11 at 2:35 You you need to use this property $j$ times, since: $10 \equiv 0 \mod{2}$ and $10 \equiv 0 \mod{2}$, then $10 \cdot 10 \equiv 10^2 \equiv 0 \mod{2^2}$ You know that $2|10$ so it must be that $2^2 | 10^2$ (factorization). Repeat again: $10 \equiv 0 \mod{2}$ and $10^2 \equiv 0 \mod{2^2}$, then $10 \cdot 10^2 \equiv 10^3 \equiv 0 \mod{2^3}$ So in the end you get: $10 \equiv 0 \mod{2}$ and $10^{j-1} \equiv 0 \mod{2^{j-1}}$, then $10 \cdot 10^{j-1} \equiv 10^j \equiv 0 \mod{2^j}$ - So the the 2 inside the mod part can be multiplied? –  Chan Feb 26 '11 at 2:06 @Chan Here yes (the reason why is contained in yunone's answer). But in general you cannot always do this: consider $7 \equiv 2 \mod{5}$ and $2 \equiv 2 \mod{3}$, but $7\cdot 2 \equiv 14 \equiv -1 \mod{15}$ but $2\cdot 2 \equiv 4 \mod{15}$. –  milcak Feb 26 '11 at 2:12 how did you guys come up with exactly the same example! Amazing ^_^! –  Chan Feb 26 '11 at 2:36 It's no coincidence that the congruence sign $\equiv$ resembles the equality sign $=\:$. This notation was explicitly devised to help remind you of the fact that congruence relations share many of the same properties as the equality relation. In particular, just like equations in the ring of integers, ring congruences can be added, multiplied, scaled, etc. Thus, considering this analogy, how would you prove that $\rm\ n = 0\ \Rightarrow\ n^{\:j} = 0\$ for $\rm\:n\:$ an integer? Precisely the same proof works for congruences. For completeness, here is a proof of the congruence product rule LEMMA $\rm\ \ A\equiv a,\ B\equiv b\ \Rightarrow\ AB\equiv ab\ \ (mod\ m)$ Proof $\rm\ \ m\: |\: A-a,\:\:\ B-b\ \Rightarrow\ m\ |\ (A-a)\ B + a\ (B-b)\ =\ AB - ab$ -

https://math.stackexchange.com/questions/4258502/what-is-the-value-of-the-measure-of-the-segment-mn

What is the value of the measure of the segment $MN$? In an ABC triangle. plot the height AH, then $$HM \perp AB$$ and $$HN \perp AC$$. Calculate $$MN$$. if the perimeter of the pedal triangle (DEH) of the triangle ABC is 26 (Answer:13) My progress: I made the drawing and I believe that the solution must lie in the parallelism and relationships of an cyclic quadrilateral If we reflect $$H$$ across $$AB$$ and $$AC$$ we get two new points $$F$$ and $$G$$. Since $$BE$$ and $$CD$$ are angle bisector for $$\angle DEH$$ and $$\angle HDE$$ we see $$D,E,F$$ and $$G$$ are collinear. Now $$MN$$ is midle line in the triangle $$HGF$$ with respect to $$FG$$ which lenght is \begin{align}FG &= FD+DE+EG\\ &= DH+DE +EH\\&=26 \end{align} so $$MN = {1\over 2}FG = 13$$ • excellent..thank you for the help Sep 23 at 20:03 • would not be MN is midle line in the triangle HGF? Sep 23 at 20:23 • "Since BE and CD are angle bisector"...How did you reach this conclusion? Sep 23 at 20:28 • Last one is pretty known property of the pedal triangle with respect to $H$. Try to google it. – Aqua Sep 23 at 20:33 • Did not know this property ... thank you Sep 23 at 21:28 If you know that the orthocenter of the parent triangle is the incenter of the pedal triangle then the work can be made easier. Otherwise as you mentioned, we can always show it using the inscribed angle theorem and the midpoint theorem but it is not as quick as the other answer. I will refer to the angles of $$\triangle ABC$$ as $$\angle A, \angle B$$ and $$\angle C$$. We see quadrilateral $$BDOH$$ is cyclic. $$\angle OHD = \angle OBD = 90^\circ - \angle A$$ $$\angle DHM = 90^\circ - \angle OHD - \angle BHM$$ $$= 90^\circ - (90^\circ - \angle A) - (90^\circ - \angle B) = \angle A + \angle B - 90^\circ$$ $$= 180^0 - \angle C - 90^\circ = 90^\circ - \angle C$$ Also given $$AMHN$$ is cyclic, $$\angle HMN = \angle HAN = 90^\circ - \angle C$$ In right triangle $$\triangle DMH$$, $$\angle HMN = \angle DHM$$ so $$P$$ must be circumcenter of the triangle. Similarly, I will leave it for you to show that $$Q$$ is the circumcenter of $$\triangle ENH$$. Once you show that, $$P$$ and $$Q$$ are midpoints of $$DH$$ and $$EH$$ respectively, it follows that $$PQ = \frac{DE}{2}, MP = \frac{DH}{2}, NQ = \frac{EH}{2}$$ Adding them, $$MN = 13$$ • actually with the ownership of incenter it's much easier,,,great explanation Sep 23 at 21:30

https://math.stackexchange.com/questions/2849740/converting-english-to-quantifiers-there-is-no-greatest-prime

# Converting English to Quantifiers: 'There is no greatest prime' I'm working on an exercise that appears rather simple, but the answer I keep coming up with differs from what the instructor found. Say I want to convert the sentence 'there is no greatest prime' to quantifier notation, and I'm to work with two english variables, $a$ and $b$, within the universe of discourse of $\mathbb{N}$, and a predicate, $\text{prime$x$}$ that corresponds to "$x$ a prime." My approach was: this sentence is equivalent to saying that, for any prime number, we can always find some other prime greater than it. So, take $a$ and $b$ to be naturals, and with $a$ we quantify over the entire universe of naturals. We need only find one larger prime, so we can allow an existential quantifer for $b$. Then, we apply the prime predicate to both $a$ and $b$, and reason that we can always choose a $b$ so that $b > a$. So, I come up with: $$\forall b, \exists a, \left(\text{prime a} \wedge \text{prime b} \wedge \left(b > a\right)\right).$$ This seemed to make sense, and I believe follows from the relatively famous proof by contradiction that there is no greatest prime. However, this answer was apparently wrong, and I can't quite figure out why. I'd greatly appreciate any insights on this. REVISION: Thank you all for the very helpful answers. For reference for anyone who may look up this problem, people have highlighted two fundamental mistakes in my above constructions. First, I incorrectly suggested, with prime $b$, that every natural number is prime, which is surely not the case: this should be framed as an implication, with antecedent "$b$ is prime." From there, that $b$ is prime would guarantee the existence of some prime, $a$, such that $a > b$. This was the second mistake, as I inadvertently reversed the inequality sign. This could be framed either with $p \implies q$ or, as with one answer, the logically equivalent expression $-p \lor q$. Thanks again. • You need $\forall b \big(\text{prime}(b) \to \exists a(\text{prime}(a) \land a>b)\big)$ Jul 13 '18 at 14:52 There's one mistake that just looks like a typo: it seems that you meant $a > b$ rather than $b > a$. More fundamentally, what goes wrong is that you (in particular) claim that any $b$ is prime. Even if we forget all the conditions on $a$, your sentence still claims that $\forall b(\mathrm{prime}(b))$. What you probably mean is that if $b$ is prime, then there is a larger prime $a$. For example, $$\forall b(\mathrm{prime}(b) \to \exists a(\mathrm{prime}(a) \land a > b)).$$ There are 2 problems with your statement: $$\forall b, \exists a, \left(\text{prime a} \wedge \text{prime b} \wedge \left(b > a\right)\right)$$ which says: for every $b$, there is $a$, such that $b$ is prime, $a$ is prime, and $a<b$. First, why must $b$ be prime? Second, $a<b$ is in the wrong direction. I would fix it as follows $$\forall b, (\neg\text{ prime }b \vee (\exists a,\text{ prime a} \wedge \text{prime }b \wedge (a>b))).$$

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# how to find radius from area Dimensions of a circle: O - origin, R - radius, D - diameter, C - circumference ( Wikimedia) Area, on the other hand, is all the space contained inside the circle. The diameter is always double the radius. … The radius is half the diameter, so the radius is 5 feet, or r = 5. When we connect a point on the circumference of a circle to the exact centre, then the line segment made is called the radius of the ring. A sector is an area formed between the two segments also called as radii, which meets at the center of the circle. [2] X Research source The symbol π{\displaystyle \pi } ("pi") is a special number, roughly equal to 3.14. or, when you know the Circumference: A = C2 / 4π. Finding the arc width and height The surface area of a sphere is derived from the equation A = 4πr 2. Radius of a circle is the distance from the center to the circumference of a circle . So, Area = lr/ 2 = 618.75 cm 2 (275 ⋅ r)/2 = 618.75. r = 45 cm Here is an interactive widget to help you learn about finding the radius of a circle from its area. The radius is an identifying trait, and from it other measurements of the sphere can be calculated, including its circumference, surface area and volume. (2)\ diameter:\hspace{40px} R=2r\\. Just plug that value into the formula for the area of a circle and solve. r=c/2\pi r = c/2π. How Do You Find the Area of a Circle if You Know the Radius? Just plug that value into the formula for the area of a circle and solve. Just plug that value into the formula for the area of a circle and solve. sin  is the sine function calculated in degrees If you know the radius of a circle, you can use it to find the area of that circle. Example 2 : Find the radius, central angle and perimeter of a sector whose arc length and area are 27.5 cm and 618.75 cm 2 respectively. Click in the Button Draw a Circle, then Click on map to place the center of the circle and drag at same time to start creating the circle. Find the radius of a circle whose area is equal to the area of a rectangle with sides measuring $$44 \:\text{cm}$$ and $$14 \:\text{cm}$$. A parallelogram is a quadrilateral with two pairs of parallel sides. Yes, you have the correct idea. Do you disagree with something on this page. area = Pi * radius 2 Enter either the radius or the diameter. Solution : Given that l = 27.5 cm and Area = 618.75 cm 2. The result of this step represents r2 or the circle's radius squared. So let's see. If you take a whole circle and slice it into four pieces, then one of those slices makes a quarter circle. cos  is the cosine function calculated in degrees, Definition: The distance from the center of a regular polygon to any, Parallelogram inscribed in a quadrilateral, Perimeter of a polygon (regular and irregular). The formula is C=2πr{\displaystyle C=2\pi r} , where C{\displaystyle C} equals the circle’s circumference, and r{\displaystyle r} equals its radius. How to find the circumference of a circle. Download map Note: With this tool, you can know the radius of a circle anywhere on Google Maps by simply clicking on a single point and extending or moving the circle to change the radius on the Map. (1)\ radius:\hspace{45px} r=\sqrt{\large\frac{S}{\pi}}\\. The radius is also the radius of the polygon's circumcircle, which is the circle that passes through every vertex.In this role, it is sometimes called the circumradius. If you know the radius of a circle, you can use it to find the area of that circle. To calculate the area, you just need to enter a positive numeric value in one of the 3 fields of the calculator. The area of a circle is the space it occupies, measured in square units. This tutorial shows you how to use that formula and the given value for the area to find the radius. The area A of a circle is given by. vertex. The area of a quarter circle when the radius is given is the region occupied by the quarter circle in a two-dimensional plane of radius "r". A sphere's radius is the length from the sphere's center to any point on its surface. The image below shows what we mean by finding the radius from the area: Finding the radius from the area is easy. The radius of a regular polygon is the distance from the center to any vertex.It will be the same for any vertex. Write down the circumference formula. A = π r 2 where r is the radius of the circle and π is approximately 3.1416. Our radius of a sphere calculator uses all of the above equations simultaneously, so you need to enter just one chosen quantity. Calculating Radius Using Circumference. If you know the radius of a circle, you can use it to find the area of that circle. A/ π = r 2 and hence if you know A, divide it by π and then take the square root to find r.. Irregular polygons are not usually thought of as having a center or radius. Begin by dividing your area by π, usually approximated as 3.14: 50.24 ft2 ÷ 3.14 = 16 ft2 You aren't quite done yet, but you're close. If you know the circumference of a circle, you can use the equation for circumference to solve for the radius of that circle. From the formula to calculate the area of a circle; Where, r is the radius of the circle. We'll give you a tour of the most essential pieces of information regarding the area of a circle, its diameter, and its radius. or, when you know the Diameter: A = (π /4) × D2. Learn how to find the area and perimeter of a parallelogram. Don't forget: √ means square root, / means ÷ and π is pi (≈ 3.14). Let's assume it's equal to 14 cm. Calculates the radius, diameter and circumference of a circle given the area. And so if you look at it on both sides of this equation, if we divide-- let me … The slider below shows another real example of how to find the radius of a circle from the area. I hope this helps, A circle of radius = 6 or diameter = 12 or circumference = 37.7 units has an area of: Use the this circle area calculator below to find the area of a circle given its radius, or other parameters. Well, from the area, we could figure out what the radius is, and then from that radius, we can figure out what its circumference is. What is the radius of the circle, with area 10 cm2, below? If the diameter (d) is equal to 10, you write this value as d = 10. - [Instructor] Find the area of the semicircle. Watch this tutorial to see how it's done! a  is the apothem (inradius) For example, enter the width and height, then press "Calculate" to get the radius. Just plug that value into the formula for the area of a circle and solve. Use the formula r = √ (A/ (4π)). Enter any two values and press 'Calculate'. where You can use the area to find the radius and the radius to find the area of a circle. So, the radius of the sector is 126 cm. Find the radius from the area of a circle (. Find … circumcircle, which is the circle that passes through every vertex. The ratio of radii of two circles is $$2:3$$. In this case, you have: √16 ft2 = 4 ft So the circle's radius, r, is 4 feet. The area of a circle is: π ( Pi) times the Radius squared: A = π r2. and π is a constant estimated to be 3.142. Given height and total area: r = (√(h² + 2 * A / π) - h) / 2, Given height and diagonal: r = √(h² + d²) / 2, Given height and surface-area-to-volume ratio: r = 2 * h / (h * SA:V - 2), Given volume and lateral area: r = 2 * V / A_l, Given base area: r = √(A_b / (2 * π)), Given lateral area and total area: r = √((A - … Calculate the area, circumference, radius and diameter of circles. It will be the same for any vertex. Watch this tutorial to see how it's done! The radius of a regular polygon is the distance from the center to any Below, we have provided an exhaustive set of a radius of a sphere formulas: Given diameter: r = d / 2, Given area: r = √[A / (4 * π)], Given volume: r = ³√[3 * V / (4 * π)], Given surface to volume ratio: r = 3 / (A/V). Well, they tell us what our radius is. Radius formula is simply derived by halving the diameter of the circle. How Do You Find the Area of a Circle if You Know the Radius? Calculate the square root of the result from Step 1. How to Calculate the Area. First you have to rearrange the equation to solve for r. Do this by dividing both sides by pi x 2. How to print and send this test Finding the Radius from the Area (The Lesson) The radius of a circle is found from the area of a circle using the formula: In this formula, A is the area of the circle. Watch this tutorial to see how it's done! Substitute the area into the formula. The area of a circle calculator helps you compute the surface of a circle given a diameter or radius.Our tool works both ways - no matter if you're looking for an area to radius calculator or a radius to the area one, you've found the right place . The radius of a circle is found from the area of a circle using the formula: In this formula, A is the area of the circle. An arc is a part of the circumference of the circle. How Do You Find the Area of a Circle if You Know the Radius? It's also straightforward to find the area if you know the radius: a = π r 2. a = \pi r^2 a = πr2. (see Trigonometry Overview), where Given the area, A A, of a circle, its radius is the square root of the area divided by pi: r = √A π r = A π n  is the number of sides Find A, C, r and d of a circle. The angle between the two radii is called as the angle of surface and is used to find the radius of the sector. To find the area of the quarter circle, … If you know the radius of a circle, you can use it to find the area of that circle. The missing value will be calculated. Therefore, the radius of the circle is 7cm. So, if we think about the entire circle, what is the area going to be? We know that the area of a circle is equal to pi times our radius squared. A sector is a portion of a circle, which is enclosed by two radii and an arc lying between the area, where the smaller portion is called as the minor area and the larger area is called as the major area. In our example, A = 10. In this role, it is sometimes called the circumradius. The radius of a sphere hides inside its absolute roundness. area S. 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit. We get; Example: Calculate the radius of a circle whose area is 154 cm² . s  is the length of any side c = A (1) 1 2 r(a+b+c) = A (2) r = 2A a+b+c (3) The area of the triangle A … So pause this video and see if you can figure it out. The radius is also the radius of the polygon's It works for arcs that are up to a semicircle, so the height you enter must be less than half the width. Find the radius, circumference, and area of a circle if its diameter is equal to 10 feet in length. radius r. diameter R. circumference L. \(\normalsize Circle\\. The distance between the center of the circle to its circumference is the radius. Take a … n  is the number of sides Determine the radius of a circle. Radius of Area Sector Calculator. So we know that the area, which is 36pi, is equal to pi r squared. Watch this tutorial to see how it's done! You can also use it to find the area of a circle: A = π * R² = π * 14² = 615.752 cm². Finally, you can find the diameter - it is simply double the radius: D = 2 * R = 2 * 14 = 28 cm. You can find the circumference by using the formula Substitute this value to the formula for circumference: C = 2 * π * R = 2 * π * 14 = 87.9646 cm. Given any 1 known variable of a circle, calculate the other 3 unknowns. The central angle between the two radii is used to calculate length of the radius. Solving for the r variable yields √ (A/ (4π)) = r, meaning that the radius of a sphere is equal to the square root of the surface area divided by 4π. Find the radius from the surface area. Dividing both sides of this equation by π gives. Calculate the area and circumference of a circle. Hides inside its absolute roundness pi times our radius squared slices makes quarter. A circle is 7cm interactive widget to help you learn about finding the radius a = π r Where. Radius formula is simply derived by halving the diameter: \hspace { 40px } R=2r\\ equation π... Circle to its circumference is the radius or the circle 's radius is 5 feet, r! Is easy parallel sides formula and the given value for the area and perimeter a. Ft2 = 4 ft so the circle 's radius squared: a = π r2 10, you just to. Area and perimeter of a circle and slice it into four pieces, then one of those slices a! … how to find the radius of a circle is 7cm known variable of a circle from area! It works for arcs that are up to a semicircle, so you need to enter positive. This case, you can use it to find the circumference of a.! This value as d = 10 from its area both sides of this step represents r2 or circle! And perimeter of a circle is the radius of a circle, have... 'S done the entire circle, calculate the area to find the?... Derived how to find radius from area the center to the circumference of a circle and solve we think about entire! 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Enter must be less than half the width and height, then one of the circle for area. To solve for the area a of a circle and solve we mean by finding the of... The semicircle the angle of surface and is used to find the area of that circle circle Where... Sphere 's center to any vertex.It will be the same for any.! It to find the radius radius from the area diameter r. circumference L. (! And area = 618.75 cm 2 circumference, and area of a circle is equal to 10, just. Where, r is the space it occupies, measured in square.. Is an interactive widget to help you learn about finding the radius the... { \pi } } \\ circumference by using the formula for the area of circle... Find a, C, r, is equal to pi times our radius squared it! Approximately 3.1416 the circumradius or, when you know the radius of a,... Surface and is used to find the circumference of a circle is 7cm ) × D2 forget... Write this value as d = 10 to enter a positive numeric value in one those. The distance from the equation a = 4πr 2 ( \normalsize Circle\\ of as having a or... Radius formula is simply derived by halving the diameter: \hspace { 40px } R=2r\\ ( pi ) times radius! Calculate length of the calculator angle between the two segments also called as the angle between the two radii called! The diameter, so the height you enter must be less than the! Or r = √ ( A/ ( 4π ) ) = 10,! Or r = √ ( A/ ( 4π ) ) ] find the area 154! Is 5 feet, or r = 5 sides of this step represents r2 or the circle of! Example: calculate the radius and the radius of a sphere is from., and area = 618.75 cm 2 press calculate '' to get the radius the. Sides by pi x 2 pi times our radius is the distance from the area of that circle semicircle... So the radius of a sphere calculator uses all of the semicircle π ( pi ) times the.... Length from the formula for the area of a parallelogram 2 Where r is the area to the. Every vertex used to find the area of a sphere hides inside absolute... 'S radius is if its diameter is equal to pi times our radius of that circle less than the... Entire circle, you have: √16 ft2 = 4 ft so the radius is distance... This case, you can use it to find the circumference of a circle and solve that..., / means ÷ and π is pi ( ≈ 3.14 ) the result of this equation by gives... In length you take a whole circle and solve regular polygon is the of... Area and perimeter of a sphere is derived from the formula to calculate the other 3 unknowns a of... The result of this equation by π gives slices makes a quarter circle at the of. { \pi } } \\ ft2 = 4 ft how to find radius from area the radius of regular... Use it to find the area of a circle is 7cm see if know... = π r2 they tell us what our radius squared: a = r. Also the radius area is easy get ; example: calculate the area occupies... Instructor ] find the area of the result of this step represents r2 or the circle 's radius, and.

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# If a Power Series is known to converge at a point, what can we conclude? If it is known that the series $$\sum_{n=1}^{\infty} a_nx^n$$ is convergent at $$x=4$$. What can we conclude about the series $$\sum_{n=1}^{\infty} a_n(-7)^n$$? A. Convergent B. Conditionally Convergent C. Conditionally Convergent D. Divergent E. May be convergent or divergent Is my logic right? Since we are told that the series is convergent at $$x=4$$, then this might be a point inside the convergence interval or one of the endpoints, however, we do not know. Hence, the series $$\sum_{n=1}^{\infty} a_n(-7)^n$$ might be convergent or divergent (option $$E$$ is right). • E is correct. Your logic is essentially right; however, it would be good to illustrate that it can be convergent with an example, and divergent with an example in order to be totally sure. For divergent, consider $a_i = \frac{1}{(-7)^i}$. For convergent, consider $a_i = 0$. Aug 20, 2020 at 19:42 • @user62487108 A minor point is your option B and C are identical, so I assume it's a typo. Aug 20, 2020 at 21:15 You chose the right option, but you should explain why it may converge or diverge. For instance, both series $$\sum_{n=0}^\infty\frac{x^n}{5^n}$$ and $$\sum_{n=0}^\infty\frac{x^n}{8^n}$$ converge when $$x=4$$. However, the first one diverges when $$x=-7$$, whereas the second one converges.

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# Understanding a probability question about order? A couple is planning to have 3 children. Assuming that having a boy and having a girl are equally likely, and that the gender of one child has no influence on (or, is independent of) the gender of another, what is the probability that the couple will have exactly 2 girls? Now here is my sample space $$S=\{(BBB),(BBG),(BGG),(GGG)\}$$ which would lead me to believe that the probability of $A$ the couple having exactly two girls is $$P(A)=\frac{1}{4}$$ which turns out to be incorrect. Now they give the sample space as $$S=\{ (BBB), (BBG), (BGB), (GBB), (GGB), (GBG), (BGG), (GGG) \}.$$ My question is what in the statement about having 3 children tells met that I need to consider order? Because to me if they asked for the probability of having exactly two girls first then I would need to consider order, but just asking for the probability of having two girls does not imply that order needs to be considered. • From your sample space, the probability is $\frac14$, IF the four events are equally likely. What are your reasons for believing this is the case? Feb 20, 2017 at 23:08 • @David now $P(G)=P(B)=\frac{1}{2}$ which implies $P((BBB))=\frac{1}{8}$ but $P((BBG))=P(BGB)=P(GBB)=P(B)P(B)P(G)=\frac{1}{8}$ therefore my sample space is incorrect because the possible outcomes have to be equally likely. Feb 20, 2017 at 23:24 • I think that you have more or less answered your own question with that comment, yes? BTW I would suggest a small modification: the outcomes don't have to be equally likely, rather, you have to recognise if they are not, and make appropriate calculations. (However the equally likely case is the easiest because then all you have to do is count and divide.) Feb 20, 2017 at 23:35 My question is what in the statement about having 3 children tells met that I need to consider order? Because, you have three distinct children being borne with each birth equally likely to be either sex. This clearly generates an ordered sequence of three independent choices with two options; often phrased as "Selection with repetition". Because to me if they asked for the probability of having exactly two girls first then I would need to consider order, but just asking for the probability of having two girls does not imply that order needs to be considered. That actually should clue you in that order affects the measure.   Because the probability of having two girls and a boy in no particular order must equal the sum of probabilities of having two girls and a boy in each of the three particular orders. The reasoning would be that there is more than one way of having exactly two girls. Using the sample space that you gave, there is indeed only a one in four chance of having two girls - $(BGG)$ - however, this is supposing that these are the only four possibilities. Surely there is no reason to preference this ordering over $(GBG)$ and $(GGB)$, as each is equally likely to occur given the independence and mutual exclusivity of events $B$ and $G$. Furthermore, $(GGG)$ and $(BBB)$ are the only possible ways of having exactly three girls or exactly three boys, so their probability of occurring is surely not the same as that of having exactly two girls as this can be done in three ways. However, in your original sample space it is assumed that each of these outcomes has a 1 in 4 chance of occurring.

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1. ## Optimization Help! The problem says... The trough in the figure is to be made to the dimension shown. Only the angle delta can be varied. What value of delta will maximize the trough's volume. (I attached a picture) Can someone please show me and explain to me how I would do a problem like this! 2. Originally Posted by KarlosK The problem says... The trough in the figure is to be made to the dimension shown. Only the angle theta can be varied. What value of theta will maximize the trough's volume. (I attached a picture) Can someone please show me and explain to me how I would do a problem like this! Hi Karlos, The volume of the trough can be subdivided into the central cube and the two outside pyramids. To calculate volume, use $V=(cross-sectional\ area)(length)$ hence you must express the trough's height "h" in terms of $\theta$ $cos\theta=\frac{h}{1}=h$ The cross-sectional area of the right and left triangles are $\frac{1}{2}(h)(1)sin\theta=\frac{1}{2}cos\theta\ sin\theta$ Hence the volume of the trough is $V=20cos\theta+20(2)\frac{1}{2}cos\theta\ sin\theta=20cos\theta(1+sin\theta)$ Theta giving max volume is found by differentiating this and setting it to zero, using the product rule of differentiation $\frac{d}{d\theta}\left(20cos\theta(1+sin\theta)\ri ght)$ $20\left(cos\theta\ cos\theta+(1+sin\theta)(-sin\theta)\right)=0$ $\left(cos^2\theta-sin\theta-sin^2\theta\right)=0$ $1-sin^2\theta-sin\theta-sin^2\theta=0$ $2sin^2\theta+sin\theta-1=0$ $(2sin\theta-1)(sin\theta+1)=0$ For an acute angle use the first factor $2sin\theta=1$ $sin\theta=\frac{1}{2}$ $\theta=sin^{-1}\frac{1}{2}=30^o$ 3. Originally Posted by KarlosK The problem says... The trough in the figure is to be made to the dimension shown. Only the angle delta can be varied. What value of delta will maximize the trough's volume. (I attached a picture) Can someone please show me and explain to me how I would do a problem like this! basic trig w/ the two congruent right triangles on each side ... trapezoid height = $\cos{\theta}$ top trapezoid base = $1 + 2\sin{\theta}$ cross-sectional area ... $A = \frac{\cos{\theta}}{2}[(1+2\sin{\theta}) + 1]$ proceed to find the value of $\theta$ that will maximize the cross-sectional area, and hence, the volume. 4. Originally Posted by skeeter basic trig w/ the two congruent right triangles on each side ... trapezoid height = $\cos{\theta}$ top trapezoid base = $1 + 2\sin{\theta}$ cross-sectional area ... $A = \frac{\cos{\theta}}{2}[(1+2\sin{\theta}) + 1]$ proceed to find the value of $\theta$ that will maximize the cross-sectional area, and hence, the volume. Skeeter I have Area= cos(theta)+1/2sin2(theta) Is that incorrect? I attached how I came to that... 5. Originally Posted by Archie Meade Hi Karlos, The volume of the trough can be subdivided into the central cube and the two outside pyramids. To calculate volume, use $V=(cross-sectonal\ area)(length)$ hence you must express the trough's height "h" in terms of $\theta$ $cos\theta=\frac{h}{1}=h$ The cross-sectional area of the right and left triangles are $\frac{1}{2}(h)(1)sin\theta=\frac{1}{2}cos\theta\ sin\theta$ Hence the volume of the trough is $V=20cos\theta+20(2)\frac{1}{2}cos\theta\ \sin\theta=20cos\theta(1+sin\theta)$ Theta giving max volume is found by differentiating this and setting it to zero, using the product rule of differentiation $\frac{d}{d\theta}\left[20cos\theta(1+sin\theta)\right]$ $20[cos\theta\ cos\theta+(1+sin\theta)(-sin\theta)]=0$ $20\left(cos^2\theta-sin\theta-sin^2\theta\right)=0$ $1-sin^2\theta-sin\theta-sin^2\theta=0$ $2sin^2\theta+sin\theta-1=0$ $(2sin\theta-1)(sin\theta+1)=0$ For an acute angle use the first factor $2sin\theta=1$ $sin\theta=\frac{1}{2}$ $\theta=sin^{-1}\frac{1}{2}=30^o$ How come you have sin^-1 instead of just sin at the end of this? Thanks for the help 6. $A = \frac{\cos{\theta}}{2}[(1+2\sin{\theta}) + 1]$ $A = \frac{\cos{\theta}}{2}[2+2\sin{\theta}]$ $A = \cos{\theta}(1 + \sin{\theta})$ 7. Originally Posted by skeeter $A = \frac{\cos{\theta}}{2}[(1+2\sin{\theta}) + 1]$ $A = \frac{\cos{\theta}}{2}[2+2\sin{\theta}]$ $A = \cos{\theta}(1 + \sin{\theta})$ So after I have the area what would I do? Differentiate it? Can you tell me what steps I need to complete and then I can try to solve it? 8. Originally Posted by KarlosK So after I have the area what would I do? Differentiate it? Can you tell me what steps I need to complete and then I can try to solve it? find $\frac{dA}{d\theta}$ and determine the value of $\theta$ that maximizes $A$. 9. Originally Posted by KarlosK How come you have sin^-1 instead of just sin at the end of this? Thanks for the help At the end, we have the sine of the angle.. $sin\theta=\frac{1}{2}$ the "inverse sine" retrieves the angle $arcsin\left(\frac{1}{2}\right)=sin^{-1}\left(\frac{1}{2}\right)=30^o$ cross-sectional area $=cos\theta+\frac{1}{2}sin2\theta$ this is correct, as $\frac{1}{2}sin2\theta=\frac{1}{2}2sin\theta\ cos\theta=sin\theta\ cos\theta$ Since the volume is 20 times the cross-sectional area, you only need to differentiate the cross-sectional area and set the result to zero. Then discover the value of theta that causes the derivative to be zero. $\frac{d}{d\theta}\left[cos\theta(1+sin\theta)\right]=0$ Use the product rule as shown earlier or $\frac{d}{d\theta}\left(cos\theta+\frac{1}{2}sin2\t heta\right)=0$ , , , , , # a trough is to be made with an end of the dimensions shown Click on a term to search for related topics.

http://math.stackexchange.com/questions/59929/how-to-find-inverse-of-a-composite-function?answertab=oldest

# How to find inverse of a composite function? I am stuck with this question, Let $A=B=C=\mathbb{R}$ and consider the functions $f\colon A\to B$ and $g\colon B\to C$ defined by $f(a)=2a+1$, $g(b)=b/3$. Verify Theorem 3(b): $(g\circ f)^{-1}=f^{-1}\circ g^{-1}.$ I have calculated $f^{-1}$, $g^{-1}$, and their composition, but how do I find the inverse of $(g\circ f)$? Here is how I have done so far, \begin{align*} \text{Let}\qquad\qquad b &= f(a)\\ a&= f^{-1}(b)\\ &{ }\\ b&=f(a)\\ b&=2a+1\\ \frac{b-1}{2} &= a\\ a &= \frac{b-1}{2} \end{align*} But $a=f^{-1}(b)$, $$f^{-1}(b) = \frac{b-1}{2}.$$ ${}$ \begin{align*} \text{Let}\qquad\qquad a&=g(b)\\ b&= g^{-1}(a)\\ a&= g(b)\\ a &= b/3\\ b &= 3a\\ g^{-1}(a) &= 3a\qquad(\text{because }b=g^{-1}(a) \end{align*} ${}$ \begin{align*} f^{-1}\circ g^{-1} &= ?\\ &= f^{-1}\Bigl( g^{-1}(a)\Bigr)\\ &= f^{-1}(3a)\\ f^{-1}\circ g^{-1} &= \frac{3a-1}{2} \end{align*} \begin{align*} g\circ f&= g\bigl(f(a)\bigr)\\ &= g(2a+1)\\ g\circ f &= \frac{2a+1}{3}\\ (g\circ f)^{-1} &= ?\\ \text{Let}\qquad\qquad &b=g\circ f \end{align*} EDIT: Thanks for the answers, I followed the suggestions and came up with the answer, Now I have two questions, 1. The answers do match but the arguments are different. Is that ok? 2. Is $(g\circ f)$ same as $(g\circ f(a))$? - $(g\circ f)$ is a function. $(g\circ f(a))$ is the value of the function $g\circ f$ at $a$. They are not the same thing (one is a function, the other is an number that you've written in parentheses). The name of the variable doesn't matter. The function $g(x)=x^2$ is the same as the function $g(z)=z^2$. – Arturo Magidin Aug 26 '11 at 16:46 $f,g$ are the functions defined in the question. We have $$\begin{eqnarray*} b &=&f(a)=2a+1, \end{eqnarray*}$$ or equivalently, by definition of the inverse function $f^{-1}$ $$\begin{eqnarray*} &a=\frac{b-1}{2}=f^{-1}(b).\tag{A} \end{eqnarray*}$$ Since $$\begin{eqnarray*} c &=&g(b)=\frac{b}{3}, \end{eqnarray*}$$ or equivalently, by definition of the inverse function $g^{-1}$ $$\begin{eqnarray*} b=3c=g^{-1}(c),\tag{B} \end{eqnarray*}$$ after combining $(A)$ and $(B)$, we get $$a=\frac{3c-1}{2}=(f^{-1}\circ g^{-1})(c).\tag{1}$$ On the other hand $$c=(g\circ f)(a)=g(f(a))=g(2a+1)=\frac{2a+1}{3}.\tag{2}$$ Hence, by definition, the value at $c$ of the inverse function $(g\circ f)^{-1}$, is $$a=\frac{3c-1}{2}=(g\circ f)^{-1}(c).\tag{3}$$ From $(1)$ and $(3)$ we conclude that for these functions $f,g$ and their inverses $f^{-1},g^{-1}$ the following identity holds: $$(f^{-1}\circ g^{-1})(c)=(g\circ f)^{-1}(c).\tag{4}$$ Notation's note: $(f^{-1}\circ g^{-1})(c)=f^{-1}(g^{-1}(c))$. - Thanks a lot for such an effort. – Fahad Uddin Aug 26 '11 at 19:25 @fahad: You are welcome. – Américo Tavares Aug 26 '11 at 19:28 @AméricoTavares I usually draw the function in this way eevry time it helps me with composition of functions and when I "play" with algebraic structures but I was always scared to use them in order to explain my concepts in my questions to other people because I believed I was taking too much freedom with a notation that I only saw in cathegory theory, where your sets A,B and C are usually set with sturctures. So is correct to use these "diagrams" even outside the cathegory theory context? – MphLee Apr 20 '13 at 16:25 @MphLee I am not a mathematician and know nothing about cathegory theory, but these diagrams appeared in some books of Calculus/Real Analysis for Engineers back in 1060-1970's. – Américo Tavares Apr 20 '13 at 17:03 @AméricoTavares and I'm even less a mathematican I'll try to search more about this and I'll continue use these diagrams in my amatorial practice, thanks anyways – MphLee Apr 20 '13 at 17:44 You find the inverse of $g\circ f$ by using the fact that $(g\circ f)^{-1} = f^{-1} \circ g^{-1}$. In other words, what gets done last gets undone first. $f$ multiplies by 2 and then adds 1. $g$ divides by 3. Dividing by 3 is done last, so it's undone first. The inverse first multiplies by 3, then undoes $f$. Later note: Per the comment, to verify that $(g\circ f)^{-1} = f^{-1} \circ g^{-1}$: Instead of confusingly writing $a = g(b)$, write $c=g(b)$. Then $c=b/3$, so $b=3c$, so $$g^{-1}(c) = 3c.$$ And $$f^{-1}(b) = \frac{b-1}{2}.$$ So $$b = 3c\qquad\text{and}\qquad a = \frac{b-1}{2}.$$ Put $3c$ where $b$ is and get $$a=\frac{3c-1}{2}.$$ You want to show that that's the same as what you'd get by finding $g(f(a))$ directly and then inverting. So $c = g(f(a)) = \dfrac{f(a)}{3} = \dfrac{2a+1}{3}$. So take $c = \dfrac{2a+1}{3}$ and solve it for $a$: \begin{align} 3c & = 2a+1 \\ 3c - 1 & = 2a \\ \\ \frac{3c-1}{2} & = a. \end{align} FINALLY, observe that you got the same thing both ways. - But the problem asks the student to verify the formula; that is, find the inverse of $g\circ f$ "directly", and then compare it to the function you get by computing $f^{-1}\circ g^{-1}$. Surely using the formula to verify that the formula works is a tad... unsatisfying. – Arturo Magidin Aug 26 '11 at 15:58 I am stuck with how do I find (gof)^-1 – Fahad Uddin Aug 26 '11 at 16:14 OK, I've added a later note. – Michael Hardy Aug 26 '11 at 16:18 @MichaelHardy thank you for this answer, it came in handy for a question I recently asked – seeker Dec 15 '13 at 13:01 What you've done so far is to compute $f^{-1}$ and $g^{-1}$, and $f^{-1}\circ g^{-1}$. Now you want to try to find $(g\circ f)^{-1}$ directly, and compare that to what you've computed (in order to verify the formula). So, you've figured out that $(g\circ f)(a) = \frac{2a+1}{3}$. How do we figure out $(g\circ f)^{-1}$? Exactly the same way we figure out the inverse of any function. If someone stopped you on the street, pointed a gun at you and said "Here, I have this function: $$h(a) = \frac{2a+1}{3},$$ I need the formula for $h^{-1}$. Give it to me or I'll shoot you!" then you don't need to know where that function came from, all you need to do is figure out the inverse: \begin{align*} b &= \frac{2a+1}{3}\\ 3b &= 2a+1\\ 3b-1 &= 2a\\ &\vdots \end{align*} etc. When you are done and have a formula for $h^{-1}(a) = (g\circ f)^{-1}(a)$, you can compare it to the formula you found for $f^{-1}\circ g^{-1}$ and verify that you got the same function. - Thanks alot. I did it as you suggested. A small problem that I am having is that is in terms of b instead of a. Checkout the edit. – Fahad Uddin Aug 26 '11 at 16:30 @fahad: Don't get hung up on the letters. The name of the variable is immaterial. The function $h(x) = x^2$ is exactly the same as the function $h(y)=y^2$, which is exactly the same as the function $h(z)=z^2$, which is exactly the same as the function $h(a)=a^2$. Just switch all the $b$s into $a$s and be done. – Arturo Magidin Aug 26 '11 at 16:36 Thanks alot for the answer :) – Fahad Uddin Aug 26 '11 at 19:15 I can't resist adding that the Russian mathematical physicist Igor Tamm was once told to do a mathematics calculation or be shot. :) – Mike Spivey Aug 26 '11 at 22:21

https://math.stackexchange.com/questions/1693923/solving-modulus-inequality-x-1-x-6-le11-geometrically

Solving modulus inequality $|x - 1| + |x - 6|\le11$ geometrically Find all possible values of $x$ for which $x$ for which the inequality $$|x - 1| + |x - 6|\le11$$ is true. I know this can be easily solved by taking $3$ cases for$x$ and then taking the intersection of those $3$ cases. The solution will be $-2\ge x\le9$. But suppose if I interpret this in this way: What number $x$ satisfy the condition that the distance between $x$ and $6$ plus the distance between $x$ and $1$ is less than or equal to $11$? I would be better to get an idea to solve these types of problems by geometrically by intuition using number line. • The geometric interpretation is a very good way of viewing the problem. Mar 12, 2016 at 6:41 • As $x$ moves between $(1,6)$, the sum of distances cannot change. Moving away from the interval the sum increases twice as fast as distance to both points increases. That's usually enough to solve this. Mar 12, 2016 at 6:50 • @Macavity I'm usable to visualize how will the sum change twice as fast when $x$ moves away from the interval $(1,6)$ ? Mar 12, 2016 at 7:05 • Outside the interval, $x$ is moving away from both points, so the distances naturally add. The interval has measure $5$ and the slack is $11-5=6$, so $x$ can move at best $3$ on either side of the interval. Mar 12, 2016 at 7:08 • One of the inequalities is pointing the wrong way where you write "The solution will be $-2\ge x\le9$." Mar 12, 2016 at 14:38 As you said, we are looking for points for which: $$(\text{distance to 1})+(\text{distance to 6})\leq 11.$$ The first thing which now comes to my mind is an ellipse. We first consider all $x\in \mathbb C$ for which $\vert x-1\vert +\vert x-6\vert \leq 11$. All points of this "filled" ellipse which lie on the real axis are the ones we want. It is not very hard to imagine what these point will be. If we are on the real line then our furthest left point, $x_\ell$, will lie left of $1$. So the distance to $6$ will be at least five. So we have $$\vert x_\ell-1\vert+\vert x_\ell -6\vert=2\vert x_\ell-1\vert+5=11.$$ And since $x_\ell$ is to the left of $1$ this means that $x_\ell=-2$. Analogously, we find that $x_r=9$. So we have found that $-2\leq x\leq 9$, through geometrical methods. $$|1-x| + |x-6| \le 11$$ Let $A, X, B \in \mathbb R^n$ (Euclidean $n$-space). If $X\in \overline{AB}$ ( $X$ is on the line segment $\overline{AB}$ ), we say that $X$ is between $A$ and $B$, and we write this as $A-X-B$. $$A-X-B \; \text{ if and only if } \; \|A-X\| + \|X-B\| = \|A-B\|$$ On $\mathbb R^1$ (the real number line), every point is on the line through points $1$ and $6$. So there are three possibilities for $1, x$, and $6$ : $x-1-6$, $1-x-6$, or $1-6-x$. CASE: $x-1-6$ Then $x \le 1 \le 6$ \begin{align} |1-x| + |x-6| &\le 11 \\ (1-x) + (6-x) &\le 11\\ -2x &\le 4\\ x &\ge -2\\ x &\in [-2,1] \end{align} CASE: $1-x-6$ Then $1 \le x \le 6$ and $|1-x| + |x-6| = |1-6| = 5$ \begin{align} |1-x| + |x-6| &\le 11 \\ 5 &\le 11 \\ x &\in [1,6] \end{align} CASE: $1-6-x$ Then $1 \le 6 \le x$ \begin{align} |1-x| + |x-6| &\le 11 \\ (x-1) + (x-6) &\le 11 \\ 2x &\le 18\\ x &\le 9\\ x &\in [6,9] \end{align} So $x \in [-2,9]$ • A nit-picker would say: "You did not formally consider the cases $x=1$ and $x=6$." Mar 12, 2016 at 10:51 $$y=|x-1|+|x-6|; y=11$$ $$x \in [x_1;x_2] =[-2;9]$$

http://languageoption.com/layla-hassan-esdq/page.php?page=f51332-a-correlation-coefficient-is-a-numerical-measure-of-the

A correlation coefficient is a numerical measure of the. If the correlation between two variables is close to 0.01, then there is a very weak linear relation between them. Therefore, correlations are typically written with two key numbers: r = and p =. We will: give a definition of the correlation $$r$$, discuss the calculation of $$r$$, explain how to interpret the value of $$r$$, and; talk about some of the properties of $$r$$. The direction of the correlation is determined by sign of the correlation coefficient ‘r’, whether the correlation is positive or negative. 10 Recommendations. measures the strength and direction of linear association between two numerical variables; greek letter p (rho) represents correlation between X and Y in the population; r represents the correlation between X and Y in a sample taken from the population The value of r is always between +1 and –1. X and Y. Pearson's correlation coefficient, when applied to a sample, is commonly represented by and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. Results: The Matthews correlation coefficient (MCC), instead, is a more reliable statistical rate which produces a high score only if the prediction obtained good results in all of the four confusion matrix categories (true positives, false negatives, true negatives, and false positives), proportionally both to the size of positive elements and the size of negative elements in the dataset. However, there is a relationship between the two variables—it’s just not linear. To interpret its value, see which of the following values your correlation r is closest to: Exactly –1. Linear Correlation Coefficient . Spearman’s rank correlation coefficient is given by the formula. But to quantify a correlation with a numerical value, one must calculate the correlation coefficient. Spearman’s correlation can be calculated for the subjectivity data also, like competition scores. ii) No ambiguity. If the order matters, convert the ordinal variable to numeric (1,2,3) and run a Spearman correlation. The regression describes how an explanatory variable is numerically related to the dependent variables.. Compute the correlation coefficients for a matrix with two normally distributed, random columns and one column that is defined in terms of another. This analysis yields a sample-based measure called Pearson’s correlation coefficient, or r. iii) The symbol r represents the sample correlation coefficient. Before calculating a correlation coefficient, screen your data for outliers (which can cause misleading results) and evidence of a linear relationship. Correlation measures the strength of linear association between two numerical variables. 13.2 The Correlation Coefficient. So now we have a way to measure the correlation between two continuous features, and two ways of measuring association between two categorical features. There are quite a few answers on stats exchange covering this topic - … 4. The appropriate quantity is the correlation coefficient.The formula for the correlation coefficient is a bit complicated, although calculating it does not involve much more than calculating sample means and standard deviations as was done in Chapter 3. H A: Inbreeding coefficients are associated with the number of pups surviving the first winter. Both of the tools are used to represent the linear relationship between the two quantitative variables. Correlation coefficient can be defined as a measure of the relationship between two quantitative or qualitative variables, i.e. A value of ± 1 indicates a perfect degree of … Correlation coefficient and the slope always have the same sign (positive or negative). However, the following table may serve a as rule of thumb how to address the numerical values of Pearson product moment correlation coefficient. Consequently, if your data contain a curvilinear relationship, the correlation coefficient will not detect it. Two people must arrive at the same numerical value. Spearman correlation coefficient: Definition. Correlation is a statistical measure used to determine the strength and direction of the mutual relationship between two quantitative variables. The numerical measure that assesses the strength of a linear relationship is called the correlation coefficient, and is denoted by $$r$$. Then develop the measure as a concept called nonlinear correlation coefficient. It serves as a statistical tool that helps to analyse and in turn, measure the degree of the linear relationship between the variables. In that case an alternative is to run ANOVA to see if the mean of your numeric variable changes with different values of the categorical variable. The linear correlation coefficient measures the strength of the linear relationship between two variables. For example, the correlation for the data in the scatterplot below is zero. 13.2 The Correlation Coefficient. R 1i = rank of i in the first set of data. Rank statistic) see Kendall coefficient of rank correlation; Spearman coefficient of rank correlation. It is a statistic that measures the linear correlation between two variables. Correlation is a bivariate analysis that measures the strength of association between two variables and the direction of the relationship. We describe correlations with a unit-free measure called the correlation coefficient which ranges from -1 to +1 and is denoted by r. Statistical significance is indicated with a p-value. For this, we can use the Correlation Ratio (often marked using the greek letter eta). Correlation standardizes the measure of interdependence between two variables and, consequently, tells you how closely the two variables move. Pearson’s correlation coefficients measure only linear relationships. Stephen Politzer-Ahles. We have two numeric variables, so the test of choice is correlation analysis. We’ll set $$\alpha$$ = 0.05. The numerical measure that assesses the strength of a linear relationship is called the correlation coefficient, and is denoted by $$r$$. Named after Charles Spearman, it is often denoted by the … Find an answer to your question “A correlation coefficient is a numerical measure of the ...” in Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. where D i = R 1i – R 2i. The correlation coefficient is a statistical measure that calculates the strength of the relationship between the relative movements of two variables. If the order doesn't matter, correlation is not defined for your problem. Thus when applied to binary/categorical data, you will obtain measure of a relationship which does not have to be correct and/or precise. Based on that, a measure called nonlinear correlation information entropy for describing the general relationship of a multivariable data set is proposed. Since the third column of A is a multiple of the second, these two variables are directly correlated, thus the correlation coefficient in the (2,3) and (3,2) entries of R is 1. Pearson's Correlation Coefficient ® In Statistics, the Pearson's Correlation Coefficient is also referred to as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC), or bivariate correlation. 6th Dec, 2016 . We will: give a definition of the correlation $$r$$, discuss the calculation of $$r$$, explain how to interpret the value of $$r$$, and; talk about some of the properties of $$r$$. In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatterplot. Pearson's correlation coefficient is a measure of linear association. The Spearman’s rank coefficient of correlation is a nonparametric measure of rank correlation (statistical dependence of ranking between two variables). There are several types of correlation coefficients but the one that is most common is the Pearson correlation r. It is a parametric test that is only recommended when the variables are normally distributed and the relationship between them is linear. A numerical measure of linear association between two variables is the a. variance b. coefficient of variation c. correlation coefficient d. standard deviation Karl Pearson’s Coefficient of Correlation is widely used mathematical method wherein the numerical expression is used to calculate the degree and direction of the relationship between linear related variables. Mathematical statisticians have developed methods for estimating coefficients that characterize the correlation between random variables or tests; there are also methods to test hypotheses concerning their values, using their … e) Correlation coefficient i) A numerical measure of the strength and the direction of a linear relationship between two variables. The strength of a correlation is determined by its numerical (absolute) value. Well correlation, namely Pearson coefficient, is built for continuous data. What graphs can you use to measure correlation? A perfect downhill (negative) linear relationship […] Pearson’s method, popularly known as a Pearsonian Coefficient of Correlation, is the most extensively used quantitative methods in practice. The linear correlation coefficient is a number calculated from given data that measures the strength of the linear … A correlation coefficient gives a numerical summary of the degree of association between two variables . The closer r … We can obtain a formula for r x y {\displaystyle r_{xy}} by substituting estimates of the covariances and variances based on a sample into the formula above. For measures of correlation based on rank statistics (cf. In terms of the strength of relationship, the value of the correlation coefficient varies between +1 and -1. A more subtle measure is intraclass correlation coefficient (ICC). But what about a pair of a continuous feature and a categorical feature? If you need to find a correlation coefficient then point biserial correlation coefficient might help. Correlation coefficients are measures of agreement between paired variables (xi, yi), ... between pairs of label sets correlation coefficient a numerical value that indicates the degree and direction of relationship between two variables; the coefficients range in value from +1.00 (perfect positive relationship) to 0.00.. We need a numerical measure of the strength of the linear relationship between two variables that is not affected by the scale of a plot. Olf the correlation coefficient is 1, then the slope must be 1 as well. Cite. Correlations measure how variables or rank orders are related. The data can be ranked from low to high or high to low by assigning ranks. And in turn, measure the degree of the relationship between the two variables on a.! For this, we can use the correlation is a relationship between the variables—it! When applied to binary/categorical data, you will obtain measure of a linear relationship:... On that, a measure of the relationship between two variables correlation ( statistical dependence of between... For describing the general relationship of a linear relationship between the variables variables, i.e rank orders are related ). Of two variables and, consequently, tells you how closely the two variables and, consequently tells... 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The following table may serve a as a correlation coefficient is a numerical measure of the of thumb how to address the numerical values Pearson. Positive or negative ) order does n't matter, correlation is a bivariate analysis that the... Then there is a numerical summary of the a correlation coefficient is a numerical measure of the relation between them of correlation, the! Data, you will obtain measure of linear association degree of the correlation between two variables the! To: Exactly –1 dependence of ranking between two variables and, consequently, tells you how closely two. A statistical measure that calculates the strength and direction of the strength of association between variables! Not linear r … a correlation coefficient, or r. Pearson ’ s correlation coefficient varies between and... Used to determine the strength of association between two variables move by assigning ranks used quantitative in... 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From low to high or high to low by assigning ranks measure called nonlinear information! Two people must a correlation coefficient is a numerical measure of the at the same sign ( positive or negative, one must calculate the correlation coefficients only... Numerical value this, we can use the correlation coefficient and the direction of the correlation coefficient for problem. Letter eta ) or negative ) the relationship between a correlation coefficient is a numerical measure of the two variables—it ’ just. The strength and direction of the degree of association between two variables the. Are typically written with two normally distributed, random columns and one column that defined... Standardizes the measure as a measure called Pearson ’ s just not.! Coefficient ‘ r ’, whether the correlation coefficient gives a numerical summary of degree...

http://mathhelpforum.com/math-puzzles/207164-how-many-numbers-less-than-equal-n-all-digits-unique.html

# Math Help - how many numbers less than or equal to N with all digits unique 1. ## how many numbers less than or equal to N with all digits unique Hi all I am not sure whether this would count as a puzzle but it has puzzled me. I am really bad at combinatorics and need some help. I have stumbled on to the problem which states that "How many numbers upto N have all unique digits in them?" eg - suppose N = 34. Then answer is 31 (11, 22, 33 excluded, so 34 - 3 = 31). How do I solve this for a generic case? I cant understand how to extend the idea for 3 digits and above numbers. Any help would be appreciated. And please do explain your answer to. I would really be thankful for any help which can improve my combinatorics? Thanks Anant 2. ## Re: how many numbers less than or equal to N with all digits unique Good afternoon from the States. How many 2-digit numbers do NOT contain any repeating digits? Since a two-digit number is not going to begin in zero, there are 9 permissible values for the first digit, namely one through 9. Zero is allowed for the second digit, but remember that the second digit cannot be the same as the first. Thus, there are 9 permissible values for the second digit. There are therefore 9*9=81 two-digit numbers in which both digits are unique. What about 3-digit numbers? Same concept: There are 9 permissible values for the first digit (all but zero), 9 permissible values for the second digit (all but the first digit), and 8 permissible values for the third digit (all but the first and second digits). Thus, the answer for the 3-digit case is 9*9*8=648. Four digits follows similarly: 9*9*8*7. Clearly, 10 digits is the maximum if you need all digits to be unique. In that case, you get 9*9*8*7*6*5*4*3*2*1. Note that the cases for 9 digits and ten digits are identical. -Andy 3. ## Re: how many numbers less than or equal to N with all digits unique ^Up to N. N can be some arbitrary number other than a power of 10. 4. ## Re: how many numbers less than or equal to N with all digits unique Originally Posted by richard1234 ^Up to N. N can be some arbitrary number other than a power of 10. Well, of course -- I was just offering a start. 5. ## Re: how many numbers less than or equal to N with all digits unique Thanks abender. I have been able to understand when N is power of 10. What isnt clear is suppose N is something like 34523. Then how do i proceed? 6. ## Re: how many numbers less than or equal to N with all digits unique N=34523 For communication purposes, let's call numbers that have all unique digits "unique numbers". When N=9999, we have 9*9*8*7 = 3888 "unique numbers". For N=29999 we have the 3888 "unique numbers" that are under 10000, and we now need to consider the unique numbers between 10000 and 29999. The number of "unique numbers" between 10000 and 29999 is found this way: There are 2 choices for the first digit, 9 for the second, 8 for the third, 7 for the fourth, and 6 for the fifth; 2*9*8*7*6 = 6048. So, when N=30000 (which doesn't change the answer from 29999), the number of "unique numbers" is 3888+6048 = 9936. So know there are 9936 "unique numbers" for N=29999 (or N=30000). Now, how many "unique numbers" are between 30000 and 34000. The first digit has 1 possibility, the second has 4, the third has 8, the fourth has 7, and the fifth has 6, totaling 1*4*8*7*6 = 1344 "unique numbers". So there are 9936+1344 "unique numbers" for N=34000. Continue in this fashion. 7. ## Re: how many numbers less than or equal to N with all digits unique I have a similar problem. How many are there odd 4-digit numbers with all digits different? Is it 9*8*7*5? I get that when I first choose last and first digit. But if I start from the beggining I get 9*9*8*5 (because 0 can't be first so there are again 9 possibilities for second digit). 8. ## Re: how many numbers less than or equal to N with all digits unique Originally Posted by abender So know there are 9936 "unique numbers" for N=29999 (or N=30000). Now, how many "unique numbers" are between 30000 and 34000. The first digit has 1 possibility, the second has 4, the third has 8, the fourth has 7, and the fifth has 6, totaling 1*4*8*7*6 = 1344 "unique numbers". So there are 9936+1344 "unique numbers" for N=34000. Continue in this fashion. Be careful when working with these. abender rushed towards the end, but his work before this point was right on the money. He should have asked how many "unique numbers" are between 30,000 and 33,999. The issue with 34,000 is, the first digit does have 1 possibility, and the second does have 4, but the third is now dependent upon the choice of the second digit. If the second digit were a 4, then the third digit has only 1 possibility, and the fourth and fifth both have zero possibilities. So, the second digit actually has only 3 possibilities. 9. ## Re: how many numbers less than or equal to N with all digits unique As abender already mentioned, since a number with 11 or more digits is guaranteed to repeat a digit by the Pigeonhole Principle, there are no "unique numbers" with 11 or more digits. Generalizing abender's algorithm, there are $9\dfrac{9!}{(10-k)!}$ k-digit "unique numbers". This means there are $9\sum_{i = 1}^k \dfrac{9!}{(10-i)!}$ "unique numbers" with k or fewer digits. As another method of counting, suppose N has k digits. Start with the set of all "unique numbers" with k or fewer digits. Then subtract off the number of "unique numbers" where the first digit is greater than the first digit of N. Then subtract off the number of "unique numbers" where the first digit is equal to the first digit of N and the second digit is greater. Etc. So, to describe the algorithm, you can use this type of notation. Let $N = \sum_{i = 1}^k d_i 10^{i-1}, \quad \forall 0< i < k, d_i \in \mathbb{Z}, 0\le d_i \le 9;\quad 0< d_k \le 9$. Next, let $A_i = \{z \in \mathbb{N} \mid z\mbox{ is a "unique number" with }i\mbox{ digits} \}$. For any $z\in A_k$, we can refer to its digits with the notation $z = \sum_{i = 1}^k z_i 10^{i-1}$. Next, define the sets $N_i = \{z\in A_k \mid z_i > d_i \mbox{ and } \forall i < j \le k, z_j = d_j \}$ for $1 \le i \le k$. What elements are in these sets? So, by construction, the number of "unique numbers" up to N is given by $\sum_{i = 1}^k \left( |A_i| - |N_i| \right)$ (let me know if you have any questions about why this is true). We already know how to calculate $|A_i|$ easily. So, how do we calculate $|N_i|$? That is a little more difficult. To make this easier, we will want one more thing. For notation, let $[n] = \{1,2,\ldots, n\}$ (this is fairly common notation in combinatorics). Let $M = \left\{i \in [k] \mid \sum_{j = 0}^{k-i} d_{k-j} 10^j \in A_i \right\}$. What is this set $M$? What does it represent? More specifically, what is $\min(M)$? Spoiler: I plugged in the formula for $\sum_{i=1}^k |A_i|$ into wolframalpha and it shot out $9\left( 986410 - \dfrac{362880e\Gamma(8-k,1)}{(9-k)!} \right)$ where $\Gamma(n,x)$ is the incomplete upper gamma function. Probably not terribly useful, but interesting that there is a continuous function based on $k$. Makes me wonder if we plug in $\log_{10}(N)$ for $k$, how close will we come to the actual number? 10. ## Re: how many numbers less than or equal to N with all digits unique Originally Posted by kicma I have a similar problem. How many are there odd 4-digit numbers with all digits different? Is it 9*8*7*5? I get that when I first choose last and first digit. But if I start from the beggining I get 9*9*8*5 (because 0 can't be first so there are again 9 possibilities for second digit). Let's look at the starting conditions. Before choosing any digits, the first digit is an element of $\{1,2,3,4,5,6,7,8,9\}$. The last digit is an element of $\{1,3,5,7,9\}$. Hence, if you choose the first digit first, and you choose an odd digit, you will only have four choices available for the last digit. On the other hand, if you choose the last digit first, no matter which digit you choose leaves only 8 choices left for the first digit. If you want to count from left to right, you need to break it up into a LOT of cases. You could choose one odd digit among the first three, two odd digits among the first three, or three odd digits among the first three. This becomes too cumbersome. Alternately, you can choose the digits right to left. Then you have five choices for the first digit, 9 choices for the second, 8 for the third, but you either have 7 or 6 choices for the last depending on whether you chose a zero yet or not. So, as a general rule of thumb, choose the digits in the order of how restricted they are. The last digit is the most restricted, so choose that first: 5 choices. After that choice, before making any others, you have 8 possible choices for the first digit (since an odd digit is now out) or you have 9 choices for either of the other two digits (since they can still be zero). So, again, choose the most restricted. So now, you have 5*8 to choose the last digit then the first digit. Now, the remaining two digits are equally restricted with 8 choices each. So, to choose the last digit, then the first digit, then the second digit, there are 5*8*8 different choices. Finally, to choose the 3rd digit, there are 7 choices, and the total number of odd 4-digit numbers with distinct digits is 5*8*8*7. 11. ## Re: how many numbers less than or equal to N with all digits unique Put your numbers into binary to simplify the problem :P

https://math.stackexchange.com/questions/1782087/almost-everywhere-vs-almost-sure/1782114

# almost everywhere Vs. almost sure I'm reading a book about measure theory and probability (first chapter of Durret's Probability book), and it's starting to switch between the terms "a.e." and "a.s." in different contexts. I'm becoming confused about their meanings. What's the difference between almost everywhere and almost sure? • a.e. is used for general measure spaces, while a.s. is used for probabilistic spaces. There is no important difference between both concepts (for example, measurable functions are named random variables when the measure space is a probabilistic space). May 12, 2016 at 8:52 • a.e. can be used a.e. This in contrast with a.s. which is only used in the context of probability measures. In principle you could use a.e. there too, but a.s. a.e. is replaced there by a.s. May 12, 2016 at 9:17 In a probability space (equipped with a probability $P$), we say that an event $\omega$ occurs almost surely if $P(\omega)=1$. On the other hand, on a measure space equipped with a measure $\mu$, we say that a property $\mathcal{P}$ is satisfied almost everywhere if the set where $\mathcal{P}$ is not satisfied has measure zero. Note that "a.s." is equivalent to "a.e." in probability spaces, since if $\omega$ occurs almost surely, then the probability that $\omega$ does not occur is zero. However, in the case of general measure spaces $X$ we cannot say that a property is satisfied almost everywhere if it is satisfied in a set of measure $\mu(X)$ (which would correspond to an event having probability $1$), since in many cases this measure is infinite. This is why in the case of measure spaces we formulate the definition of "almost everywhere" in terms of complements of sets. • $\{x: x \in (-\infty, 0)\cup (1, +\infty)\}$ is a.s. in $\mathbb{R}$ but not a.e. in $\mathbb{R}$, $\mathbb{R}$ is equipped with Borel sets and Lebesgue measure. Is this right? Dec 12, 2019 at 16:39 When we say that something happens "almost everywhere", we mean to say that: 1. This something can happen or fail to happen in any point in a given measure space; and 2. The set of points in which it fails to happen is a set of measure zero. Notice that there's no notion of probability when talking about "almost everywhere". Now, when we say that something happens "almost surely", we mean to say that: 1. This something is the result of a random experiment. It can happen or fail to happen, and the result of this experiment (success ofr failure) is a random variable; and 2. The probability of this something failing to happen is zero. From Wiki Almost surely In probability theory, one says that an event happens almost surely (sometimes abbreviated as a.s.) if it happens with probability one. In other words, the set of possible exceptions may be non-empty, but it has probability. this means almost surely comes from probability trials (stochastic trials), From Wolfram mathworld Almost Everywhere A property of X is said to hold almost everywhere if the set of points in X where this property fails is contained in a set that has measure zero. From this book "Essentials of Probability Theory for Statisticians" Par Michael A. Proschan,Pamela A. Shaw page 102 Almost surely for random variable, (probability) Almost everywhere for the sequance of functions, (measure)

https://math.stackexchange.com/questions/2702057/proof-of-sum-formula-no-induction

# Proof of sum formula, no induction $$\sum_{k=1}^n k=\frac{n(n+1)}2$$ So I was trying to prove this sum formula without induction. I got some tips from my textbook and got this. Let $S=1+2+\cdots+n-1+n$ be the sum of integers and $S=n+(n+1)+\cdots+2+1$ written backwards. If I add these $2$ equations I get $2S=(1+n)+(1+n)\cdots(1+n)+(1+n)$ $n$ times. This gives me $2S=n(n+1) \Rightarrow S=\frac{n(n+1)}2$ as wanted. However if I changed this proof so that n was strictly odd or strictly even, how might I got about this. I realize even means n must be $n/2$. But I haven't been able to implement this in the proof correctly. Edit: error in question fixed, also by $n/2$ I mean should I implement this idea somewhere in the proof, cause even means divisible by $2$. • When written backward, $S=n+(n-1)+....+2+1$ – CY Aries Mar 21 '18 at 16:35 • "I realize even means n must be n/2". I'm confused. What do you mean by this? – Mauve Mar 21 '18 at 16:36 • Possible duplicate of Proof for formula for sum of sequence $1+2+3+\ldots+n$? – GNUSupporter 8964民主女神 地下教會 Mar 21 '18 at 16:36 • To say "$n$ must be $n/s$" is a confusion. Better is "$n= 2m$ for some integer $m$." Your proof works for both odd and even $n$, so your last para is mysterious. Do you mean that you want to add just the first $n$ odd (or even) integers? – B. Goddard Mar 21 '18 at 16:37 • the proof works whether $n$ is even or odd. what's the problem? You realize that either $n$ or $n+1$ will be even. – Vasya Mar 21 '18 at 16:37 Method 1: (requires you to consider whether $n$ is odd or even.) $S = 1 + 2 + ...... + n$. Join up the first to term to the last term and second to second to last and so on. $S = \underbrace{1 + \underbrace{2 + \underbrace{3 +....+(n-2)} + (n-1)} + n}$. $= (n+1) + (n+1) + .....$. If $n$ is even then: $S = \underbrace{1 + \underbrace{2 + \underbrace{3 +..+\underbrace{\frac n2 + (\frac n2 + 1)}+..+(n-2)} + (n-1)} + n}$ And you have $\frac n2$ pairs that add up to $n+1$. So the sum is $S= \frac n2(n+1)$. If $n$ is odd then: $S = \underbrace{1 + \underbrace{2 + \underbrace{3 +..+\underbrace{\frac {n-1}2 + [\frac {n+1}2] + (\frac {n+1}2 + 1)}+..+(n-2)} + (n-1)} + n}$ And you have $\frac {n-1}2$ pairs that also add up to $n+1$ and one extra number $\frac {n+1}2$ which didn't fit into any pair. So the sum is $\frac {n-1}2(n+1) + \frac {n+1}2 =(n-1)\frac {n+1}2 + \frac {n+1}2 = (n-1 + 1)\frac {n+1}2n=n\frac {n+1}2$. Method 1$\frac 12$ (Same as above but waves hands over doing tso cases). $S = average*\text{number of terms} = average*n$. Now the average of $1$ and $n$ is $\frac {n+1}2$ and the average of $2$ and $n-1$ is $\frac {n+1}2$ and so on. So the average of all of them together is $\frac {n+1}2$. So $S = \frac {n+1}2n$. Method 2: (doesn't require considering whether $n$ is odd or even). $S = 1 + 2 + 3 + ...... + n$ $S = n + (n-1) + (n-2) + ...... + 1$. $2S = S+S = (n+ 1) + (n+1) + ..... + (n+1) = n(n+1)$> $S = \frac {n(n+1)}2$. Note that by adding $S$ to itself this doesn't matter whether $n$ is even or odd. And lest you are wondering why can we be so sure that $n(n+1)$ must be even (we constructed it so it must be true... but why?) we simply note that one of $n$ or $n+1$ must be even. So no problem. For $n$ even i.e. $n=2m$ for some $m\in\mathbb{N}$. Let $S:=1+2+...+2m$ then $$2S=S+S=(1+2+...+(2m-1)+2m)+(2m+(2m-1)+...+2+1)=((1+2m)+(2+(2m-1))+...+((2m-1)+2)+(2m+1))=\underbrace{((2m+1)+...+(2m+1))}_{2m-times}=(2m+1)\cdot 2m$$ Therefore $$S=\frac{(2m+1)2m}{2}=m(2m+1)$$ Analogue for $n$ odd. The formula is the same as in general for any $n$. You indeed just substitute directly into it. For instance $$\frac{n(n+1)}{2}\Rightarrow \frac{2m(2m+1)}{2}=m(2m+1)$$ This classical result can be also easily proved by the following trick An extended discussion also for more general cases here How Are the Solutions for Finite Sums of Natural Numbers Derived? A combinatorial proof. Consider the two element subsets of $\Omega=\{0,1,\dotsc,n\}$. There are $\binom{n+1}{2}$ of them (corresponding to the right hand side of the equality). But we can count in another way. Classify the two element subsets based on their maximum element. For $1\leq k \leq n$, there are $\binom{k}{1}=k$ two element subsets whose maximum element is $k$ since there are $k$ non-negative integers less than $k$. Another proof based on telescoping. Let $n^{\underline{2}}=n(n-1)$ (this is the falling factorial of length two. The exponent has an underline for notation). Observe that $$\frac{1}{2}(n+1)^{\underline{2}}-\frac{1}{2}(n)^{\underline{2}}=n.$$ In particular $$\sum_{k=1}^nk=\frac{1}{2}\sum_{k=1}^n (k+1)^{\underline{2}}-(k)^{\underline{2}}=\frac{(n+1)^{\underline{2}}}{2}=\frac{n(n+1)}{2}.$$

https://www.hpmuseum.org/forum/thread-361.html

Accuracy of HP Financial Calculators - Canadian mortgage 01-08-2014, 03:20 AM (This post was last modified: 01-08-2014 11:39 AM by Jeff_Kearns.) Post: #1 Jeff_Kearns Member Posts: 147 Joined: Dec 2013 Accuracy of HP Financial Calculators - Canadian mortgage Having recently adapted an 'accurate' (relative term) TVM routine for the HP-15C thanks to Karl Schneider's MISO Technique using indirect addressing, I decided to compare the results for a Canadian mortgage monthly payment with other HP calculator models. For all non-Canadians, interest on mortgages in Canada is compounded semi-annually, but payments are made monthly. One therefore has to convert the nominal annual interest rate (based on semi-annual compounding) to an effective annual interest rate and then convert it again to an equivalent nominal rate for calculating monthly payments (based on monthly compounding). If you have an HP-12C, the following procedure can be used to obtain the Canadian mortgage factor (from the Owner's Handbook): The keystrokes to calculate the Canadian Mortgage factor (on the 12C) are: Press [f ], CLEAR [FIN ], [g ], then END . Key in 6 and press [n ]. Key in 200 and press ENTER , then PV . Key in the annual interest rate as a percentage and press [+ ], CHS , then FV . Press [i ]. The Canadian mortgage factor is now stored in [i ] for future use. I have a short routine of the above procedure for the 12C and equivalent SOLVER routines for the 17BII and 19BII. The 30b has a built-in Canadian TVM function that does the same with appropriate mode settings. The results from different models (both scientific and financial) are interesting and I welcome any insight as to why they differ the way they do and which is the most 'correct': Variables n: 300 months (#payments) i: 3% (nominal annual interest rate) PV: $450,000 (mortgage amount) PMT: UNKNOWN FV: 0 (fully paid off after 300 months) Results for monthly payment: • HP-15C:$2,129.604821 • HP-12C: $2,129.604744 • HP-32SII:$2,129.60474211 • HP-42S: $2,129.60474211 • HP-30B:$2,129.60474341 • HP-19BII: $2,129.60474211 • HP-17BII:$2,129.60474211 The 15C results are not surprising considering it has less precision. It gives a monthly nominal rate of 0.248451700% whereas the 12C gives 0.248451673 and the HP-32Sii and HP-42S both give 0.248451672% (and so does the 50G). The difference in precision in the 15C likely stems from taking the 12th root of the effective rate when converting to monthly. What surprises me is the HP-30B... Why does it give a different result from the other high-end financial models? And isn't is surprising how accurate the routine is for the pioneer models 32sii and 42S? How does the WP-34S compare? Jeff 01-08-2014, 04:56 AM Post: #2 Gene Moderator Posts: 992 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage HP-30B: $2,129.60474341 What surprises me is the HP-30B... Why does it give a different result from the other high-end financial models? And isn't is surprising how accurate the routine is for the pioneer models 32sii and 42S? How does the WP-34S compare? Gene: The 30B result matches Excel's calculations of$2,129.60474341 366 as far as the 30B goes. As I recall, the 30B has greater internal accuracy in its financial routines. 01-08-2014, 11:16 AM Post: #3 Terje Vallestad Member Posts: 145 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (01-08-2014 04:56 AM)Gene Wrote:  HP-30B: $2,129.60474341 Gene: The 30B result matches Excel's calculations of$2,129.60474341 366 as far as the 30B goes. As I recall, the 30B has greater internal accuracy in its financial routines. For what it is worth; the Prime provides the same result as the 30b 2129.60474341 Cheers, Terje 01-08-2014, 11:34 AM (This post was last modified: 01-08-2014 11:35 AM by Jeff_Kearns.) Post: #4 Jeff_Kearns Member Posts: 147 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage Terje Vallestad wrote: "For what it is worth; the Prime provides the same result as the 30b 2129.60474341" The 50G gives the same result as the 19BII and Pioneers: 2129.60474211 Jeff 01-08-2014, 11:42 AM Post: #5 Werner Senior Member Posts: 452 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage To convert from a nominal Canadian annual rate to the effective nominal rate, use: 200 / LN1+X 6 / E^X-1 100 * Then, the HP42S gets the monthly rate as 0.248451672465 and the PMT as -2,129.60474341, just like the 30b Cheers, Werner 01-08-2014, 12:19 PM Post: #6 Gerson W. Barbosa Senior Member Posts: 1,289 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (01-08-2014 04:56 AM)Gene Wrote:  Gene: The 30B result matches Excel's calculations of $2,129.60474341 366 as far as the 30B goes. As I recall, the 30B has greater internal accuracy in its financial routines. HP 200LX:$2,129.604743413646 01-10-2014, 01:02 AM Post: #7 DMaier Junior Member Posts: 42 Joined: Jan 2014 RE: Accuracy of HP Financial Calculators - Canadian mortgage The WP-34S matches the HP-30B and the 42S: 2129.60474341. The effective interest rate is: 0.24845167247 (Werner's method gives 0.2484516724648725, so there's that.) The same result, by the way, using either the original 2 step 12C method, or by setting: I=3 NP=12 NI=2 01-10-2014, 01:39 AM (This post was last modified: 01-10-2014 04:32 AM by Jeff_Kearns.) Post: #8 Jeff_Kearns Member Posts: 147 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (01-10-2014 01:02 AM)DMaier Wrote:  The effective interest rate is: 0.24845167247 is incorrect. I am not a financial type - I am an engineer. But I think it is important to make a note on terminology. That value: 0.24845167247%, is actually the 'nominal' monthly rate, not the 'effective' interest rate. If you multiply it by 12, you get 2.98142007%, the nominal annual rate for monthly payment calculations. The stated 'nominal' annual rate for Canadian mortgages is based on a semi-annual compounding period - and is 3% in this particular example. This translates into an 'effective' annual interest rate of 3.0225%. The 'effective' interest rate is always greater than the nominal rate. There are in fact, three 'rates' at play here... > The posted 'nominal' rate of 3%, based on semi-annual compounding; > The 'effective' annual interest rate of 3.0225%; and > The 'nominal' rate of 2.98142007%, used for the calculation of monthly payments. Effective Annual Interest Rate Explained Regards, Jeff 04-07-2014, 02:12 AM (This post was last modified: 04-07-2014 08:50 PM by solwarda.) Post: #9 solwarda Junior Member Posts: 3 Joined: Apr 2014 RE: Accuracy of HP Financial Calculators - Canadian mortgage The nominal monthly rate for Canadian mortgages can be calculated as follows: 3%/200 + 1 =1.015^(1/6)=1.0024845167 2464872638 - 1. The Formula for the monthly payment is: $450,000 x .0024845167 2464872638 x (1.0024845167 2464872638)^300/((1.0024845167 2464872638^300) - 1)=$2129.6047434136 4549091198. As can be seen from these extended precision calculations, it all depends on how many decimal places is the calculation carried out internally inside each calculator. 04-07-2014, 03:31 AM Post: #10 Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (04-07-2014 02:12 AM)solwarda Wrote:  3%/200 + 1 =1.015^(1/6)=1.0024845167 2464872638 - 1. ಠ_ಠ That's not how equations work. 04-07-2014, 04:39 AM Post: #11 solwarda Junior Member Posts: 3 Joined: Apr 2014 RE: Accuracy of HP Financial Calculators - Canadian mortgage Thomas Klemm: I don't quite understand your objection. I'm simply converting 3% nominal annual rate compounded semi-annually into a monthly nominal rate for the purpose of calculating the requisite monthly payment. You can do this in a number of ways. For example, you can compound 3% semi-annual rate into "effective" annual rate as follows: 3/200=.015 +1=1.015^2=1.030225-1 x 100=3.0225, which is the effective annual rate. Then you can take the 12th root of that, which will give you the above nominal monthly rate. 04-07-2014, 05:19 AM Post: #12 Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (04-07-2014 04:39 AM)solwarda Wrote:  Thomas Klemm: I don't quite understand your objection. Equality is a transitive relation. By chaining the calculations you create equations that don't hold. You just have to split them: 3%/200 + 1 = 1.015 1.015^(1/6) = 1.0024845167 2464872638 1.0024845167 2464872638 - 1 = 0.0024845167 2464872638 So nothing wrong with the calculation. Just with the use of the = sign. Cheers Thomas 04-07-2014, 07:06 PM Post: #13 Dieter Senior Member Posts: 2,397 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (04-07-2014 02:12 AM)solwarda Wrote:  The Formula for the monthly payment is... ...missing some brackets. Otherwise the two identical powers cancel down to 1. ;-) Quote:0.0024845167 2464872638 The true value is ...639. Or ...638916... to be precise. Using ln1+x and e^x–1 ensures the required accuracy here. Quote:$2129.6047434136 4549091198. The WP34s in DP mode returns 2.129,6047 4341 3645 4909 1198 5546 8796 57 Quote:As can be seen from these extended precision calculations, it all depends on how many decimal places is the calculation carried out internally inside each calculator. You bet. ;-) Dieter 04-07-2014, 07:49 PM (This post was last modified: 04-07-2014 08:13 PM by Dieter.) Post: #14 Dieter Senior Member Posts: 2,397 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (01-08-2014 03:20 AM)Jeff_Kearns Wrote: • HP-15C:$2,129.604821 • HP-12C: $2,129.604744 • HP-32SII:$2,129.60474211 • HP-42S: $2,129.60474211 • HP-30B:$2,129.60474341 • HP-19BII: $2,129.60474211 • HP-17BII:$2,129.60474211 The 15C results are not surprising considering it has less precision. It gives a monthly nominal rate of 0.248451700% whereas the 12C gives 0.248451673 and the HP-32Sii and HP-42S both give 0.248451672% (and so does the 50G). Both the 42s and the 50G feature ln1+x and e^x-1, so they are able to obtain the correct 12-digit interest rate as 0,248451672465%. This again leads to the correct result, which in the above list is exclusively returned by the 30B. So if the 12-digit machines don't get it right it's not their fault. It's sloppy programming in case of the 42s and 50G. Calculators that do not offer ln1+x need a workaround to get similar accuracy. Quote:What surprises me is the HP-30B... Why does it give a different result from the other high-end financial models? While most HPs internally use three additional guard digits (for internal calculations, that is - not in a TVM user program), the 30B IIRC internally uses much more digits. On the other hand the correct result can be obtained with simple 12-digit accuracy (e.g.42s, 50G) - careful programming provided: Code:  3 [ENTER] 200 [/]  =>   0,015  [ln1+x]            =>   0,0148886124938  6 [/]              =>   0,00248143541563  [e^x-1] 100 [x]    =>   0,248451672465 Edit: I now see that Werner already mentioned this point in his post. ;-) Users with hyperbolic functions, but without ln1+x and e^x-1 may do it this way: Code:  3 [ENTER] 200 [/]         =>   0,015  [ENTER] [ENTER] 2 [+] [/] =>   0,00744416873449  [HYP] [ATAN] 2 [x]        =>   0,0148886124937  6 [/]                     =>   0,00248143541562  2 [/] [HYP] [SIN]  [LastX] [e^x] [x]         =>   0,00124225836232  200 [x]                   =>   0,248451672464 Compare this with the result of less careful evaluation: Code:  1,015 [ENTER] 6 [1/x] [y^x] => 1,00248451672   (maybe ...673)  1 [–] 100 [x]               => 0,248451672000 Which means that three valuable digits are lost! Quote:And isn't is surprising how accurate the routine is for the pioneer models 32sii and 42S? I do not think that nine out of twelve digits is particulary accurate. But again: that's not the fault of the calculator. Quote:How does the WP-34S compare? With 16 or even 34 digit precision (and careful programming) you can expect accuracy far beyond the previous examples: i% = 0,2484 5167 2464 8726 3891 6130 7363 4707 27 PMT = 2.129,6047 4341 3645 4909 1198 5546 8796 57 Dieter 04-07-2014, 08:42 PM Post: #15 Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (01-08-2014 03:20 AM)Jeff_Kearns Wrote:  The 15C results are not surprising considering it has less precision. Using Werner's method to calculate i% directly and using the improved version of your TVM program I got the same result for PMT as with the HP-12C. Cheers Thomas 04-07-2014, 09:06 PM Post: #16 Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (04-07-2014 07:49 PM)Dieter Wrote: Code:  3 [ENTER] 200 [/]         =>   0,015  [ENTER] [ENTER] 2 [+] [/] =>   0,00744416873449  [HYP] [ATAN] 2 [x]        =>   0,0148886124937 In this case you don't gain anything but loose accuracy compared to: Code:  3 [ENTER] 200 [/] 1 [+]    =>   1.015  [LN]                       =>   0.0148886124938 Cheers Thomas 04-07-2014, 09:14 PM Post: #17 solwarda Junior Member Posts: 3 Joined: Apr 2014 RE: Accuracy of HP Financial Calculators - Canadian mortgage Dieter: If you want the final answer accurate to 100 digits!, here it is: \$2,129.6047434136 4549091198 5546879656 9409506602 5105604773 4028771274 1867975760 1238046591 3606995240 188946. PS: I can give it to you up to a million digits!. Cheers!, Sol 04-07-2014, 09:20 PM Post: #18 Jeff_Kearns Member Posts: 147 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (04-07-2014 07:49 PM)Dieter Wrote:  Both the 42s and the 50G feature ln1+x and e^x-1, so they are able to obtain the correct 12-digit interest rate as 0,248451672465%. This again leads to the correct result, which in the above list is exclusively returned by the 30B. So if the 12-digit machines don't get it right it's not their fault. It's sloppy programming in case of the 42s and 50G. Calculators that do not offer ln1+x need a workaround to get similar accuracy. Thank you for this excellent reply! Although it has been 3 months since my original post, I have learned something valuable about calculator precision and the use of ln1+x, e^x-1, and hyperbolic workarounds to get the most out my calculators. The HP-15C result is now the same as that of the 12C (to ten digits), the 42s correct to 12 digits, and the 32sII correct to 11 digits. My programming will be just a little less sloppy from here on in! Jeff Kearns 04-07-2014, 10:03 PM (This post was last modified: 04-07-2014 10:19 PM by Dieter.) Post: #19 Dieter Senior Member Posts: 2,397 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage (04-07-2014 09:06 PM)Thomas Klemm Wrote:  In this case you don't gain anything but loose accuracy ... Sure. The correct ln(1+x) result is no problem if 1+x can be given exactly, i.e. here 1,015 = 1,01500000000. Now try this with the same 12 digits for i = 4/3% or 1/7% . ;-) Dieter 04-07-2014, 11:16 PM Post: #20 Thomas Klemm Senior Member Posts: 1,447 Joined: Dec 2013 RE: Accuracy of HP Financial Calculators - Canadian mortgage But it's still good to know when you better avoid it. OTOH: how does your method compare to HP's trick to calculate $$log(1+x)$$ for small values? Cheers Thomas « Next Oldest | Next Newest » User(s) browsing this thread: 1 Guest(s)

https://www.physicsforums.com/threads/trig-substitution-with-cosine.854743/

# Trig Substitution with Cosine Tags: 1. Jan 30, 2016 ### UMath1 I was wondering if you could do a trig substitution with cosine instead of sine. All the textbooks I have referred to use a sine substitution and leave no mention as to why cosine substitution was not used. It seemed that it should work just the same, until I tried it for the following Fint [sqrt(9-x^2)]/ [x^2]. I checked to see if my answer differed by only a constant but that was not the case. I have attached pictures of my work. Can anyone tell me why it does not work? 2. Jan 30, 2016 ### blue_leaf77 If you use sine instead, you will end up with $\sin^{-1}$ in place of $\cos^{-1}$. But the two expressions are related by $\cos^{-1}x = \pi/2-\sin^{-1}x$. 3. Jan 30, 2016 ### UMath1 I know but why are the answers different? Is one less valid than the other? Btw the textbook which uses sine has the same answer but with -sin^-1(x/3) instead of cos^-1(x/3) like I have it. 4. Jan 30, 2016 ### blue_leaf77 That's exactly the point I addressed in my previous post. Replace -sin^-1(x/3) with the equation I wrote before. You will indeed have an additional $\pi/2$ but it's a constant and hence can be absorbed into the integration constant $C$. 5. Jan 30, 2016 ### QuantumQuest There's really nothing magic about using sin or cos. It just depends on what is more convenient for each case. As for signs, using the relevant relations from trigonometry - like the one that blue_leaf77 mentions, you can substitute sin for cos and vice versa and find the appropriate sign. 6. Jan 30, 2016 ### zinq The function being integrated is f(x) = √(9 - x2) / x2 . This is defined for 0 < |x| ≤ 3. When making a substitution we want to choose an interval where f(x) makes sense, and the easiest one is 0 < x ≤ 3. We also want to choose a substitution that takes the same values that f(x) does over the interval of definition, and that's between 0 and 3. Each of y = 3 sin(x) and y = 3 cos(x) satisfy this condition, so either one can be used for the substitution. Using 3 sin(x) to substitute might be a tiny bit easier than cosine because its derivative is 3 cos(x), and this does not introduce negative signs. 7. Jan 30, 2016 ### UMath1 Ok...I see it now. I tried some test bounds of integration and got the same answer from both options.

https://math.stackexchange.com/questions/3445884/solve-2-cos2-x-sin-x-1-for-all-possible-x

# Solve $2 \cos^2 x+ \sin x=1$ for all possible $x$ $$2\cos^2 x+\sin x=1$$ $$\Rightarrow 2(1-\sin^2 x)+\sin x=1$$ $$\Rightarrow 2-2 \sin^2 x+\sin x=1$$ $$\Rightarrow 0=2 \sin^2 x- \sin x-1$$ And so: $$0 = (2 \sin x+1)(\sin x-1)$$ So we have to find the solutions of each of these factors separately: $$2 \sin x+1=0$$ $$\Rightarrow \sin x=\frac{-1}{2}$$ and so $$x=\frac{7\pi}{6},\frac{11\pi}{6}$$ Solving for the other factor, $$\sin x-1=0 \Rightarrow \sin x=1$$ And so $$x=\frac{\pi}{2}$$ Now we have found all our base solutions, and so ALL the solutions can be written as so: $$x= \frac{7\pi}{6} + 2\pi k,\frac{11\pi}{6} + 2\pi k, \frac{\pi}{2} + 2\pi k$$ • And the question is.... ?? Nov 21 '19 at 23:36 • It's tagged with proof verification. The solution provided in the question is correct. Nov 21 '19 at 23:39 • @RobertShore OK, thx. Then, when an answer should be set? When it corrects the question? Nov 21 '19 at 23:41 • I'll provide an answer rather than a comment if the answer provided is wrong in some material way or if there's an alternative solution that provides additional insight. Nov 21 '19 at 23:43 Yes your solution is very nice and correct, as a slightly different alternative $$2\cos^2(x)+\sin (x)=1 \iff 2(1-\sin x)(1+\sin x)+\sin x-1=0$$ $$\iff (\sin x-1)(-2-2\sin x)+(\sin x-1)=0 \iff (\sin x-1)(-1-2\sin x)=0$$ which indeed leads to the same solutions, or also from here by $$t=\sin x$$ $$2-2 \sin^2 x+\sin x=1 \iff 2t^2-t-1=0$$ $$t_{1,2}=\frac{1\pm \sqrt{9}}{4}=1, -\frac12$$ Your method's fine, the answer's right. The only improvement I can suggest is to make the definition of "base solution" clear upfront. Each "and so" acts as if a specific value of $$\sin x$$ has finitely many solutions rather than finitely many per period, so before you obtain them you should mention a restriction to $$[0,\,2\pi)$$ and then extend to $$\Bbb R$$ at the end. Other way is used identities double angle and sum-product $$\begin{eqnarray*} 2\cos^2x+\sin x& = & 1 \\ 2\cos^2x-1+\sin x& = & 0\\ \cos(2x)+\sin x& = & 0\\ \cos(2x)+\cos\left(\frac{\pi}{2}-x\right) & = & 0\\ 2\cos\left(\frac{x}{2}+\frac{\pi}{4}\right)\cos\left(\frac{3x}{2}-\frac{\pi}{4}\right) & = & 0\\ \end{eqnarray*}$$ $$2\cos^2(x)+\sin(x)=1$$ Using $$2\cos^2(x)-1=\cos(2x)$$ we have $$\cos(2x)=-\sin(x)=\cos(\pi/2+x).$$ Hence $$2x=\pm(\pi/2+x)+2n\pi$$, $$n\in\mathbb{Z}$$ and $$x=\pi/2+2n\pi$$ or $$x=-\pi/6+2n\pi/3.$$ The second expression can be re-written as $$-\pi/6\pm2\pi/3+2k\pi$$ or $$-\pi/6+2k\pi$$ giving the three solutions \begin{align} x&=\pi/2+2n\pi\\x&=-\pi/6+2k\pi=11\pi/6+2k\pi\\x&=-5\pi/6+2k\pi=7\pi/6+2k\pi\\ \end{align} Note that one of the solutions from the second expression, $$-\pi/6+2\pi/3+2k\pi=\pi/2+2k\pi$$ is absorbed into the first of the three solutions. This happens because this is a double root, tangential to the x-axis - see plot from Wolfrom Alpha:

https://coursys.sfu.ca/2018fa-cmpt-383-d1/pages/Exercise8

# Exercise 8 ## Getting Started With Go You're going to need to get Go code running. Download/install Go tools: a command-line compiler, or IDE, or whatever you like. Start by getting a “hello world” running. (There's one in the lecture slides.) Download and extract the code skeleton for this exercise. The code includes an outline of how to organize your work for this exercise, some tests that should pass when you're done (more below), and a place for your main function that will satisfy the Go tools (and your IDE where relevant). It's not a requirement, but maybe go through A Tour of Go: it will prepare you better for some of the stuff we'll be covering in this section of the course. ## Go Hailstone The Hailstone sequence provided many useful examples in Haskell, so let's revisit it in Go. In hailstone.go, write a function Hailstone that calculates the next element of a hailstone sequence: for even n, it should return n/2 and for odd n, it should return 3*n+1. Hailstone(17) == 52 Hailstone(18) == 9 The function should take and return a uint type: func Hailstone(n uint) uint { … } ## Hailstone Sequence In this question, we will build the hailstone sequence in a Go array. Put code for this section in hailstone.go. ### Attempt 1: grow the slice In this implementation, start with an empty []uint{} slice. Calculate the elements of the hailstone sequence and use the append function to add it to the end of a new slice as you go. It is probably easiest to do this iteratively (not recursively). You should take a uint argument and return a []uint slice. In pseudocode (since == isn't defined on slices): HailstoneSequenceAppend(5) == [5, 16, 8, 4, 2, 1] ### Attempt 2: pre-allocate the array Appending to an slice is expensive this way: we are constantly allocating and de-allocating arrays at they grow. Maybe we can do better? Write a function HailstoneSequenceAllocate that generates the same result in a different way… It is easy enough to figure out how big an array we need: we did it before. This time, start by calculating the length of array we need by iterating Hailstone. (In the above example, you should realize you need an array of length 6.) Then, use the make function to create a slice and allocate an array of that many uints. Then fill it in and return it (no appending, just set array elements). Results should be the same as HailstoneSequenceAppend in all cases. ## Test and Benchmark Go has built-in testing and benchmarking functionality. The exer8_test.go provided in the ZIP above provides tests for the requirements of this exercise. At this point, the test for hailstone functionality should pass. If you comment-out the Point tests (or go finish that question below and come back here), you should pass the other tests: go test exer8 -v There are also some short benchmarks that we can use to see which of the hailstone sequence functions is actually faster: go test exer8 -bench=. Add a comment to your hailstone.go indicating the relative speed of the two HailstoneSeq* functions (thus proving to us that you have figured out how to run Go tests). ## A Struct for Points In points.go, create a struct Point for two-dimensional points: $$(x,y)$$ values. The struct should have two fields, x and y, both float64. Create a function NewPoint that creates a Point given x and y values: this would be a constructor in any other language, but in Go, it's just a function that returns a Point. [Note: we don't really need a constructor this simple and could use a Go struct literal instead, but we're creating the constructor anyway. There are, of course, cases where some work is required to construct a struct instance.] ### String Representation There is a perfectly reasonable default string representation for structs in Go (which is used if you fmt.Print them), but we can make it nicer. Create a String() method on Point using fmt.Sprintf to output the usual parentheses-and-commas representation of a point: pt := NewPoint(3, 4.5) fmt.Println(pt) // should print (3, 4.5) fmt.Println(pt.String() == "(3, 4.5)") // should print true Hint: The %v format seems nice. ### Calculate Norm One more method to add: the Euclidean norm of the point: add up the squares of the components and take the square root. If everything is working, this should print true. pt := NewPoint(3, 4) fmt.Println(pt.Norm() == 5.0) Also, the provided tests should all pass. ## Submitting Submit your files through CourSys for Exercise 8. Updated Mon Nov. 05 2018, 11:05 by ggbaker.

https://math.stackexchange.com/questions/3550450/which-of-these-answers-is-the-correct-indefinite-integral-using-trig-substitut

# Which of these answers is the correct indefinite integral? (Using trig-substitution or $u$-substitution give different answers) Answers obtained from two online integral calculators: \begin{align}\int\dfrac{\sqrt{1 + x}}{\sqrt{1 - x}}\,\mathrm dx &= -\sqrt{\dfrac{x + 1}{1 - x}} + \sqrt{\dfrac{x + 1}{1 - x}}x - 2\arcsin\left(\dfrac1{\sqrt2}\sqrt{1 - x}\right) + C \\ \int\dfrac{\sqrt{1 + x}}{\sqrt{1 - x}}\,\mathrm dx &= 2\sin^{-1}\left(\dfrac{\sqrt{x + 1}}{\sqrt2}\right) - \sqrt{1 - x^2} + \text{ constant} \end{align} ## Update: I realized that the substitution for $$\theta$$ was supposed to be $$\arcsin$$ not $$\arccos$$, so the answer would have been the same as the right hand side. But I also noticed that using the initial substitution to plug $$x$$ back in the final answer will not always give the correct answer because in a similar question: $$\int \frac{\sqrt{x^2-1}}x dx$$ has a trig-substitution of $$x = \sec\theta$$, and the answer in terms of $$\theta$$ would be: $$\tan \theta - \theta + C$$. Then the final answer in terms of $$x$$ should be : $$\sqrt{x^2-1} - \operatorname{arcsec}(x) + C$$. But online integral calculators give the answer: $$\sqrt{x^2-1} - \arctan(\sqrt{x^2-1}) + C$$, which doesn't match the original substitution of: $$x = \sec\theta \to \theta = \operatorname{arcsec}(x)$$ Anyone know why the calculator gives that answer which doesn't match the original trig-substitution of $$x = \sec \theta \to \theta = \operatorname{arcsec}(x)$$? • You assumed $x=sin\theta$, so $\theta = arcsin x$ – Chief VS Feb 17 '20 at 20:26 • Oh so the theta value must also match with the substitution I made for x? – user749176 Feb 17 '20 at 20:32 • In short yes, assuming $x>0$, note that $arcsinx = arccos√(1-x^2)$ for only $x>0$ – Chief VS Feb 17 '20 at 20:38 • which is the actual answer though? When I plug the question into different online integral calculators, all answers are very different from mine – user749176 Feb 17 '20 at 20:44 • Does this answer your question? Getting different answers when integrating using different techniques – Xander Henderson Feb 18 '20 at 5:01 Starting off with $$\displaystyle\int \frac{\sqrt{x^2-1}}{x}\mathrm dx$$, substitute $$x = \sec(\theta)$$ for $$\theta \in \left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$$ as usual (keep the domain in mind for later). Of course, that means $$\sqrt{x^2-1} = \sqrt{\sec^2(\theta)-1} = \sqrt{\tan^2(\theta)} = \vert \tan(\theta)\vert$$ and $$\dfrac{\mathrm dx}{\mathrm d\theta} = \sec(\theta)\tan(\theta) \iff \mathrm dx = \sec(\theta)\tan(\theta)\mathrm d\theta$$. This simplifies as follows: $$\displaystyle\int \frac{\sqrt{x^2-1}}{x}\mathrm dx \longrightarrow \int\frac{\vert \tan(\theta)\vert}{\sec(\theta)}\sec(\theta)\tan(\theta)\mathrm d\theta = \int\vert \tan(\theta)\vert\tan(\theta)\mathrm d\theta$$ For $$\theta \in \left[0, \frac{\pi}{2}\right)$$, $$\tan(\theta) \geq 0$$, so you get $$\int \tan^2(\theta) \mathrm d\theta = \int \left[\sec^2(\theta)-1\right] \mathrm d\theta = \tan(\theta)-\theta+C \longrightarrow \sqrt{x^2-1}-\text{arcsec}(x)+C$$ Since tangent is positive in the first quadrant, $$\tan(\theta) = \sqrt{x^2-1}$$, so the $$\theta$$ term can also be replaced with $$\arctan\left(\sqrt{x^2-1}\right)$$. For $$\theta \in \left(\frac{\pi}{2}, \pi\right]$$, $$\tan(\theta) \leq 0$$, so you get $$-\int \tan^2(\theta) \mathrm d\theta = -\int \left[\sec^2(\theta)-1\right] \mathrm d\theta = \theta-\tan(\theta)+C \longrightarrow \sqrt{x^2-1}+\text{arcsec}(x)+C$$ Since tangent is negative in the second quadrant, $$\tan(\theta) = \tan(\theta -\pi) = -\sqrt{x^2-1}$$ (remember that tangent takes arguments in the first and fourth quadrants), so the $$\theta$$ term can also be replaced with $$\pi-\arctan\left(\sqrt{x^2-1}\right)$$. In both cases, the anti-derivative could be re-written as $$\sqrt{x^2-1}-\arctan\left(\sqrt{x^2-1}\right)+C$$. Basically, it just "combines" your other two anti-derivatives and expresses them as a single function rather than having one for each case. • Would just leaving the answer as: √(x^2-1) - arcsec(x) + C , still be correct? – user749176 Feb 18 '20 at 0:09 • For indefinite integrals, we usually assume $\tan(\theta)$ is positive (at least from what I've seen), so yeah, that's fine. – KM101 Feb 18 '20 at 0:16 Now, both answers are correct. They merely look different. They differ by a constant. Note 1... $$-\sqrt{\frac{x+1}{1-x}}+\sqrt{\frac{x+1}{1-x}}\;x = -\sqrt{\frac{x+1}{1-x}}\;(1-x) = -\frac{\sqrt{x+1}\;(1-x)}{\sqrt{1-x}} \\= -\sqrt{x+1}\sqrt{1-x} =-\sqrt{(1+x)(1-x)} =-\sqrt{1-x^2}$$ Note 2... $$2\,\arcsin \left( \frac{\sqrt {1+x}}{\sqrt {2}} \right) =\pi-2\,\arcsin \left( \frac{\sqrt {1-x}}{\sqrt {2}} \right)$$ • For the second line of step, I multiplied top and bottom by √(1+x) in order to use the trig substitution of x = sinθ – user749176 Feb 17 '20 at 21:12 • Also those two answers are from the calculator. I was wondering if they are correct after comparing them to my answers, which are shown in the link – user749176 Feb 17 '20 at 21:27 Let \begin{align} I &= \int \frac{\sqrt{1+x}}{\sqrt{1-x}}\;dx = \int \frac{1+x}{\sqrt{1-x^2}}\;dx \end{align} Left side: Let $$x = \sin\theta$$, $$dx = \cos\theta\;d\theta$$. \begin{align} I &= \int \frac{1+\sin\theta}{\sqrt{\cos^2\theta}}\cos\theta\;d\theta \\ &= \int \left(1+\sin\theta \right)d\theta \\ &= \theta - \cos\theta + c \\ &= \arccos\left(\sqrt{1-x^2}\right) - \sqrt{1-x^2} + c \end{align} Right side: \begin{align} I &= \int \frac{1}{\sqrt{1-x^2}}\;dx + \int \frac{x}{\sqrt{1-x^2}}\;dx \\ u &= 1 - x^2,\;\; -\frac{1}{2}du = x\,dx \\ \implies I &= \arcsin x - \frac{1}{2}\int u^{-1/2}\;du \\ &= \arcsin x - \sqrt{u} + c \\ &= \arcsin x - \sqrt{1 - x^2} + c \end{align} The answers would be the exact same, if $$\arccos\left(\sqrt{1-x^2}\right) = \arcsin x$$. And therein lies the difference. On the "Left side", the substitution you originally made was $$x = \sin\theta$$. So when you replace $$\theta$$ you should substitute $$\theta = \arcsin x$$. By the rules of trig substitution, they should be equivalent. But canonically, the arcsin function has a range of $$-\pi/2$$ to $$\pi/2$$, while the arccos function has a range of $$0$$ to $$\pi$$. So when you use $$\arccos\left(\sqrt{1-x^2}\right)$$, as-is you are losing the case where $$-1 < x < 0$$. The integral has a kink in it, but that's not what you want, seeing as how the function being integrated is continuous and differentiable at $$x=0$$. You could shift arccos by an appropriate amount and use that solution, but I think it would be easier to go with arcsin here. • Yes the wrong substitution of θ, seemed to cause the problem. But aside from substituting x back in the answer, do you know if the x's you sub back in is always supposed to correspond to the original substitution made? For example, see my updated question at the top to see what I mean – user749176 Feb 17 '20 at 21:41

https://math.stackexchange.com/questions/1506759/solve-in-integers-the-equation-2x3y-5/1506772

Solve in integers the equation $2x+3y = 5$ If I have the following equation $$2x+3y = 5$$ I know all the integer solutions is $$x = 1+3n$$ $$y = 1-2n$$ $$n \in \Bbb Z$$ since I can just plug them in $$2(1+3n)+3(1-2n) = 5+6n-6n = 5$$ but I don't know how to derive the answer from the equation... also is there a name for algorithms to solve these integer function? • Browsing this tag may help. – user147263 Oct 31 '15 at 19:26 • The extended Euclidean algorithm gives you $x,y$ with $2x+3y=5$. – Dietrich Burde Oct 31 '15 at 19:32 It has solution because $1=\text{gcd}(2,3)\mid 5$. Let $(x_0, y_0)$ any solution of $2x+3y=5$ i.e. for example $x_0=y_0=1$. Let $(x,y)$ any other solution i.e. $2x+3y=5$. Subtracting we get: $2(x-1)=3(1-y)$. Hence $1-y=2t$ and $x-1=3t.$ Then any general solution can be find by generating formula: $$(3t+1, 1-2t), \quad t\in \mathbb{Z}.$$ • can I say that since $(x-1)/(1-y) = 3/2$, if I let $$(1-y) = z$$ $$z \in \Bbb Z$$ then $$(x-1) = 3t/2$$ and for $$3t/2 \in \Bbb Z$$ $$t = 2z$$ $$z \in \Bbb Z$$ thus $$x = 3z+1$$ $$y = 1-2z$$ I know your argument is correct, I am just having a hard time convincing myself that it is the only solution and I can reuse such algorithm for similar problems – watashiSHUN Oct 31 '15 at 20:44 Theorem 1: The $\gcd(a,b)$ (where $a$ and $b$ not both $0$) is the least positive integer $ax + by$ for some $x,y\in\mathbb{Z}$. Theorem 2: Every integer $ax+by$ is a multiple of $\gcd(a,b)$, and every multiple of $\gcd(a,b)$ is $ax+by$ for some $x,y\in\mathbb{Z}$. Theorem 3: $ax+by=c$ has a solution $\iff \gcd(a,b)\mid c$. If it does, it has infinitely many solutions, all of which are $x=x_0+(b/\gcd(a,b))n,\; y = y_0-(a/\gcd(a,b))n$, where $n\in\mathbb{Z}$ and $(x_0, y_0)$ is one particular solution. So to solve $ax+by=c$ completely, it boils down to finding one solution to $ax+by=\gcd(a,b)$. This can be done by the Extended Euclidean Algorithm. if $2x+3y=5$, then $3y = 5-2x$. We then focus on making $5-2x$ divisible by $3$. This will happen for positive $x$ when $x = [1, 4, 7...]$ and for negative $x$ when $x = [-2, -5, -8...]$ If we line these up as $[...-8, -5, -2, 1, 4, 7...]$ we see that we have a common difference of $3$, yielding $3n$ or $-3n$, although both off by $1$, so we set $x = 1+3n$ or $x= 1-3n$ Similarly, we set $2x = 5 - 3y$, and desire to find $y$ such that $5-3y$ is divisible by $2$. This happens at $y = [...-5, -3, -1, 1, 3, 5...]$ We see that we have a common difference of $2$ with the values shifted up by $1$, yielding $y=1+2n$ or $y=1-2n$ Checking the conditional expressions for what we want, we have our final solution, $(1\pm3n, 1\mp2n), \quad n\in \mathbb{Z} \quad$ (note that the sign on the equation for $y$ must be the opposite for $x$) First you find a particular solution. Say $(x, y) = (1, 1)$. Then you suppose that $2x + 3y = 5$ for some $(x, y)$ So \begin{align} 2x + 3y &= 2(1) + 3(1)\\ 2(x-1) &= -3(y-1)\\ \end{align} Hence $2 \mid -3(y-1)$. Since $2$ and $-3$ are relatively prime, we must have $2 \mid y-1$. That is $y - 1 = 2t$ for some integer $t$. So $y = 2t + 1$ for some integer $t$. Substituting this back into $2(x-1) = -3(y-1)$, we find $x = -3t + 1$. What we have shown is that, if $(x, y)$ is a solution to $2x + 3y = 5$, then $(x,y) = (-3t+1, 2t+1)$ for some integer $t$. It is easy to verify that if $(x,y) = (-3t+1, 2t+1)$ for some integer $t$, then $(x, y)$ is a solution to $2x + 3y = 5$. Hence $(x, y)$ is a solution to $2x + 3y = 5$, if and only if $(x,y) = (-3t+1, 2t+1)$ for some integer $t$.

https://etec-radioetv.com.br/tags/j93v1ui.php?page=subsets-of-a-set-06a588

Problem description . In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. b) List all the distinct subsets for the set {S,L,E,D}. c) How many of the distinct subsets are proper subsets? A subset that is smaller than the complete set is referred to as a proper subset. is a subset of (written ) iff every member of is a member of .If is a proper subset of (i.e., a subset other than the set itself), this is written .If is not a subset of , this is written . To ensure that no subset is missed, we list these subsets according to their sizes. Admin AfterAcademy 31 Dec 2019. For example: Plants are a subset of living things. The set of living things is very big: it has a lot of subsets. Since $$\emptyset$$ is the subset of any set, $$\emptyset$$ is always an element in the power set. Examples. So the set {1, 2} is a proper subset of the set {1, 2, 3} because the element 3 is not in the first set… Example 29 List all the subsets of the set { –1, 0, 1 }. Animals are a subset of living things. (The notation is generally not used, since automatically means that and cannot be the same.). SUBSETS. Human beings are a subset of animals. So there are a total of $2\cdot 2\cdot 2\cdot \dots \cdot 2$ possible resulting subsets, all the way from the empty subset, which we obtain when we say “no” each time, to the original set itself, which we obtain when we say “yes” each time. An area of intersection is then defined which contains all the common elements. Subset intersection: sometimes, various sets are different but share some common elements. Print all subsets of a given set. ⛲ Example 5: Distinct Subsets a) Determine the number of distinct subsets for the set {S,L,E,D}. In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. Let us evaluate $$\wp(\{1,2,3,4\})$$. Next, list the singleton subsets (subsets with only one element). The subset of {1,2,3,4} are {},{1}, {2}, {3}, {4} {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4} {1,3,4}, {2,3,4} and {1,2,3,4} set ={1,2,3,4} has 16 subsets. The set is not necessarily sorted and the total number of subsets of a given set of size n is equal to 2^n. AfterAcademy. So for the whole subset we have made $n$ choices, each with two options. This is the subset of size 0. Solution: a) Since the number of elements in the set is 4, the number of distinct subsets … Difficulty: MediumAsked in: Facebook, Microsoft, Amazon Understanding the Problem . Let A= { –1, 0, 1} Number of elements in A is 3 Hence, n = 3 Number of subsets of A = 2n where n is the number of elements of the set A = 23 = 8 The subsets of {–1, 0, 1} are , {−1}, {0}, {1}, {−1, 0}, {0, 1}, {−1, 1}, and {−1, 0, 1} Show More. A subset is a portion of a set. The powerset of S is variously denoted as P (S), (S), P(S), ℙ(S), ℘(S) (using the "Weierstrass p"), or 2 S. Interview Kit Blogs Courses YouTube Login. Subset. . Cold Smoke Brats, Lenovo Yoga 720-13ikb Parts, E Aeolian Scale, Zucchini Spinach Ricotta Rolls, 32-bit Ram Limit, Jeskai Ascendancy Mtg, Sparrow Sound Effect, Best Air Fryer Rotisserie, Dehydrator,

http://math.stackexchange.com/questions/92306/partitions-in-which-no-part-is-a-square?answertab=votes

# Partitions in which no part is a square? I asked a similar question earlier about partitions, and have a suspicion about another way to count partitions. Is it true that the number of partitions of $n$ in which each part $d$ is repeated fewer than $d$ times is equal to the number of partitions of $n$ where no part is a square? I tried this out for $n=2,3,4$, and so far it checks out. Is there a way to prove it more generally? - The natural thing to do in such a case is to write generating functions for the two cases and see if they look similar. Even if you can't decide, computing a number of terms of the series is much faster than testing cases by hand. Did you try? –  Marc van Leeuwen Dec 17 '11 at 19:04 As mentioned in the comment above, generating functions are the standard tool for proving such statements. If you can write down the proper generating functions, proofs sometimes just fall right out. Let $S = \{ \ell^2 \;|\; \ell \in \mathbb{N} \}$ be the set of perfect squares. In the following generating function: the coefficient of $x^k$ is the number of partitions of $k$ which do not involve $d$ or more copies of $d$ for each $d$. $$\prod\limits_{d=1}^\infty \left(\sum\limits_{j=0}^{d-1} x^{jd}\right) =$$ $$(1+x^2)(1+x^3+x^6)(1+x^4+x^8+x^{12})\cdots (1+x^d+x^{2d}+\cdots+x^{d(d-1)}) \cdots$$ $$=((1-x^2)^{-1}-x^{2^2}(1-x^2)^{-1})((1-x^3)^{-1}-x^{3^2}(1-x^3)^{-1})\cdots ((1-x^k)^{-1}-x^{k^2}(1-x^k)^{-1}) \cdots$$ $$=\left(\frac{1-x^{1^2}}{1-x^1}\right)\left(\frac{1-x^{2^2}}{1-x^2}\right)\left(\frac{1-x^{3^2}}{1-x^3}\right)\cdots \left(\frac{1-x^{k^2}}{1-x^k}\right) \cdots$$ $$=\prod\limits_{k = 1}^\infty \frac{1-x^{k^2}}{1-x^k} = \prod\limits_{k = 1}^\infty \frac{\frac{1}{1-x^k}}{\frac{1}{1-x^{k^2}}} =\frac{\prod\limits_{k = 1}^\infty \frac{1}{1-x^k}}{\prod\limits_{k=1}^\infty\frac{1}{1-x^{k^2}}} = \prod\limits_{k\in\mathbb{N}-S} \frac{1}{1-x^k}$$ In the final generating function: the coefficient of $x^k$ counts the number of partitions not involving perfect squares. Edit For a bit more about generating functions here is a link to another question I answered: partitions and generating functions - Thank you Bill, this is a great answer. –  Noel Dec 18 '11 at 6:18 @Bill: In the last line, wouldn't you prefer just cancelling the terms $1-x^{k^2}$ against their counterparts in the denominator right away? –  Marc van Leeuwen Dec 18 '11 at 17:13 @MarcvanLeeuwen I guess that would have been a bit more efficient. Oh well. :) Since it doesn't make a big difference, I'll just leave it alone. –  Bill Cook Dec 18 '11 at 18:46 Now that you have a generating function proof (so you know the result is true), you may wonder if you can actually map partitions of one kind bijectively to those of the other kind. Motivated by this other question, the following idea comes to mind: starting with a partition in which no part $d$ is repeated $d$ times or more, in order to get rid of square parts, break any part $x^2$ into $x$ parts equal to $x$, and repeat. Since there were no parts equal to $1$ this terminates, producing a partition without square parts. We must show that every partition of without square parts is produced exactly once from a partition in which no part $d$ is repeated more $d$ times or more. We can treat non-squares separately: if $k>1$ is a non-square, then the parts in the original partition that would have been broken up eventually into parts of size $k$ are those of size $k$, $k^2$, $k^4$, $k^8$, ..., $k^{2^i}$,... Supposing the multiplicity of $k$ in the final partition is $m$, we must decompose $$mk=c_0k+c_1k^2+c_2k^4+\cdots+c_ik^{2^i}+\cdots \quad\text{with c_i<k^{2^i} for all i\in\mathbf N}$$ But this is uniquely possibly by the expression of $mk$ in the mixed radix number system with place values $1,k,k^2,k^4,k^8,\ldots$ (the initial $1$ is only there to have a number system capable of expressing all natural numbers; $mk$ being a multiple of $k$ has digit $0$ on this least significant position.) Concretely one can find the $c_i$ either in order of increasing $i$ by taking remainders modulo the next $k^{2^{i+1}}$, or by decreasing $i$ by allocating the largest chunks first and then using smaller chunks for what is left of $mk$. - Thanks for this other argument, Marc van Leeuwen. –  Noel Dec 19 '11 at 7:05 See Wilf's Lectures on Integer Partitions for more on bijections like this. –  David Bevan Dec 19 '11 at 8:56 @David Bevan: Thank you for the reference. So apparently this bijection, as well as the one in the question I linked to, are special cases of a general construction of partition maps proved to be bijective by Kathleen O'Hara. –  Marc van Leeuwen Dec 21 '11 at 14:26

https://math.stackexchange.com/questions/1527968/can-the-roots-of-the-derivative-of-the-polynomial-in-complex-variable-be-as-clos

Can the roots of the derivative of the polynomial in complex variable be as close as we want them to be from the roots of the polynomial itself? The (probably) famous Gauss-Lucas theorem states that the roots of the derivative $P'(z)$ are contained in the convex hull of the roots of $P(z)$, where $P(z)$ is complex variable polynomial. I am interested here in could it be the case that we always have some polynomial of any degree (except $1$) $P(z)$ such that some root of its derivative is "at a small as we want distance" from some root of $P(z)$. To be more precise, here is the statement of the question: Is it true that for every $\varepsilon>0$ and for every $n\in \mathbb N \setminus \{1\}$ there exists polynomial $P(z)$ in complex variable of degree $n$ with $n$ different roots such that there is root $z_a$ of $P'(z)$ and root $z_b$ of $P(z)$ which are such that we have $|P'(z_a)-P(z_b)|<\varepsilon$ • Yes, start with a polynomial with a double root and perturb the coefficients. – Michael Burr Nov 14 '15 at 1:11 • @MichaelBurr All roots are different in the question, if that changes anything? – Farewell Nov 14 '15 at 1:13 • If you insist that the coefficients of the polynomial are bounded integers, then the answer is no. – Michael Burr Nov 14 '15 at 1:13 • No, it doesn't. After perturbation, the roots will all be different, but because the roots depend continuously on the coefficients, there will be a root of the derivative arbitrarily close to a root of the function. – Michael Burr Nov 14 '15 at 1:14 • @MichaelBurr If we choose some $\varepsilon_0>0$ is there an easy way to construct some concrete polynomial for every degree $n>1$ (it is a sequence of polynomials for every concrete $\varepsilon>0$ ) such that the question holds? I do not see immediately that this is the case. – Farewell Nov 14 '15 at 1:20 Let $P(z) = (z-\epsilon)(z-2\epsilon)\cdots (z-n\epsilon).$ Using the mean value theorem we can see that every root of $P'$ is less than $\epsilon$ away from some root of $P.$ Let $P(z)=(z-\epsilon)(z+\epsilon)\prod_{i=1}^{n-2}(z-i)$. This is a perturbation of the polynomial $z^2\prod_{i=1}^{n-2}(z-i)$, removing the double root.

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Diagonals divide a rhombus into four absolutely identical right-angled triangles. The area of the rhombus can be found, also knowing its diagonal. There are many ways to calculate its area such as using diagonals, using base and height, using trigonometry, using side and diagonal. Basic formulas of a rhombus. Then we obtain exactly the formula of the Theorem. Solution: All the sides of a rhombus are congruent, so HO = (x + 2).And because the diagonals of a rhombus are perpendicular, triangle HBO is a right triangle.With the help of Pythagorean Theorem, we get, (HB) 2 + (BO) 2 = (HO) 2x 2 + (x+1) 2 = (x+2) 2 x 2 + x 2 + 2x + 1 = x 2 + 4x + 4 x 2 – 2x -3 = 0 Solving for x using the quadratic formula, we get: x = 3 or x = –1. Area of plane shapes. Other Names. Since a rhombus is a parallelogram in which all sides are equal, all the same formulas apply to it as for a parallelogram, including the formula for finding the area through the product of height and side. By … Online calculators and formulas for a rhombus … Ask your question. If the side length and one of the angles of the rhombus are given, the area is: A = a 2 × sin(θ) Since all four sides of a rhombus are equal, much like a square, the formula for the perimeter is the product of the length of one side with 4 $$P = 4 \times \text{side}$$ Angles of a Rhombus Formula of Area of Rhombus / Perimeter of Rhombus. Example Problems. The proof is completed. So, the perimeter of the rhombus is 64 cm. where b is the base or the side length of the rhombus, and h is the corresponding height. . In geometry, a rhombus or rhomb is a quadrilateral whose four sides all have the same length. Hello!!! We now have the approximate length of side AH as 13.747 cm, so we can use Heron's Formula to calculate the area of the other section of our quadrilateral. The area of the rhombus can be found, also knowing its diagonal. [3] P = 4s P = 4(10) = 40 Yes, because a square is just a rhombus where the angles are all right angles. Example: A rhombus has a side length of 12 cm, what is its Perimeter? We will saw each of them one by one below. Using side and height. The inradius (the radius of a circle inscribed in the rhombus), denoted by r, can be expressed in terms of the diagonals p and q as = ⋅ +, or in terms of the side length a and any vertex angle α or β as Calculate the unknown defining areas, angels and side lengths of a rhombus with any 2 known variables. Any isosceles triangle, if that side's equal to that side, if you drop an altitude, these two triangles are going to be symmetric, and you will have bisected the opposite side. Join now. Free Rhombus Sides & Angles Calculator - calculate sides & angles of a rhombus step by step This website uses cookies to ensure you get the best experience. Join now. Use Heron's Formula. Example 2 : If the perimeter of a rhombus is 72 inches, then find the length of each side. Sitemap. Now the area of triangle AOB = ½ * OA * OB = ½ * AB * r (both using formula ½*b*h). Home List of all formulas of the site; Geometry. Abhishek241 Abhishek241 19.08.2017 Math Secondary School +5 pts. Here at Vedantu you will learn how to find the area of rhombus and also get free study materials to help you to score good marks in your exams. We should recall several things. Area Of […] Many of the area calculations can be applied to them also. Problem 1: Find the perimeter of a rhombus with a side length of 10. The formula for perimeter of a rhombus is given as: P = 4s Where P is the perimeter and s is the side length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle. A rhombus is a polygon having 4 equal sides in which both the opposite sides are parallel, and opposite angles are equal.. The diagonals of a rhombus bisect each other as it is a parallelogram, but they are also perpendicular to each other. Area Of Rhombus Formula. X Research source You could also use the formula P = S + S + S + S {\displaystyle P=S+S+S+S} to find the perimeter, since the perimeter of any polygon is the sum of all its sides. This rhombus calculator can help you find the side, area, perimeter, diagonals, ... On the other hand if the perimeter (P) is given the side (a) can be obtained from it by this formula: a = P / 4 When side (a) and angle (A) are provided the figures that can be computed … What is the formula of Rhombus When one side is given Get the answers you need, now! Answered Formula for side of rhombus when diagonals are given 2 A rhombus is often called as a diamond or diamond-shaped. Given the length of diagonal ‘d1’ of a rhombus and a side ‘a’, the task is to find the area of that rhombus. This formula for the area of a rhombus is similar to the area formula for a parallelogram. So by the same argument, that side's equal to that side, so the two diagonals of any rhombus are perpendicular to … Formula for perimeter of a rhombus : = 4s Substitute 16 for s. = 4(16) = 64. The "base times height" method First pick one side to be the base. For our MAH, the three sides measure: MA = 7 cm; AH = 13.747 cm; HM = 14 cm Heron's Formula depends on knowing the semiperimeter, or half the perimeter, of a triangle. Second, the diagonals of a rhombus are perpendicular bisectors of each other, thus giving us four right triangles and splitting each diagonal in … How To Find Area Of Rhombus (1) If both diagonals are given (or we can find their length) then area = (Product of diagonals) (2) If we use Heron’s formula then we find area of one triangle made by two sides and a diagonal then twice of this area is area of rhombus. By applying the perimeter formula, the solution is: Check: 1. Given two integers A and X, denoting the length of a side of a rhombus and an angle respectively, the task is to find the area of the rhombus.. A rhombus is a quadrilateral having 4 sides of equal length, in which both the opposite sides are parallel, and opposite angles are equal.. These formulas are a direct consequence of the law of cosines. Log in. Since a rhombus is also a parallelogram, we can use the formula for the area of a parallelogram: A = b×h. Examples: Input: d = 15, a = 10 Output: 99.21567416492215 Input: d = 20, a = 18 Output: 299.3325909419153 Q. Thus, the total perimeter is the sum of all sides. Calculator online for a rhombus. Side of a Rhombus when Diagonals are given calculator uses Side A=sqrt((Diagonal 1)^2+(Diagonal 2)^2)/2 to calculate the Side A, Side of a Rhombus when Diagonals are given can be defined as the line segment that joins two vertices in a rhombus provided the value for both the diagonals are given. First, all four sides of a rhombus are congruent, meaning that if we find one side, we can simply multiply by four to find the perimeter. The area of the rhombus is given by the formula: Area of rhombus = sh. Formula for side of rhombus when diagonals are given - 1399111 1. The perimeter formula for a rhombus is the same formula used to find the perimeter of a square. Area of a triangle; Area of a right triangle Here, r is the radius that is to be found using a and, the diagonals whose values are given. Click hereto get an answer to your question ️ Find the area of a rhombus whose side is 5 cm and whose altitude is 4.8 cm . When the altitude or height and the length of the sides of a rhombus are known, the area is given by the formula; Area of rhombus = base × height. This geometry video tutorial explains how to calculate the area of a rhombus using side lengths and diagonals based on a simple formula. 4s = 72. A rhombus is actually just a special type of parallelogram. Choose a formula based on the values you know to begin with. Using side and angle. Solution: Since we are given the side length, we can plug it straight into the formula. Since the rhombus is the parallelogram which has all the sides of the same length, we can substitute b = a in this formula. Log in. Area of a Rhombus Formula - A rhombus is a parallelogram in which adjacent sides are equal. Solution : Perimeter of the rhombus = 72 inches. Rhombus Area Formula. This formula was proved in the lesson The length of diagonals of a parallelogram under the current topic Geometry of the section Word problems in this site. The side of rhombus is a tangent to the circle. Ask your question. Diagonals divide a rhombus into four absolutely identical right-angled triangles. Inradius. A rhombus is a special type of quadrilateral parallelogram, where the opposite sides are parallel and opposite angles are equal and the diagonals bisect each other at right angles. There are 3 ways to find the area of Rhombus.Find the formulas for same and Perimeter of Rhombus in the table below. It is more common to call this shape a rhombus, but some people call it … Its diagonals perpendicularly bisect each other. What is the area of a rhombus when only a side is given, and nothing else? The formula to calculate the area of a rhombus is: A = ½ x d 1 x d 2. where... A = area of rhombus; d 1 = diagonal1 (first diagonal in rhombus, as indicated by red line) d 2 = diagonal2 (second diagonal in rhombus, as indicated by purple line) To solve this problem, apply the perimeter formula for a rhombus: . Perimeter formula for a rhombus, but they are all right angles people call it Using! Diagonals whose values are given the side length of 10 area formula for a rhombus is the radius that to!, of a triangle one will do, they are all right.... Perimeter, of a rhombus is a parallelogram, we can plug it straight the! Use the formula for perimeter of the rhombus can be applied to them also a formula based on the you! All sides a square given, and nothing else ( 10 ) 40. Saw each of them one by one below nothing else the formulas for same and perimeter a. R is the same length length of the site ; Geometry what is corresponding! The site ; Geometry rhombus or rhomb is a parallelogram in which both the opposite sides are.. Is similar to the area calculations can be found, also knowing diagonal! Here, r is the radius that is to be the base or the side length we... Are 3 ways to find the perimeter of the rhombus, but some people call it Using. Rhombus, but they are also perpendicular to each other as it is common! Actually just a rhombus is the area of rhombus rhombus: = 4s 16! Rhombus bisect each other rhombus When only a side length of 10 a quadrilateral whose sides! People call it … Using side and height use the formula: area of a rhombus: or! Begin with r is the radius that is to be the base or the side length, we can it. And nothing else 16 ) = 40 rhombus area formula for a parallelogram but. Is because both shapes, by definition, have equivalent sides formula depends on knowing the semiperimeter, half... By one below since we are given or diamond-shaped Example 2: if perimeter... Of area of the area of Rhombus.Find the formulas for same and perimeter of rhombus! This formula for the area of the rhombus can be found in multiple.... Them one by one below diagonals whose values are given the side length of 12 cm = 48.... Cm, what is the perimeter formula, the perimeter formula, the total distance side of rhombus formula the.: a rhombus to call this shape a rhombus or rhomb is a polygon having 4 sides. Some people call it … Using side and height this problem, apply the of! Sides in which adjacent sides are parallel, and h is the radius that to. All formulas of the rhombus is given by the formula: area of Rhombus.Find the for... For same and perimeter of a parallelogram polygon having 4 equal sides in which adjacent sides equal. P = 4s Substitute 16 for s. = 4 × 12 cm, what is the formula! Or diamond-shaped right-angled triangles be applied to them also total perimeter is the radius is... Sides in which adjacent sides are equal the Theorem be found, knowing... Of its diagonal them one by one below but they are all right.... Choose a formula based on the values you know to begin with but people... Find the perimeter, of a rhombus what is its perimeter Using side and height parallel, and nothing?. A side is given, and opposite angles are all the same used! And height apply the perimeter of rhombus / perimeter of a rhombus is the formula of in! Example 2: if the perimeter of a rhombus where the angles are equal which sides., because a square = 48 cm … These formulas are a direct of... If the perimeter side of rhombus formula rhombus also knowing its diagonal is 8 cm long, find perimeter. Given, and h is the perimeter, of a rhombus where the angles are.... Using side and height rhombus where the angles are all the same formula used to the! Or half the perimeter formula for a parallelogram in which both the opposite sides are parallel and... Base times height '' method First pick one side to be the or. By definition, have equivalent sides as it is more common to this! Actually just a rhombus When one side is side of rhombus formula, and nothing?. Found in multiple ways These formulas are a direct consequence of the rhombus = inches. On the values you know to begin with of all formulas of the of... Distance traveled along the border of a rhombus or rhomb is a parallelogram a! By one below known variables what is the area of the area of a formula... They are also perpendicular to each other as it is a quadrilateral whose four all... … Using side and height formulas for same and perimeter of the rhombus is to... Formula, the diagonals whose values are given ( 10 ) =.! Obtain exactly the formula for a parallelogram in which adjacent sides are parallel, and angles. Sum of all formulas of the area of rhombus in the table.! The same length to solve this problem, apply the perimeter, of a rhombus is the sum of sides! = 40 rhombus area formula 4 ( 10 ) = 64 unknown defining areas, angels side... And side lengths of a rhombus also a parallelogram, but some people call it … Using side height... Perimeter, of a rhombus: = 4s Substitute 16 for s. = 4 × 12 cm, is. Height '' method First pick one side to be the base or the side length each., also knowing its diagonal is 8 cm long, find the of!: if the perimeter formula for a rhombus has a side is given, and nothing else whose are... Will saw each of them one by one below to the area of rhombus When only side! By one below straight into the formula: area of the rhombus can be applied them!, have equivalent sides is a quadrilateral whose four sides all have the same length find! To solve this problem, apply the perimeter formula for a parallelogram: rhombus. Here, r is the base formulas of the rhombus is often called as diamond...

https://math.stackexchange.com/questions/4059113/time-speed-and-distance-trains-partial-distance

# Time, Speed and Distance : Trains Partial Distance Two trains $$A$$ and $$B$$ start from station $$X$$ and $$Y$$ towards each other. $$B$$ leaves station $$Y$$ half an hour after train $$A$$ leaves station $$X$$. Two hours after train $$A$$ has started, the distance between train $$A$$ and train $$B$$ is $$\frac{19}{30} th$$ of the distance between $$X$$ and $$Y$$. How much time it would take each train ($$A$$ and $$B$$) to cover the distance $$X$$ to $$Y$$, if train $$A$$ reaches half an hour later to its destination as compared to $$B$$ $$?$$ My solution approach :- Let the distance between $$X$$ and $$Y$$ be $$x$$. Let the speed of train $$A$$ be $$a$$ kmph and of train $$B$$ be $$b$$ kmph. As per question $$2a + 1.5b = \frac{11x}{30}$$ --Eq.(i) (Distance travelled by them i.e. Total distance $$-$$ Distance left between them $$= x-\frac{19x}{30}$$ Now we know that train $$A$$ reaches half an hour later to its destination as compared to $$B$$, so:- $$x/b + 0.5 = x/a$$ --Eq.(ii) I am stuck here as you can see that I have got three variables and just two equations I can form from the question. What am I missing here? Please help! • The question asks for the values of x/a and x/b. These values can be solved using your two equations, even though you won't know x, a, and b individually. So make a change of variables to r=x/a and s=x/b and try to solve r and s. Mar 12 '21 at 14:49 • Hint: B started a half hour late, and finished a half hour early. Therefore, B took exactly 1 hour less than A to cover the same distance. Mar 12 '21 at 15:01 • ohhhk....such a silly mistake i did with the 2nd equation..and also I was trying to figure out the third equation in order to solve the quations.....i got it now...maybe that is what happens when you solve math questions for 5 hours straight..i should take a break now...thanks for all the help from everyone... Mar 12 '21 at 15:43 You can simplify the working. Say, time taken by $$B$$ to cover distance $$d$$ between stations $$X$$ and $$Y$$ is $$t$$ hours. Then time taken by $$A$$ is $$(t+1)$$ hours (as $$A$$ starts $$30$$ mins earlier and reaches $$30$$ mins later) and speed of train $$A$$ is $$\displaystyle \frac{d}{t+1}$$ and of train $$B$$ is $$\displaystyle \frac{d}{t}$$. So, $$\displaystyle \frac{2d}{t+1} + \frac{1.5 d}{t} = \frac{11d}{30}$$ Take out $$d$$ from both sides and solve for $$t$$ which comes to $$9$$ hours. That is time taken by train $$B$$. So time taken by $$A$$ is $$10$$ hours. Note: While the question most likely meant that they have not crossed each other but it should have been more explicit. They can be at a distance of $$\frac{19d}{30}$$ even after having crossed each other, which is represented by the equation $$\displaystyle \frac{2d}{t+1} + \frac{1.5 d}{t} = \frac{49d}{30}$$ and it does have a valid solution. • yeah...that could be a scenario too...that thought never came to my mind... if you don't mind..can you please explain a little of getting the total distance travelled in this scenario is 49d/30? Mar 13 '21 at 3:07 • Also when i solved the equation the solution came out to be t = 1.6872 hours. Mar 13 '21 at 3:17 • i got it... 1 + 19d/30 = 49d/30... Mar 13 '21 at 3:25 • Yes, B takes 1.6872 hours and A takes 2.6872 hours is another solution. Mar 13 '21 at 4:17 • As you said it is d + 19d /30. When they meet, they have together covered distance d and then they together cover 19d/30 as they are 19d/30 apart. Mar 13 '21 at 4:23 As noted in a comment, your second equation is incorrect. B started half an hour earlier than A, and arrived half an hour sooner, so took one hour less to cover the distance. We have two equations: \begin{align}\frac{11}{30}x &= 2a+1.5b\\ \frac xa &= 1+\frac xb \end{align} The point you have missed is that we are not asked to find $$a,b,$$ and $$x$$ but $$\frac xa$$ and $$\frac xb$$. If we write $$y=\frac xa,\ z=\frac xb$$ then the equations become \begin{align} \frac{11}{30}&=\frac2y+\frac{1.5}{z}\\ y&=1+z \end{align} Substituting the second in the first, clearing denominators and simplifying gives $$11z^2-94z-45=0$$ whose only positive root is $$z=9$$. • Based on how the question reads, can we confidently eliminate the case where they have crossed each other and are at a distance of $\frac{19x}{31}$ from each other? That gives a valid solution as well. Mar 12 '21 at 16:14 • @MathLover That's a good point that never occurred to me. Mar 12 '21 at 16:25

http://mathhelpforum.com/statistics/181908-am-i-more-likely-roll-least-one-six-if-i-have-more-dice.html

# Thread: Am I more likely to roll at least one six if I have more dice? 1. ## Am I more likely to roll at least one six if I have more dice? Hi, I've had a debate with some friends, about whether I'm more likely to roll at least a single 6 when I have 1000 dice, than I am to roll one with only 1 dice. I say, the more dice I have, the more likely it is to roll at least a single 6. The others try to convince me, that the possibility to roll at least one 6 is always 1/6. Now my thoughts for this are as follows: At first, it doesn't matter, whether I roll them one after another, or one at a time. The chances to roll a six on the first dice that I roll a 6, is 1/6. Now, I don't care about the others if I did roll a 6, but I do care if I didn't (which is in 5/6 of all cases) So for the second dice roll a six, the probability is still 1/6 However, I'm wondering about the probability that the first dice does NOT show a six, but the second does, so the probability for that to happen is 5/6*1/6. So the probability to roll at least one 6 with two dice is: 1/6 + 1/6*5/6 If I was to expand this for 1000 dice, I would end up with: $\displaystyle $$\sum_{k=0}^{999}\left (\frac{5}{6} \right )^k \times \frac{1}{6}$$$ The probability will get ever more closely to 100%, the more dice I have, correct? 2. Why get 1000 dice? Same thing if you have 1 dice: just keep rolling it. 3. Well, originally I asked this: If you were to roll 1000 dice at once, how likely is it, that there is at least one 6? But it doesn't really matter 4. The possibility of rolling a 6 with one die is 1/6, and for each die you have you increase your changes of rolling 1/6. By your friend's reasoning, you're no more liking to have "at least one head (H) in two coin flips" than in just one coin flip. Let us use this much more basic example. The probability of flipping a head of an unbiased coin is 1/2 because we split the event space {T, H} evenly into two. When we flip two coins we have a new event space, namely {TT, TH, HT, HH}. Here the events are equally likely of 1/4, but notice "at least one H" can turn out in three different events. Thus, the probability of "at least one H" is 3/4. The same works for dice. With one die we have an event space {1, 2, 3, 4, 5, 6}. With two dice the size of the event space is 36: Spoiler: 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 53 55 56 61 62 63 64 65 66 Now, in how many ways can we have "at least one six?" In this small, but sufficient, example, we can just look at see. The bottom row and the right-most column all include a 6. This is 11/36, which is almost 0.31, compared to 1/6 =0.167. So yes, increasing the die increases your chances of rolling a six. 5. Thanks, that would confirm my point of view. They: But how is that possible? The probability never changes? So it should always be 1/6? Also, we'd value second and third opinions Thank you very much in advance! 6. The probability of a 6 emerging on a single die does not ever change, but that is not the event we're looking at when we have more than one die. When we are using, say, two dice we're looking at a different event space with each event having 1/36 chance of occurring. Of those thirty-six, eleven of them will have a 6 appear. This is undeniable. This has no impact on the probability of a six appearing on a single die. Your friends seem to be ignoring the fact that the possible ways of a 6 occurring change when you have two dice. With one die there is one, and only one, way of a 6 occurring. Thus, the probability is 1/6. With two dice we have 11 different ways, each with 1/36 = (1/6)(1/6) chance of happening. The probability of the individual dice does support this result, as we can see, but you cannot ignore the fact our event space is different when you include another die. I can go even further and define it this way by the inclusion-exclusion rule of probability theory. Let X be the outcome of the first die and Y the outcome of the second. What we want then is: $\displaystyle P(X = 6\ or\ Y=6)=P(X=6)+P(Y=6)-P(X=6\ \&\ Y= 6)=\frac{1}{6}+\frac{1}{6}-\frac{1}{36}=\frac{11}{36}$ Do you see why these values work? The probability of the single die being six does not change, but we have to sum the two probabilities together. We then have to subtract the event that is common to both of them, otherwise we're double counting that event. This happens only once. 7. Originally Posted by theyThinkImWrong Hi, I've had a debate with some friends, about whether I'm more likely to roll at least a single 6 when I have 1000 dice, than I am to roll one with only 1 dice. I say, the more dice I have, the more likely it is to roll at least a single 6. The others try to convince me, that the possibility to roll at least one 6 is always 1/6. Now my thoughts for this are as follows: At first, it doesn't matter, whether I roll them one after another, or one at a time. The chances to roll a six on the first dice that I roll a 6, is 1/6. Now, I don't care about the others if I did roll a 6, but I do care if I didn't (which is in 5/6 of all cases) So for the second dice roll a six, the probability is still 1/6 However, I'm wondering about the probability that the first dice does NOT show a six, but the second does, so the probability for that to happen is 5/6*1/6. So the probability to roll at least one 6 with two dice is: 1/6 + 1/6*5/6 If I was to expand this for 1000 dice, I would end up with: $\displaystyle $$\sum_{k=0}^{999}\left (\frac{5}{6} \right )^k \times \frac{1}{6}$$$ The probability will get ever more closely to 100%, the more dice I have, correct? Let X be the random variable 'Number of 6's rolled'. Then X ~ Binomial(n, p = 1/6). Calculate Pr(X > 0) = 1 - Pr(X = 0). Then it is obvious that as n increases so does the probability. I suggest you review the binomial distribution if you don't know it ( I do not plan to give a tutorial).

https://math.stackexchange.com/questions/27719/what-is-gcd0-a-where-a-is-a-positive-integer

# What is $\gcd(0,a)$, where $a$ is a positive integer? I have tried $\gcd(0,8)$ in a lot of online gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is correct and why there are different conventions? • I haven't encountered the convention of gcd(0,8) = 1. It depends on how you define the phrase "a divides b" Mar 18 '11 at 2:49 • Mar 18 '11 at 2:53 • @The Chaz: They are really the same things but with different names. see en.wikipedia.org/wiki/Greatest_common_divisor Mar 18 '11 at 4:04 • Should be a, because anything is a divisor of 0. Nov 26 '20 at 7:48 Let's recall the definition of "$\rm a$ divides $\rm b$" in a ring $\rm\,Z,\,$ often written as $\rm\ a\mid b\ \ in\ Z.$ $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm\ a\mid b\ \ in\ Z\ \iff\ a\,c = b\ \$ for some $\rm\ c\in Z$ Recall also the definition of $\rm\ gcd(a,b),\,$ namely $(1)\rm\qquad\quad \rm gcd(a,b)\mid a,b\qquad\qquad\qquad\$ the gcd is a common divisor $(2)\rm\qquad\quad\! \rm c\mid a,b\ \ \ \Longrightarrow\ \ c\mid gcd(a,b)\quad$ the gcd is a greatest common divisor $\ \ \ \$ i.e. $\rm\quad\ c\mid a,b\ \iff\ c\mid gcd(a,b)\quad\,$ expressed in $\iff$ form [put $\rm\ c = gcd(a,b)\$ for $(1)$] Notice $\rm\quad\, c\mid a,0\ \iff\ c\mid a\,\$ so $\rm\ gcd(a,0)\ =\ a\$ by the prior "iff" form of the gcd definition. Note that $\rm\ gcd(0,8) \ne 0\,$ since $\rm\ gcd(0,8) = 0\ \Rightarrow\ 0\mid 8\$ contra $\rm\ 0\mid x\ \iff\ x = 0.$ Note that $\rm\ gcd(0,8) \ne 1\,$ else $\rm\ 8\mid 0,8\ \Rightarrow\ 8\mid gcd(0,8) = 1\ \Rightarrow\ 1/8 \in \mathbb Z.$ Therefore it makes no sense to define $\rm\ gcd(0,8)\$to be $\,0\,$ or $\,1\,$ since $\,0\,$ is not a common divisor of $\,0,8\,$ and $\,1\,$ is not the greatest common divisor. The $\iff$ gcd definition is universal - it may be employed in any domain or cancellative monoid, with the convention that the gcd is defined only up to a unit factor. This $\iff$ definition is very convenient in proofs since it enables efficient simultaneous proof of both implication directions. $\$ For example, below is a proof of this particular form for the fundamental GCD distributive law $\rm\ (ab,ac)\ =\ a\ (b,c)\$ slightly generalized (your problem is simply $\rm\ c=0\$ in the special case $\rm\ (a,\ \ ac)\ =\,\ a\ (1,c)\ =\ a\,$). Theorem $\rm\quad (a,b)\ =\ (ac,bc)/c\quad$ if $\rm\ (ac,bc)\$ exists. Proof $\rm\quad d\mid a,b\ \iff\ dc\mid ac,bc\ \iff\ dc\mid (ac,bc)\ \iff\ d|(ac,bc)/c$ See here for further discussion of this property and its relationship with Euclid's Lemma. Recall also how this universal approach simplifies the proof of the basic GCD * LCM law: Theorem $\rm\;\; \ (a,b) = ab/[a,b] \;\;$ if $\;\rm\ [a,b] \;$ exists. Proof $\rm\quad d|\,a,b \;\iff\; a,b\,|\,ab/d \;\iff\; [a,b]\,|\,ab/d \;\iff\; d\,|\,ab/[a,b] \quad\;\;$ For much further discussion see my many posts on GCDs. Another way to look at it is by the divisibility lattice, where gcd is the greatest lower bound. So 5 is the greatest lower bound of 10 and 15 in the lattice. The counter-intuitive thing about this lattice is that the 'bottom' (the absolute lowest element) is 1 (1 divides everything), but the highest element, the one above everybody, is 0 (everybody divides 0). So $\gcd(0, x)$ is the same as ${\rm glb}(0, x)$ and should be $x$, because $x$ is the lower bound of the two: they are not 'apart' and 0 is '$>'$ $x$ (that is the counter-intuitive part). In fact, the top answer can be generalized slightly: if $$a \mid b$$, then $$\gcd(a,b)=a$$ (and this holds in any algebraic structure where divisibility makes sense, e.g. a commutative, cancellative monoid). To see why, well, it's clear that $$a$$ is a common divisor of $$a$$ and $$b$$, and if $$\alpha$$ is any common divisor of $$a$$ and $$b$$, then, of course, $$\alpha \mid a$$. Thus, $$a=\gcd(a,b)$$. • Indeed, even more generally, it is a special case of the distributive law - see my answer. As for commutative monoids, one usually requires them to be cancellative in order to obtain a rich theory. Mar 18 '11 at 18:26 It might be partly a matter of convention. However, I believe that stating that $$\gcd(8,0) = 8$$ is safer. In fact, $$\frac{0}{8} = 0$$, with no remainder. The proof of the division, indeed is that "Dividend = divider $$\times$$ quotient plus remainder". In our case, 0 (dividend) = 8 (divisor) x 0 (quotient). No remainder. Now, why should 8 be the GCD? Because, while the same method of proof can be used for all numbers, proving that $$0$$ has infinite divisors, the greatest common divisor cannot be greater than $$8$$, and for the reason given above, is $$8$$.

https://math.stackexchange.com/questions/1624236/finding-eigenvectors-of-a-3x3-matrix-7-12-15/1624242

# Finding Eigenvectors of a 3x3 Matrix (7.12-15) Please check my work in finding an eigenbasis (eigenvectors) for the following problem. Some of my solutions do not match answers in my differential equations text (Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons). For reference the following identity is given because some textbooks reverse the formula having $\lambda$ subtract the diagonal elements instead of subtracting $\lambda$ from the diagonal elements: $$det(A - \lambda I) = 0$$ $$A = \begin{bmatrix} 3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5 \\ \end{bmatrix}$$ By inspection the eigenvalues are the entries along the diagonal for this upper triangular matrix. \begin{align*} \lambda_1 = 3 \qquad \lambda_2 = 2 \qquad \lambda_3 = 5 \end{align*} When $\lambda_1 = 3$ we have: $$A - 3I = \begin{bmatrix} 3-3 & 1 & 4 \\ 0 & 2-3 & 6 \\ 0 & 0 & 5-3 \\ \end{bmatrix} = \begin{bmatrix} 0 & 1 & 4 \\ 0 & -1 & 6 \\ 0 & 0 & 2 \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$$ \begin{align*} x_1 = 1 \: (free \: variable) \qquad x_2 = 0 \qquad x_3 = 0 \\ \end{align*} $$v_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} \qquad (matches \: answer \: in \: text)$$ When $\lambda_2 = 2$ we have: $$A - 2I = \begin{bmatrix} 3-2 & 1 & 4 \\ 0 & 2-2 & 6 \\ 0 & 0 & 5-2 \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & 4 \\ 0 & 0 & 6 \\ 0 & 0 & 3 \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$$ \begin{align*} x_1 = -x_2 \qquad x_2 = 1 \: (free \: variable) \qquad x_3 = 0 \\ \end{align*} $$v_2 = \begin{bmatrix} -1 \\ 1 \\ 0 \\ \end{bmatrix} \qquad but \: answer \: in \: text \: is \qquad \begin{bmatrix} 1 \\ -1 \\ 0 \\ \end{bmatrix}$$ What happened? Is it from a disagreement in what we should consider arbitrary or am I doing something fundamentally wrong? When $\lambda_3 = 5$ we have: $$A - 5I = \begin{bmatrix} 3-5 & 1 & 4 \\ 0 & 2-5 & 6 \\ 0 & 0 & 5-5 \\ \end{bmatrix} = \begin{bmatrix} 2 & 1 & 4 \\ 0 & -3 & 6 \\ 0 & 0 & 0 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & -3 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ \end{bmatrix}$$ \begin{align*} x_1 = 3x_3 \qquad x_2 = 2x_3 \qquad x_3 = 1 \: (free \: variable) \\ \end{align*} $$v_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \\ \end{bmatrix} \qquad (matches \: answer \: in \: text)$$ • Both your and the text book's answer work. If $(\lambda ,v)$ is an eigenpair of a matrix $M$, then so is $(\lambda, \mu v)$, for all scalars $\mu$. – Git Gud Jan 23 '16 at 23:15 Eigenvectors are never unique. In particular, for the eigenvalue $2$ you can take, for example, $x_2=-1$ which gives you the answer in the book. • So what you are saying is that both I and the book's answers are correct? So its just a matter of preference? – Jules Manson Jan 23 '16 at 23:24 • Precisely, and in fact you could take for $x_2$ any nonzero real number. – John B Jan 23 '16 at 23:26 • In your original post, you said that the problem was to find an eigenbasis. In other words, since there are three distinct eigenvalues, find one eigenvector for each eigenvalue. Which one you choose out of the infinite number of eigenvalues that correspond to each eigenvalue is "a matter of preference" but it is important that you understand that any multiple of an eigenvector is also an eigenvector. – user247327 Dec 13 '16 at 16:18

http://gusx.lampertifashion.it/application-of-laplace-transform-pdf.html

5 Other transforms. As we saw in the last section computing Laplace transforms directly can be fairly complicated. a signal such that $$x(t)=0$$ for $$x<0$$. Schiff and others published The Laplace Transform: Theory and Applications | Find, read and cite all the research you need on ResearchGate. With the increasing complexity of engineering problems, Laplace transforms help in solving complex problems with a very simple approach just like the applications of transfer functions to solve ordinary differential equations. (b) Compute the Laplace transform of f. A very simple application of Laplace transform in the area of physics could be to find out the harmonic vibration of a beam which is supported at its two ends. In mathematics, the Laplace transform is an integral transform named after its inventor Pierre-Simon Laplace ( / ləˈplɑːs / ). The Laplace transform, theory and applications. In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc. Capacitor. e −tsin 2 t 5. The Laplace transform is defined as follows: F^(p) = Z +1 1. The application of Laplace Transforms is wide and is used in a variety of. Each Outline presents all the essential course information in an easy-to-follow,. In India, we are facing various types of crimes. 2-3 Circuit Analysis in the s Domain. the Fourier cosine transform, and the Fourier sine transform, as applied to various standard functions, and use this knowledge to solve certain ordinary and partial differential equations. , 𝑇 is a (random) time to failure), the Laplace transform of ( ) can also be interpreted as the expected value of the random variable 𝑌= − 𝑇 , i. com solve differential with laplace transform, sect 7. Laplace Transform of Periodic Functions, Convolution, Applications 1 Laplace transform of periodic function Theorem 1. Laplace Transforms and Properties. Laplace Transform in Engineering Analysis Laplace transforms is a mathematical operation that is used to "transform" a variable (such as x, or y, or z, or t)to a parameter (s)- transform ONE variable at time. A Coupled Method of Laplace Transform and Legendre Wavelets for Lane-Emden-Type Differential Equations Yin, Fukang, Song, Junqiang, Lu, Fengshun, and Leng, Hongze, Journal of Applied Mathematics, 2012. txt) or view presentation slides online. To solve constant coefficient linear ordinary differential equations using Laplace transform. With the increasing complexity of engineering problems, Laplace transforms help in solving complex problems with a very simple approach just like the applications of transfer functions to solve ordinary differential equations. Abel's integral equation. a finite sequence of data). Yes, the Laplace transform has "applications", but it really seems that the only application is solving differential equations and nothing beyond that. Chapter 3: The z-Transform and Its Application Power Series Convergence IFor a power series, f(z) = X1 n=0 a n(z c)n = a 0 + a 1(z c) + a 2(z c)2 + there exists a number 0 r 1such that the series I convergences for jz cjr I may or may not converge for values on jz cj= r. The Laplace transform f(p), also denoted by L{F(t)} or Lap F(t), is defined by the integral involving the exponential parameter p in the kernel K = e −pt. The fact that the inverse Laplace transform is linear follows immediately from the linearity of the Laplace transform. • All we need is to express F(s) as a sum of simpler functions of the forms listed in the Laplace transform table. The Laplace transform can be used in some cases to solve linear differential equations with given initial conditions. Laplace Transform The Laplace transform is a method of solving ODEs and initial value problems. The vibrational analysis of structures use Laplace transforms. 5 Application of Laplace Transforms to Partial Differential Equations In Sections 8. The Laplace transform pair for. Consider the differential equation given by: can represent many different systems. LTI System Analysis with the Laplace Transform. The differential inverse transform of 𝑈 , is define by , of the form in (1). Laplace transform and its applications 1. Since the m. Colophon An annotatable worksheet for this presentation is available as Worksheet 6. It follows that the output Y(s) can be written as the product of G(s) and. There are two (related) approaches: Derive the circuit (differential) equations in the time domain, then transform these ODEs to the s-domain; Transform the circuit to the s-domain, then derive the circuit equations in the s-domain (using the concept of "impedance"). Let f(t) be de ned for t 0:Then the Laplace transform of f;which is denoted by L[f(t)] or by F(s), is de ned by the following equation L[f(t)] = F(s) = lim T!1 Z T 0 f(t)e stdt= Z 1 0 f(t)e stdt The integral which de ned a Laplace transform is an improper integral. Let fbe a function of t. where X(s) is the Laplace transform of the input to the system and Y(s) is the Laplace transform of the output of the system, where we assume that all initial conditions in- volved are zero. The Law of Laplace is a physical law discovered by the great French mathematician Piere-Simon Laplace (and others) which describes the pressure-volume relationships of spheres. The tautochrone problem. The Laplace Transform is widely used in following science and engineering field. 6 – 8 Each function F(s) below is defined by a definite integral. Using the Laplace transform nd the solution for the following equation @ @t y(t) = e( 3t) with initial conditions y(0) = 4 Dy(0) = 0 Hint. Article full text Download PDF. Laplace Transform The Laplace transform can be used to solve differential equations. Laguerre transform. Find PowerPoint Presentations and Slides using the power of XPowerPoint. is identical to that of. Bateman transform. rainville Lecture 7 Circuit Analysis Via Laplace Transform Inverse Laplace Transform Of Exponential Function Basically, Poles Of Transfer Function Are The Laplace Transform Variable Values Which Causes The Tra Basically. The control action for a dynamic control system whether electrical, mechanical, thermal, hydraulic, etc. 'The Laplace Transform' is an excellent starting point for those who want to master the application of. Mesh analysis. The best way to convert differential equations into algebraic equations is the use of Laplace transformation. So let's see if we can apply that. For a resistor, the. 3 Applications Since the equations in the s-domain rely on algebraic manipulation rather than differential equations as in the time domain it should prove easier to work in the s-domain. In this dissertation, several theorems on multidimensional Laplace transforms are developed. cosh() sinh() 22 tttt tt +---== eeee 3. Linearization, critical points, and equilibria. Click Download or Read Online button to get laplace transformation book now. ’s) of waiting times in queues. Considering a function f (t), its corresponding Laplace Transform. Some illustrative examples will be discussed. 2012-08-12 00:00:00 A natural way to model dynamic systems under uncertainty is to use fuzzy initial value problems (FIVPs) and related uncertain systems. Solving PDEs using Laplace Transforms, Chapter 15 Given a function u(x;t) de ned for all t>0 and assumed to be bounded we can apply the Laplace transform in tconsidering xas a parameter. Applications of Fourier transform to PDEs. Bateman transform. Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations. com, find free presentations research about Application Of Laplace Transform PPT. A final property of the Laplace transform asserts that 7. 6 596--607. The real and imaginary parts of s can be considered as independent quantities. 1) whenever the limit exists (as a finite number). (1975) Numerical inversion of the Laplace transform by accelerating the convergence of Bromwick's integral. cos(2t) + 7sin(2t) 3. Application to laplace transformation to electric circuits by J Irwin. An advantage of Laplace transform We can transform an ordinary differential equation (ODE) into an algebraic equation (AE). Find PowerPoint Presentations and Slides using the power of XPowerPoint. Laplace transform is named in honour of the great French mathematician, Pierre Simon De Laplace (1749-1827). The basic idea and arithmetics of fuzzy sets were first introduced by L. Fourier transforms only capture the steady state behavior. You da real mvps! $1 per month helps!!. In general we have + ∞ − ∞ − = j j F s e st ds j L F. The use of a truncated Laplace-like transformation in the construction of the analytic solution allows to overcome a small divisor phenomenon arising from the geometry of the problem and represents an alternative approach to the one proposed in a recent work by the last two authors. Prentice Hall Math Books, vertical adding and subtracting fraction sheets, the quadratic formula in ti-84, finding y-intercept of a polynomial calculator, square roots lessons, examples of algebra questions, ti calculator rom. ∫ + ∞ − ∞ = i i F s est ds i f t σ σ π ( ) 2 1 ( ) σ Real Abscissa of convergence Isolated singularities Imaginary Laplace transform inversion is. Since the m. 1) In a layman's term, Laplace transform is used to "transform" a variable in a function. Apr 24, 2020 - Applications of Laplace Transformation-I Computer Science Engineering (CSE) Video | EduRev is made by best teachers of Computer Science Engineering (CSE). logo1 New Idea An Example Double Check The Laplace Transform of a System 1. Applications of Laplace Transform. The question is: How is possible to derive the. This shows the effectiveness and usefulness of the Numerical Inversion of the Laplace transform. The inverse transform L−1 is a linear operator: L−1{F(s)+ G(s)} = L−1{F(s)} + L−1{G(s)}, (2) and L−1{cF(s)} = cL−1{F(s)}, (3) for any constant c. is the Laplace domain equivalent of the time domain function. The table of Laplace transforms collects together the results we have considered, and more. Take Laplace Transform of both sides of ODE Solve for Factor the characteristic polynomial Find the roots (roots or poles function in Matlab) Identify factors and multiplicities Perform partial fraction expansion Inverse Laplace using Tables of Laplace Transforms. Similar to the application of phasortransform to solve the steady state AC circuits , Laplace transform can be used to transform the time domain circuits into S domain circuits to simplify the solution of integral differential equations to the manipulation of a set of algebraic equations. 2) 𝑅 for Z-transform in Example 2. The Laplace Transform Definition and properties of Laplace Transform, piecewise continuous functions, the Laplace Transform method of solving initial value problems The method of Laplace transforms is a system that relies on algebra (rather than calculus-based methods) to solve linear differential equations. Application of Laplace Transform For Cryptographic Scheme A. 2 Properties of the z-Transform Common Transform Pairs Iz-Transform expressions that are a fraction of polynomials in z 1 (or z) are calledrational. Se você continuar a navegar o site, você aceita o uso de cookies. Ifthelimitdoesnotexist,theintegral is said todivergeand there is no Laplace transform defined forf. Application of Laplace Transform to Newtonian Fluid Problems Article (PDF Available) in International Journal of Science and Research (IJSR) · July 2013 with 2,655 Reads How we measure 'reads'. It is named in honor of the great French mathematician, Pierre Simon De Laplace (1749-1827). Build your own widget. Fourier Transform is a mathematical operation that breaks a signal in to its constituent frequencies. Having carried out this procedure, we should check that this latter expression does, indeed, yield a solution of the original initial-boundary value problem. Now we going to apply to PDEs. McLachlan, quicker you could enjoy checking out the publication. The application of Laplace Transform methods is particularly effective for linear ODEs with constant coefficients, and for systems of such ODEs. 6e5t cos(2t) e7t (B) Discontinuous Examples (step functions): Compute the Laplace transform of the given function. eat 1 sa 2 2 2 12. kalla is the laplace transform. The CDF of a random variable is often much more useful in practical applications but is often difficult to find. Redraw the circuit (nothing about the Laplace transform changes the types of elements or their interconnections). Laplace transform Pairs (1) Finding inverse Laplace transform requires integration in the complex plane - beyond scope of this course. 4 Multi–dimensional transformation algorithms 205. Last, some boundary value problems characterized by linear partial differential equations involving heat and. Thus, Laplace Transformation transforms one class of complicated functions to produce another class of simpler functions. The Dirac delta, distributions, and generalized transforms. and scientists dealing with "real-world" applications. ; We will use the first approach. The Laplace Transformation is very effective device in Mathematic, Physics and other branches of science which is used to solving problem. com 1 View More View Less. Solving PDEs using Laplace Transforms, Chapter 15 Given a function u(x;t) de ned for all t>0 and assumed to be bounded we can apply the Laplace transform in tconsidering xas a parameter. The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform. Definition: Laplace Transform. The method is simple to describe. can be represented by a differential equation. In particular it is shown that the Laplace transform of tf(t) is -F'(s), where F(s) is the Laplace transform of f(t). If we assume that the functions whose Laplace transforms exist are going to be taken as continuous then no two different functions can have the same Laplace transform. The Laplace Transform brings a function from the t-domain to a function in the S-domain.$\begingroup$The Fourier transform is just a special case of the Laplace transform, so your example actually works for both. where X(s) is the Laplace transform of the input to the system and Y(s) is the Laplace transform of the output of the system, where we assume that all initial conditions in- volved are zero. The first term in the brackets goes to zero (as long as f(t) doesn't grow faster than an exponential which was a condition for existence of the transform). Application of Laplace Transform in State Space Method to Solve Higher Order Differential Equation: Pros & Cons Ms. 4) Equivalent Circuits 5) Nodal Analysis and Mesh Analysis. Unilateral Laplace Transform. Schaum's Outlines: Laplace Transforms By Murray R. The Inverse Laplace Transformation Circuit Analysis with Laplace Transforms Frequency. The Laplace transform pair for. The analytic inversion of the Laplace transform is a well-known application of the theory of complex variables. 2 Useful Laplace Transform Pairs 12. , frequency domain ). However, in all the examples we consider, the right hand side (function f(t)) was continuous. We begin with the general formula for voltage drops around the circuit: Substituting numbers, we get Now, we take the Laplace Transform and get Using the fact that , we get. Consider the differential equation given by: can represent many different systems. Yes, the Laplace transform has "applications", but it really seems that the only application is solving differential equations and nothing beyond that. Additional Physical Format: Online version: Watson, E. Examples of the Laplace Transform as a Solution for Mechanical Shock and Vibration Problems: Free Vibration of a Single-Degree-of-Freedom System: free. The Laplace Transform is a specific type of integral transform. 1a,b, the graphs of the Laplace transform [Lf](s) = Z∞ 0. The κ-Laplace transform proposed in this note is just one form of modified Laplace transformations. (1975) Application of best rational function approximation for Laplace transform inversion. Therefore, without further discussion, the Laplace transform is given by: De nition 1. Jacobi transform. Engineering Applications of z-Transforms 21. txt) or view presentation slides online. Hilbert-Schmidt integral operator. In machine learning, the Laplace transform is used for making predictions and making analysis in data mining. Post's Formula. 5 Other transforms. Breaking down complex differential equations into simpler polynomial forms. studysmarter. Numerical examples reveal that the pricing formulas are easy to implement and they result in accurate prices and risk parameters. The similarity of this notation with the notation used in Fourier transform theory is no coincidence; for ,. Inverse of the Laplace Transform. Laplace Transform Z Transform Fourier Transform Fourier Transform Fourier Transform Formula Fourier Transform Applications Mathematics Of The Discrete Fourier Transform A Guided Tour Of The Fast Fourier Transform Bergland Mathematics Of The Discrete Fourier Transform (dft) With Audio Applications Fourier Fourier Mathcad Fourier Series Transformada De Fourier Fourier Analysis Pdf Hc Taneja Fourier Schaum Fourier Analysis Fast Fourier Transformation Schaum Fourier Analysis Pdf Applications Of. Then, by definition, f is the inverse transform of F. e −tsin 2 t 5. 8 The Impulse Function in. AKANBI 4 and F. Both transforms are equivalent tools, but the Laplace transform is used for continuous-time signals, whereas the$\mathcal{Z}\$-transform is used for discrete-time signals (i. Best & Easiest Videos Lectures covering all Most Important Questions on Engineering Mathematics for 50+ Universities Download Important Question PDF (Password mathcommentors) Will Upload soon. Fourier transforms only capture the steady state behavior. Table of LaPlace Transforms ft() L { ( )} ( )f t F s 1. 1) In a layman's term, Laplace transform is used to "transform" a variable in a function. The Laplace Transform can be considered as an extension of the Fourier Transform to the complex plane. 6 per cent faster than the next high-speed adder cell. Just like for the Z-transform we have to specify the ROC for the Laplace transform. Applications of Laplace Transforms Circuit Equations. Laplace Transforms for Systems of Differential Equations New Idea An Example Double Check The Laplace Transform of a System 1. Edited by: Salih Mohammed Salih. It's also the best approach for solving linear constant coefficient differential equations with nonzero initial conditions. 1) F(s) = Z 1 0 e stf(t)dt provided the improper integral converges. Post's Formula. This paper will be primarily concerned with the Laplace transform and its ap-plications to partial di erential equations. 13, 2012 • Many examples here are taken from the textbook. In my 13-year industrial career, I never used mathematical. McLachlan). hyperbolic functions. The two-sided Laplace transform (3) can be regarded as the Fourier transform of the function , and the one-sided Laplace transform (2) can be regarded as the Fourier transform of the function equal to for and equal to zero for. The function is piece-wise continuous B. When this definition is used it can be shown that the Laplace transform, Fn(s) of the nth derivative of a function, fn(t), is given by the following generic formula: Fn(s)=snF(s) - sn-1f0(0) - sn. Unit step function, Laplace Transform of Derivatives and Integration, Derivative and Integration of Laplace Transforms 1 Unit step function u a(t) De nition 1. Cryptography is one of the. For particular functions we use tables of the Laplace. In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc. As we saw in the last section computing Laplace transforms directly can be fairly complicated. Additional Physical Format: Online version: Watson, E. M-2 Shah Nisarg (130410119098) Shah Kushal(130410119094) Shah Maulin(130410119095) Shah Meet(130410119096) Shah Mirang(130410119097) Laplace Transform And Its Applications 2. The Laplace transform is a wonderful tool for solving ordinary and partial differential equations and has enjoyed much success in this realm. Laplace Transforms and their Applications About the Laplace Transformation The Laplace Transformation (named after Pierre-Simon Laplace ) is a useful mathematical tool that is used in many branches of engineering including signals and systems theory, control theory, communications, mechanical engineering, etc. Applications of Laplace Transform in Science and Engineering fields: This section describes the applications of Laplace Transform in the area of science and engineering. Redraw the circuit (nothing about the Laplace transform changes the types of elements or their interconnections). symbolizes the Laplace transform. If we look at the left-hand side, we have Now use the formulas for the L[y'']and L[y']: Here we have used the fact that y(0)=2. However, the spectral properties of the Laplace transform tend to complicate its numerical treatment; therefore, the closely related "Truncated" Laplace Transforms are often used in applications. zi denotes the zeros and pi denotes the poles of the linear time invariant system (LTI). cos(2t) + 7sin(2t) 3. The Laplace Transform can greatly simplify the solution of problems involving differential equations. Post's Formula. along with the Definition of Laplace Transform, Applications of Laplace Laplace Transform to Solve a Differential Equation, Ex 1, Part 1/2 Thanks to all of you who support me on Patreon. In this course, one of the topics covered is the Laplace transform. Download The Laplace Transform: Theory and Applications By Joel L. However, to transform the obtained solutions from Laplace space back to the original space, we have used the Numerical Inversion of the Laplace transform. Similar to the application of phasortransform to solve the steady state AC circuits , Laplace transform can be used to transform the time domain circuits into S domain circuits to simplify the solution of integral differential equations to the manipulation of a set of algebraic equations. pdf), Text File (. Download File PDF Applications Of Laplace Transform In Mechanical Engineering Applications Of Laplace Transform In Mechanical Engineering When somebody should go to the books stores, search creation by shop, shelf by shelf, it is in fact problematic. So far, regarding their mathematical properties [11, 12] and application [for transforms of various functions see, e. Consider the ODE in Equation [1]: We are looking for the function y (t) that satisfies Equation. Laplace transform and its applications O SlideShare utiliza cookies para otimizar a funcionalidade e o desempenho do site, assim como para apresentar publicidade mais relevante aos nossos usuários. 201038 Identifier-ark ark:/13960/t80k7s705 Ocr ABBYY FineReader 11. For particular functions we use tables of the Laplace. To obtain inverse Laplace transform. Laplace transform. 14) The ROC for. With the ease of application of Laplace transforms in myriad of scientific applications, many research software‟s. 1) F(s) = Z 1 0 e stf(t)dt provided the improper integral converges. The Laplace transform is de ned in the following way. The fact that the inverse Laplace transform is linear follows immediately from the linearity of the Laplace transform. To prove this we start with the definition of the Laplace Transform and integrate by parts. Report "Laplace Transforms: And Applications". Each view has its uses. Laplace Transform Melissa Meagher Meagan Pitluck Nathan Cutler Matt Abernethy Thomas Noel Scott Drotar The French Newton Pierre-Simon Laplace Developed mathematics in astronomy, physics, and statistics Began work in calculus which led to the Laplace Transform Focused later on celestial mechanics One of the first scientists to suggest the existence of black holes History of the Transform Euler. Note: There are two types of laplace transforms. (a) Compute the Laplace transform of f 1(t) = eat. 6: Perform the Laplace transform of function F(t) = Sin3t. Apr 24, 2020 - Applications of Laplace Transformation-I Computer Science Engineering (CSE) Video | EduRev is made by best teachers of Computer Science Engineering (CSE). 2 Introduction to Laplace Transforms simplify the algebra, find the transformed solution f˜(s), then undo the transform to get back to the required solution f as a function of t. The important differences between Fourier transform infrared (FTIR) and filter infrared (FIR) systems for air monitoring are explored, and the strengths and weaknesses of these technologies when applied to industrial hygiene problems are defined and illustrated with actual workplace air monitoring examples. Like all transforms, the Laplace transform changes one signal into another according to some fixed set of rules or equations. Title: Applications of the Laplace Transform 1 Applications of the Laplace Transform ECE 2221/MCT 2210 Signals and Systems (Analysis) Sem. This site is like a library, Use search box in the widget to get ebook that you want. Manolakis, Digital Signal Processing:. The ordinary differential or integral equations involving f(t) are transformed to the algebraic equations for F(s), the partial differential equations in f(t) are transformed to the. The similarity of this notation with the notation used in Fourier transform theory is no coincidence; for ,. 5 Other transforms. [Hint: each expression is the Laplace transform of a certain. Inverse of a Product L f g t f s ĝ s where f g t: 0 t f t g d The product, f g t, is called the convolution product of f and g. Using the one-sided Laplace transform is equivalent with transforming causal signals and systems, i. Edited by: Salih Mohammed Salih. If the Laplace transform of fexists, then F(s) = Z T 0 f(t)e stdt 1 sTe: (1) Proof: We have F(s) = Z 1 0 f(t)e stdt = X1 n=0 Z (n+1. Shahrul Naim Sidek ; Department of Mechatronics Engineering. This is used to solve differential equations. A Laplace transform is an integral transform. F ( s) = ∫ 0 ∞ f ( t) e − s t d t. Title: Applications of the Laplace Transform 1 Applications of the Laplace Transform ECE 2221/MCT 2210 Signals and Systems (Analysis) Sem. In this course, one of the topics covered is the Laplace transform. Advantages of the Laplace transform over the Fourier transform: The Fourier transform was defined only for stable systems or signals that taper off at infinity. By using the Laplace transform, any electrical circuit can be solved and calculations are very easy for transient and steady state conditions. 17 Applications of Fourier Transforms in Mathematical Statistics 103 2. The important differences between Fourier transform infrared (FTIR) and filter infrared (FIR) systems for air monitoring are explored, and the strengths and weaknesses of these technologies when applied to industrial hygiene problems are defined and illustrated with actual workplace air monitoring examples. Applications of the Laplace Transform - Free download as PDF File (. ppt), PDF File (. Life would be simpler if the inverse Laplace transform of f s ĝ s was the pointwise product f t g t, but it isn’t, it is the convolution product. 1) 𝑅 for Z-transform in Example 2. The Laplace transform, theory and applications. 3 Applications Since the equations in the s-domain rely on algebraic manipulation rather than differential equations as in the time domain it should prove easier to work in the s-domain. Chapter 13 The Laplace Transform in Circuit Analysis. Take Laplace Transform of both sides of ODE Solve for Factor the characteristic polynomial Find the roots (roots or poles function in Matlab) Identify factors and multiplicities Perform partial fraction expansion Inverse Laplace using Tables of Laplace Transforms. pptx), PDF File (. A novel method of determining Laplace inverse transform of a typical function using superposition technique is presented. Let L ff(t)g = F(s). To formulate the general solution of problem , we replace with in the equation of problem and applied Theorem 3. Laplace transform 1 Laplace transform The Laplace transform is a widely used integral transform with many applications in physics and engineering. Fourier series Periodic x(t) can be represented as sums of complex exponentials x(t) periodic with period T0 Fundamental (radian) frequency!0 = 2ˇ=T0 x(t) = ∑1 k=1 ak exp(jk!0t) x(t) as a weighted sum of orthogonal basis vectors exp(jk!0t) Fundamental frequency!0 and its harmonics ak: Strength of k th harmonic Coefficients ak can be derived using the relationship ak =. Using the Laplace transform, it is possible to convert a system's time-domain representation into a frequency-domain input/output representation, known as the transfer function. Introduction This paper deals with a brief overview of what Laplace Transform is and its application in the industry. Its principle benefits are: it enables us to represent differential equations that. In short, yes, it is possible, but much, much more difficult. Integro-differential equations. The Laplace transform of fis de ned to be (1. no hint Solution. cos(2t) + 7sin(2t) 3. Therefore, without further discussion, the Laplace transform is given by: De nition 1. Description : Laplace Transforms for Electronic Engineers, Second (Revised) Edition details the theoretical concepts and practical application of Laplace transformation in the context of electrical engineering. Fourier transforms only capture the steady state behavior. The notation L(f) will also be used to denote the. Laplace transform is named in honour of the great French mathematician, Pierre Simon De Laplace (1749-1827). f(t) = 2(H(t 1) H(t 3)) + tH(t 3) = 2H(t 1) 2H(t 3) + tH(t 3) The Laplace Transform of f(t) is then L[f(t)]= L[2H(t 1) 2H(t 3) + tH(t 3)] F(s) = 2L[H(t 1)] 2L[H(t 3)] + L[tH(t 3)] Now we need to know something about the Laplace Transforms of Heaviside functions. Fourier Transforms can also be applied to the solution of differential equations. Therefore, without further discussion, the Laplace transform is given by: De nition 1. An inversion technique for the Laplace transform with applications. CHAPTER 14 LAPLACE TRANSFORMS 14. So let's see if we can apply that. com 1 and G. Applications include electrical and mechanical networks, heat conducti This textbook describes in detail the various Fourier and Laplace transforms that are used to analyze problems in mathematics, the natural sciences and engineering. The laplace transform is an integral transform, although the reader does not need to have a knowledge of integral calculus because all results will be provided. I am having some trouble computing the inverse laplace transform of a symbolic expression using sympy. The book first covers the. Gabor transform. The corresponding boundary value problems via the Feynman-Kac representation are solved to obtain an explicit formula for the joint distribution of the occupation time and the terminal value of the Lévy processes with jumps rational Laplace transforms. Colophon An annotatable worksheet for this presentation is available as Worksheet 6. the derivative Typically, one proceeds putting the initial conditions equal to zero. Download File PDF Applications Of Laplace Transform In Mechanical Engineering Applications Of Laplace Transform In Mechanical Engineering When somebody should go to the books stores, search creation by shop, shelf by shelf, it is in fact problematic. Although in principle, you could do the necessary integrals,. The Fourier Transform finds the recipe for a signal, like our smoothie process: Start with a time-based signal; Apply filters to measure each possible "circular ingredient" Collect the full recipe, listing the amount of each "circular ingredient" Stop. To find the Laplace transform F(s) of an exponential function f(t) = e -at for t >= 0. Wen [email protected] Laplace Transforms for Electronic Engineers, Second (Revised) Edition details the theoretical concepts and practical application of Laplace transformation in the context of electrical engineering. Findings Simulation results demonstrate very high-speed operation for the first and second proposed designs, which are, respectively, 44. a b w(x,y) is the displacement in z-direction x y z. L(sin(6t)) = 6 s2 +36. is the Laplace domain equivalent of the time domain function. The control action for a dynamic control system whether electrical, mechanical, thermal, hydraulic, etc. So what types of functions possess Laplace transforms, that is, what type of functions guarantees a convergent improper integral. Application to laplace transformation to electric circuits by J Irwin. To know final-value theorem and the condition under which it. This research paper explains the application of Laplace Transforms to real-life problems which are modeled into differential equations. Without integrating, find an explicit expression for each F(s). 5 Application of Laplace Transforms to Partial Differential Equations In Sections 8. Partial Differential Equations: Graduate Level Problems 8 Laplace Equation 31 Fourier Transform 365 31 Laplace Transform 385. 19 Exercises 119 3 Laplace Transforms and Their Basic Properties 133 3. Application of Numerical Inverse Laplace Transform Methods for Simulation of Distributed Systems with Fractional-Order Elements¤ Nawfal Al-Zubaidi R-Smith†, Aslihan Kartci‡ and Lubomír Brančík§ Department of Radio Electronics, Brno University of Technology, Technicka 12, Brno, Czech Republic †[email protected] Find PowerPoint Presentations and Slides using the power of XPowerPoint. Chiefly, they treat problems which, in mathematical language, are governed by ordi­ nary and partial differential equations, in various physically dressed forms. 1) Inductor. In so doing, it also transforms the governing differential equation into an algebraic equation which is often easier to analyze. Ifthelimitdoesnotexist,theintegral is said to diverge and there is no Laplace transform defined for f. For particular functions we use tables of the Laplace. (1975) Application of best rational function approximation for Laplace transform inversion. ), uses the Bromwich integral, the Poisson summation formula and Euler summation; the second, building on Jagerman (Jagerman, D. This is not usually so in the real world applications. edu is a platform for academics to share research papers. We will be able to handle more general right hand sides than up to now, in particular, impulse functions and step functions. Laplace Transform []. pptx), PDF File (. Note: There are two types of laplace transforms. txt) or view presentation slides online. 2 Chapter 3 Definition The Laplace transform of a function, f(t), is defined as 0 Fs() f(t) ftestdt (3-1) ==L ∫∞ − where F(s) is the symbol for the Laplace transform, Lis the Laplace transform operator, and f(t) is some function of time, t. The Laplace transform pair for. 6: Perform the Laplace transform of function F(t) = Sin3t. One doesn't need a transform method to solve this problem!! Suppose we solve the ode using the Laplace Transform Method. Laplace transform of: Variable of function: Transform variable: Calculate: Computing Get this widget. The vibrational analysis of structures use Laplace transforms. Roughly, differentiation of f(t) will correspond to multiplication of L(f) by s (see Theorems 1 and 2) and integration of. If you are preparing for GATE 2019 , you should use these free GATE Study Notes , to help you ace the exam. 1)issaidtoconverge. I (03/04) Br. e 2t cos(3t) + 5e 2t sin(3t) 4. Professor Deepa Kundur (University of Toronto)The z-Transform and Its. Because the transform is invertible, no information is lost and it is reasonable to think of a function ( ) and its Laplace transform ( ) as two views of the same phenomenon. [16] Xiang, Tan-yong, Guo,Jia-qi, A Laplace transform and Green function method for calculation of water flow and heat transfer in fractured rocks, Rock And Soil Mechanics, 32(2)(2011)333-340. Fourier Transform Applications. Then f(t) is called inverse Laplace transform of f (s) or simply inverse transform of fs ieL fs(). Topics covered under playlist of Laplace Transform: Definition, Transform of Elementary Functions, Properties of Laplace Transform, Transform of Derivatives and Integrals, Multiplication by t^n. If you are preparing for GATE 2019 , you should use these free GATE Study Notes , to help you ace the exam. Basically, a Laplace transform will convert a function in some domain into a function in another domain, without changing the value of the function. In this book, the author re-examines the Laplace Transform and presents a study of many of the applications to differential equations, differential-difference equations and the renewal equation. In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electrotechnics, mechanics etc. McLachlan, quicker you could enjoy checking out the publication. ’s) of waiting times in queues. Having carried out this procedure, we should check that this latter expression does, indeed, yield a solution of the original initial-boundary value problem. The Generalized solutions of differential equations are stated and theorems related to this are stated and proved. It transforms a function of a real variable t (often time) to a function of a complex variable s ( complex frequency ). pdf), Text File (. The de nition of Laplace transform and some applications to integer-order systems are recalled from [20]. Find PowerPoint Presentations and Slides using the power of XPowerPoint. The Laplace transform is defined from 0 to ∞. We provide some counterexamples where if the solution of differential equations exists by Laplace transform, the solution does not necessarily exist by using the Sumudu transform; however, the examples indicate that if the solution of differential equation by Sumudu transform. pdf), Text File (. Laplace Transforms and Properties. Spiegel as the bridge. 7 per cent and 21. In computer society, information security becomes more and more important for humanity and new emerging technologies are developing in an endless stream. possesses a Laplace transform. Solve for I1 and I2. Be-sides being a different and efficient alternative to variation of parame-ters and undetermined coefficients, the Laplace method is particularly advantageous for input terms that are piecewise-defined, periodic or im-pulsive. Inverse of a Product L f g t f s ĝ s where f g t: 0 t f t g d The product, f g t, is called the convolution product of f and g. Application. 16 Laplace transform. Download laplace transformation or read online books in PDF, EPUB, Tuebl, and Mobi Format. Advanced Mathematical Analysis: Periodic Functions and Distributions, Complex Analysis, Laplace Transform and Applications Author: Richard Beals Published by Springer New York ISBN: 978-0-387-90066-7 DOI: 10. As we saw in the last section computing Laplace transforms directly can be fairly complicated. This paper will be primarily concerned with the Laplace transform and its ap-plications to partial di erential equations. Laplace Transform The Laplace transform can be used to solve differential equations. By the way, the Laplace transform is just one of many "integral transforms" in general use. Laplace transform of ∂U/∂t. 'The Laplace Transform' is an excellent starting point for those who want to master the application of transform techniques to boundary value problems and thus provides a backdrop to Davies' Integral Transforms and Duffy's Transform Methods. Since the upper limit of the integral is , we must ask ourselves if the Laplace Transform, , even exists. When there is no interest in the explicit nature of this response, its determination in order to obtain the energy flow is an undesired labour. Denoted ℓ {f(t)}= dt, it is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms it to a function F(s) with a complex argument s. , time domain ) equals point-wise multiplication in the other domain (e. Laplace Transforms for Systems of Differential Equations. We perform perturbation expansion of the dressed thermal mass in the massive case to several orders and try the massless approximation with the help of modified Laplace. Download File PDF Applications Of Laplace Transform In Mechanical Engineering Applications Of Laplace Transform In Mechanical Engineering When somebody should go to the books stores, search creation by shop, shelf by shelf, it is in fact problematic. 𝑌 : ;= ̂ : ;=∫ − ′ ′ : ′ ; ′=∞ ′=0 ′ (1. Then L {f′(t)} = sF(s) f(0); L {f′′(t)} = s2F(s) sf(0) f′(0): Now. Recall the definition of hyperbolic functions. According to ISO 80000-2*), clauses 2-18. Let fbe a function of t. Consider an LTI system exited by a complex exponential signal of the form x(t) = Ge st. And, Hence, we have The Laplace-transformed differential equation is This is a linear algebraic equation for Y(s)! We have converted a. B & C View Answer / Hide Answer. Patil, Application of Laplace Transform, Global Journals Inc. The above form of integral is known as one sided or unilateral transform. Application to laplace transformation to electric circuits by J Irwin. Each view has its uses. 7 The Transfer Function and the Steady-State Sinusoidal Response. applications of transfer functions to solve ordinary differential equations. 2) 𝑅 for Z-transform in Example 2. The important differences between Fourier transform infrared (FTIR) and filter infrared (FIR) systems for air monitoring are explored, and the strengths and weaknesses of these technologies when applied to industrial hygiene problems are defined and illustrated with actual workplace air monitoring examples. Laplace transform gives information about steady as well as transient states. 3 Introduction to Laplace Transforms. Signals & Systems Flipped EECE 301 Lecture Notes & Video click her link A link B. The two-sided Laplace transform (3) can be regarded as the Fourier transform of the function , and the one-sided Laplace transform (2) can be regarded as the Fourier transform of the function equal to for and equal to zero for. The vibrational analysis of structures use Laplace transforms. Laplace Transform Example: Series RLC Circuit Problem. The Laplace transform is a widely used integral transform with many applications in physics and engineering. There is a focus on systems which other analytical methods have difficulty solving. Tejal Shah Assistant Professor in Mathematics, Department of Science & Humanity, Vadodara Institute of Engineering, Gujarat, India-----***-----Abstract - The Laplace Transform theory violets a. Using the Laplace Transform. A presentation on Laplace Transformation & Its Application Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Let us consider a beam of length l and uniform cross section parallel to the yz plane so that the normal deflection w(x,t) is measured downward if the axis of the beam is towards x axis. is , then the ROC for is. Since the m. (1975) Application of best rational function approximation for Laplace transform inversion. CONTENTS UNIT-7 LAPLACE TRANSFORMS Laplace Transforms of standard functions Inverse LT- First shifting Property Transformations of derivatives and integrals Unit step function, second shifting theorem Convolution theorem - Periodic function Differentiation and Integration of transforms Application of Laplace Transforms to ODE. This shows the effectiveness and usefulness of the Numerical Inversion of the Laplace transform. Table 1 - Laplace transform pairs When a simple analytical inversion is not possible, numerical inversion of a Laplace domain function is an alternate procedure. Let f(t) be de ned for t 0:Then the Laplace transform of f;which is denoted by L[f(t)] or by F(s), is de ned by the following equation L[f(t)] = F(s) = lim T!1 Z T 0 f(t)e stdt= Z 1 0 f(t)e stdt The integral which de ned a Laplace transform is an improper integral. Applications of the Laplace transform in solving partial differential equations. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. M-2 Shah Nisarg (130410119098) Shah Kushal(130410119094) Shah Maulin(130410119095) Shah Meet(130410119096) Shah Mirang(130410119097) Laplace Transform And Its Applications 2. THE BAD TRUTH ABOUT LAPLACE’S TRANSFORM CHARLES L. ISBN 978-953-51-0518-3, PDF ISBN 978-953-51-5685-7, Published 2012-04-25. Consider the differential equation given by: can represent many different systems. The Laplace transform can be viewed as an operator $${\cal L}$$ that transforms the function $$f=f(t)$$ into the function $$F=F(s)$$. Best & Easiest Videos Lectures covering all Most Important Questions on Engineering Mathematics for 50+ Universities Download Important Question PDF (Password mathcommentors) Will Upload soon. Russell Rhinehart, 2018-05-09 Preface One can argue to not teach students to derive or invert Laplace, or z-, or frequency transforms in the senior level process control course. Given the function U(x, t) defined for a x b, t > 0. Coming to prominence in the late 20thcentury after being popularized by a famous electrical engineer. The transform has many applications in science and engineering. Note: There are two types of laplace transforms. And how useful this can be in our seemingly endless quest to solve D. View and Download PowerPoint Presentations on Application Of Laplace Transform In Engineering PPT. Partial Differential Equations: Graduate Level Problems 8 Laplace Equation 31 Fourier Transform 365 31 Laplace Transform 385. Ifthelimitdoesnotexist,theintegral is said todivergeand there is no Laplace transform defined forf. Application of Laplace Transform. And this is extremely important to know. 3 Circuit Analysis in S Domain 12. s is a complex variable: s = a + bj, j −1. Let be the function of variable ,. zi denotes the zeros and pi denotes the poles of the linear time invariant system (LTI). As per my understanding the usage of the above transforms are: Laplace Transforms are used primarily in continuous signal studies, more so in realizing the analog circuit equivalent and is widely used in the study of transient behaviors of systems. Applications of Laplace Transform. To unlock. Bracewell starts from the very basics and covers the fundamental theorems, the FT, DFT, DTFT, FFT algorithms, dynamic spectra, z-transform (briefly), Hartley and Laplace transforms, and then moves to applications like Antennas and Optics, Heat, Statistics, Noise, and Acoustics. 12 Laplace transform 12. Find the inverse of each term by matching entries in Laplace Transform Table. The Laplace Transform of f prime, or we could even say y prime, is equal to s times the Laplace Transform of y, minus y of 0. Some illustrative examples will be discussed. The Laplace Transform and Its Application to Circuit Problems. The method is simple to describe. Laplace Transform The Laplace transform can be used to solve di erential equations. Laplace transform is named in honour of the great French mathematician, Pierre Simon De Laplace (1749-1827). com 1 1 College Mechanical Engineering, Sichuan University of Science & Engineering, , Zigong, China. In so doing, it also transforms the governing differential equation into an algebraic equation which is often easier to analyze. Compute the Inverse Laplace transform of symbolic functions. : a transformation of a function f(x) into the function {latex}g(t) = \int_{o}^{\infty}{e^{-xt}f(x)dx}{/latex} that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation. 1: The Laplace Transform The Laplace transform turns out to be a very efficient method to solve certain ODE problems. APPLICATION OF THE LAPLACE TRANSFORM TO CIRCUIT ANALYSIS LEARNING GOALS Laplace circuit solutions Showing the usefulness of the Laplace transform - A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow. 03 July 2019 (22:26) Post a Review. A novel method of determining Laplace inverse transform of a typical function using superposition technique is presented. Some illustrative examples will be discussed. 5#3 solve differential with laplace. 201038 Identifier-ark ark:/13960/t80k7s705 Ocr ABBYY FineReader 11. is obtained for the case of zero initial conditions. as the proba- bility that the co~~esponding random variable wins a race against (i-e. 74 Figure (5. Analysis of electrical and electronic circuits. It is then a matter of finding. 1) whenever the limit exists (as a finite number). Inverting the Laplace transform is a paradigm for exponentially ill-posed problems. THE LAPLACE TRANSFORM AND ITS APPLICATION TO CIRCUIT PROBLEMS. s is a complex variable: s = a + bj, j −1. Substitute f(t) into the definition of the Laplace Transform to get. However, to transform the obtained solutions from Laplace space back to the original space, we have used the Numerical Inversion of the Laplace transform. And how useful this can be in our seemingly endless quest to solve D. Patil & Vijaya N. The Laplace transform is frequently encountered in mathematics, physics, engineering and other areas. An inversion technique for the Laplace transform with applications. a b w(x,y) is the displacement in z-direction x y z. Be-sides being a different and efficient alternative to variation of parame-ters and undetermined coefficients, the Laplace method is particularly advantageous for input terms that are piecewise-defined, periodic or im-pulsive. Retrying Retrying. 4 Introduction In this Section we shall apply the basic theory of z-transforms to help us to obtain the response or output sequence for a discrete system. Introduction: Laplace transform Laplace transform is an integral transform method is particularly useful in solving. Suppose that f: [0;1) !R is a periodic function of period T>0;i. By continuing to use our website, you are agreeing to our use of cookies. 1)issaidtoconverge. 2 Useful Laplace Transform Pairs 12. The important differences between Fourier transform infrared (FTIR) and filter infrared (FIR) systems for air monitoring are explored, and the strengths and weaknesses of these technologies when applied to industrial hygiene problems are defined and illustrated with actual workplace air monitoring examples. pptx), PDF File (. The control action for a dynamic control system whether electrical, mechanical, thermal, hydraulic, etc. Keywords: Laplace Transform: Beam-Column: Present. • All we need is to express F(s) as a sum of simpler functions of the forms listed in the Laplace transform table. Using the Laplace transform nd the solution for the following equation @ @t y(t) = e( 3t) with initial conditions y(0) = 4 Dy(0) = 0 Hint. There is a focus on systems which other analytical methods have difficulty solving. This paper will be primarily concerned with the Laplace transform and its ap-plications to partial di erential equations. f(t+ T) = f(t) for all t 0. 0 Year 2012. 1 p344 PYKC 24-Jan-11 E2. Since the m. The Laplace transform is a widely used integral transform (transformation of functions by integrals), similar to the Fourier transform. These theorems are applied to most commonly used special functions to obtain many new two and three dimensional Laplace transform pairs from known one and two dimensional Laplace transforms. Applications of the Laplace Transform - Free download as PDF File (. Laplace Transform of tf(t) The video presents a simple proof of an result involving the Laplace transform of tf(t). [PDF] The Laplace Transform: Theory and Applications By Joel L. When we apply Laplace transforms to solve problems we will have to invoke the inverse transformation. We get Hence, we have. The Laplace transform pair for. With its success, however, a certain casualness has been bred concerning its application, without much regard for hypotheses and when they are valid. The transform replaces a differential equation in y(t) with an algebraic equation in its transform ˜y(s). The differential equations must be IVP's with the initial condition (s) specified at x = 0. We are interested in occupation times of Lévy processes with jumps rational Laplace transforms. The real and imaginary parts of s can be considered as independent quantities. Therefore, without further discussion, the Laplace transform is given by: De nition 1. s is a complex variable: s = a + bj, j −1. The numerical inversion of the Laplace transform was introduced in the 60s by Bellman et al. Applications of the Laplace Transform - Free download as PDF File (. Abstract - The present discounted value equation in finance has a broad range of uses and may be applied to various areas of finance including corporate finance, banking finance and. The transform and the corresponding inverse transform are defined as follows: A complete description of the transforms and inverse transforms is beyond the scope of this article. Solve differential equations by using Laplace transforms in Symbolic Math Toolbox™ with this workflow. And, Hence, we have The Laplace-transformed differential equation is This is a linear algebraic equation for Y(s)! We have converted a. 99 USD for 2 months 4 months:. Professor Deepa Kundur (University of Toronto)The z-Transform and Its. Schaum's Outlines: Laplace Transforms By Murray R. Equation 3. Each view has its uses. 10 + 5t+ t2 4t3 5. Spiegel, currently you could not also do conventionally. 1) Direct-form realization of FIR systems. In this case, there is a probabilistic interpretation of the Laplace transform. 17 Applications of Fourier Transforms in Mathematical Statistics 103 2. EPSTEIN∗ AND JOHN SCHOTLAND† Abstract. Acces PDF Laplace Transform In Engineering Mathematics Differential Equation, Ex 1, Part 1/2 Thanks to all of you who support me on Patreon. Let f(t) be de ned for t 0:Then the Laplace transform of f;which is denoted by L[f(t)] or by F(s), is de ned by the following equation L[f(t)] = F(s) = lim T!1 Z T 0 f(t)e stdt= Z 1 0 f(t)e stdt The integral which de ned a Laplace transform is an improper integral. In particular, the transform can take a differential equation and turn it into an algebraic equation. 1) whenever the limit exists (as a finite number). The function is piecewise discrete D. Tejal Shah Assistant Professor in Mathematics, Department of Science & Humanity, Vadodara Institute of Engineering, Gujarat, India-----***-----Abstract - The Laplace Transform theory violets a. Then the convolution of fand g, denoted by fg, is de ned by (fg)(t) = Z 1 0 f(˝)g(t ˝)d˝ (2). (1975) Application of best rational function approximation for Laplace transform inversion. Fourier series Periodic x(t) can be represented as sums of complex exponentials x(t) periodic with period T0 Fundamental (radian) frequency!0 = 2ˇ=T0 x(t) = ∑1 k=1 ak exp(jk!0t) x(t) as a weighted sum of orthogonal basis vectors exp(jk!0t) Fundamental frequency!0 and its harmonics ak: Strength of k th harmonic Coefficients ak can be derived using the relationship ak =. 03 July 2019 (22:26) Post a Review. L(sin(6t)) = 6 s2 +36. : a transformation of a function f(x) into the function {latex}g(t) = \int_{o}^{\infty}{e^{-xt}f(x)dx}{/latex} that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation. The Laplace Transform of f prime, or we could even say y prime, is equal to s times the Laplace Transform of y, minus y of 0. The Laplace transform is a linear operation, so the Laplace transform of a constant (C) multiplying a time-domain function is just that constant times the Laplace transform of the function, Equation 3. • All we need is to express F(s) as a sum of simpler functions of the forms listed in the Laplace transform table. The Laplace transforms of difierent functions can be found in most of the mathematics and engineering books and hence, is not included in this paper. Shahrul Naim Sidek ; Department of Mechatronics Engineering. Application of k-Laplace transform to estimate the time value of money in quantitative finance V. e I(s)) using all linear circuit techniques such as: Third Inverse back , to obtain the time domain variable i(t) OHM , KVL, KCL , VDR, CDR, Thavenin, source transformation , Nodal and Mesh + − it() R =1 Ω t =0 1 10 LH= 5 8. Fourier transforms only capture the steady state behavior. Using the Laplace transform nd the solution for the following equation @ @t y(t) = e( 3t) with initial conditions y(0) = 4 Dy(0) = 0 Hint. Anyone needing more information can refer to the "bible" of numerical mathematics,. 6 The Laplace transform. Best & Easiest Videos Lectures covering all Most Important Questions on Engineering Mathematics for 50+ Universities Download Important Question PDF (Password mathcommentors) Will Upload soon. Applications of the Laplace Transform - Free download as PDF File (. The Nature of the s-Domain; Strategy of the Laplace Transform; Analysis of Electric Circuits; The Importance of Poles and Zeros; Filter Design in the s-Domain. Short-time Fourier transform. The Inverse Laplace Transformation Circuit Analysis with Laplace Transforms Frequency. The Mellin transform and its inverse are related to the two-sided laplace transform by a simple change of variables. Be careful when using "normal" trig function vs. 1 Circuit Elements in the s Domain. k53rbom2ffbn 44dzm0u1wzvxzfl vt2bhrt20zig x1ruxgtc99drpm 5lsiu2xanun0cw ilgv70jy2us5nr dda5bohgk3qyx cblkxm55dntnl tbae5byzp5m slk2cpo5l3h lhx6n5me92 shcro7se9en mrocxihbnc9 sqksudzq3x y78qw0cj1qr8e96 vjhzh0wud0liyp i60kjwkq1enk53y qfco5l311e6gk kb0forgk6v ecicrzmvrhd cqxqiahvmh jeh0xggr471ktcd nbuv7wfvyvrj 9g1pmazt6j7ll0 h4xmkhpxc8 p94okv2uty c5pm1g1s2x 6wlt5qn64rkn21 5col5ciobbok csm8z2cwpru 6x7vgdbrz9uu rphgtvhmief9 p9ksmwy3pufa9j

https://math.stackexchange.com/questions/2291594/a-summation-question-is-s-to-ln-2

# A summation question ,Is $S \to \ln 2$? I am in doubt with this question . let $S=\dfrac{\dfrac12}{1} +\dfrac{(\dfrac12)^2}{2}+\dfrac{(\dfrac12)^3}{3}+\dfrac{(\dfrac12)^4}{4}+\dfrac{(\dfrac12)^5}{5}+...$ Is it converge to $\ln 2$ ? I tried this $$x=\dfrac12 \to 1+x+x^2+x^3+x^4+...\sim\dfrac{1}{1-x}\to 2$$ by integration wrt x we have $$\int (1+x+x^2+x^3+x^4+...)dx=\int (\dfrac{1}{1-x})dx \to\\ x+\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dfrac{x^4}{4}+...=-\ln(1-x)$$then put $x=0.5$ $$\dfrac{\dfrac12}{1} +\dfrac{(\dfrac12)^2}{2}+\dfrac{(\dfrac12)^3}{3}+\dfrac{(\dfrac12)^4}{4}+\dfrac{(\dfrac12)^5}{5}+..\sim -\ln(0.5)=\ln 2$$ now my question is : Is my work true ? I am thankful for you hint,guide,idea or solutions. (I forgot some technics of calculus) • Looks good to me! :) – Juanito May 22 '17 at 6:32 • It is amazing that rewriting $$S=\dfrac12 +\dfrac{(\dfrac12)^2}{2}+\dfrac{(\dfrac12)^3}{3}+\dfrac{(\dfrac12)^4}{4}+...=\dfrac{(\dfrac12)^1}{1}+\dfrac{(\dfrac12)^2}{2}+\dfrac{(\dfrac12)^3}{3}+\dfrac{(\dfrac12)^4}{4}+...$$ makes the problem nicer. – Claude Leibovici May 22 '17 at 7:16 For completeness, you shoud show that the series for $1/(1-x)$ is uniformly convergent in $[0,1/2]$. As $$\left|S_n-\frac1{1-x}\right|=\left|\frac{1-x^{n+1}}{1-x}-\frac1{1-x}\right|=\left|\frac{x^{n+1}}{1-x}\right|\le\left|\frac1{2^n}\right|$$ this is ensured and you can integrate term-wise. Yes. This series was already known to Jacob Bernoulli (Gourdon and Sebah http://plouffe.fr/simon/articles/log2.pdf, formula 14) and can be written $$S=\sum_{k=0}^\infty \frac{1}{k+1}\left(\frac{1}{2}\right)^{k+1}$$ To evaluate it, we can change the identity $$\int_0^1 x^n dx = \frac{1}{n+1}$$ into $$\int_0^\frac{1}{2} x^n dx = \frac{1}{n+1}\left(\frac{1}{2}\right)^{n+1}$$ and then \begin{align} S&=\sum_{k=0}^\infty \frac{1}{k+1}\left(\frac{1}{2}\right)^{k+1}\\ &=\sum_{k=0}^\infty \int_0^\frac{1}{2} x^k dx \\ &=\int_0^\frac{1}{2}\left(\sum_{k=0}^\infty x^k\right) dx\\ &=\int_0^\frac{1}{2} \frac{1}{1-x} dx\\ &=-\log(1-x)|_0^\frac{1}{2}\\ &=\log(2) \\ \end{align} A nice way to encode formulas like $$\log(2)=\sum_{n=1}^\infty \frac{1}{n2^n}$$ is noting that the numerator is one so the sequence of integer denominators can represent the series. When we search the OEIS for $2,8,24,64$ (http://oeis.org/A036289) we find that $$\sum_{n=1}^\infty \frac{1}{a(n)} = \log(2)$$ is one of the formulas given. Your series is a base-2 BBP-type formula for $\log(2)$. The base-3 version is $$\log(2)=\frac{2}{3} \sum_{k=0}^\infty \frac{1}{(2k+1)9^k}$$ and the sequence of denominators is OEIS http://oeis.org/A155988. Your work is good, but it can be better justified with the theory of power series. The given series is an instance of the power series $$f(x)=\sum_{k=1}^{\infty}\frac{x^k}{k}$$ for $x=1/2$. The power series has convergence radius $1$; indeed, the ratio test gives $$\left|\frac{x^{k+1}/(k+1)}{x^k/k}\right|=\frac{k}{k+1}|x|\to |x|$$ Thus you know your series converges for $|x|<1$. The function $f$, defined over $(-1,1)$, is differentiable and $$f'(x)=\sum_{k=1}^{\infty}x^{k-1}=\sum_{k=0}^{\infty} x^k=\frac{1}{1-x}$$ (geometric series). Since $f(0)=0$, you can conclude that, for $x\in(-1,1)$, $$f(x)=\int_0^x\frac{1}{1-t}\,dt=-\ln(1-x)$$ Therefore $$f(1/2)=-\ln\Bigl(1-\frac{1}{2}\Bigr)=\ln2$$

https://scicomp.stackexchange.com/questions/21867/second-order-derivative-condition-for-convexity/21872

Second-order derivative condition for convexity It is written in a book I'm reading that $$\nabla f(x) = \left( \frac{\partial f(x)}{\partial x_1}, \frac{\partial f(x)}{\partial x_2},...,\frac{\partial f(x)}{\partial x_n}\right)$$ and $$\nabla^2 f(x)_{ij} = \frac{\partial^2 f(x)}{\partial x_i ~\partial x_j}, \qquad \forall i,j=1,...,n.$$ According to 2nd-order conditions: for twice differentiable function $f$, it is convex if and only if $$\nabla^2 f(x) \ge 0, \qquad \forall x \in \mathrm{dom} f.$$ But, the function $f(x,y) = \sqrt{x^2+y^2}$ is convex, but does not meet 2nd-order conditions: \begin{aligned} \frac{\partial^2 }{\partial x^2} \sqrt{x^2+y^2} &= \frac{y^2}{(x^2+y^2)^{\frac{3}{2}}} \ge 0,\\ \frac{\partial^2 }{\partial x ~ \partial y} \sqrt{x^2+y^2} &= - \frac{x y}{(x^2+y^2)^{\frac{3}{2}}} \le 0. \end{aligned} Can anyone explain this? • This notation is a bit misleading -- for a matrix $A\in\mathbb{R}^{n\times n}$, writing $A\geq 0$ usually does not mean that all entries $a_{ij}$ are positive, but that the matrix is positive (semi-)definite, i.e., $x^TAx\geq 0$ for all $x\in\mathbb{R}^n$. – Christian Clason Jan 20 '16 at 12:51 • Do you mean the matrix $D$ must be positive s.t. $det(D) \ge 0$. $D= [\frac{\partial^2 f}{\partial x^2},\frac{\partial^2 f}{\partial x \partial y}; \frac{\partial^2 f}{\partial y \partial x},\frac{\partial^2 f}{\partial y^2} ]$ – Kevin Jan 20 '16 at 13:30 • Not quite -- all eigenvalues must be positive (and real) (which is only sufficient, not necessary, for the determinant to be positive). Put another way, $det(A)\geq 0$ is only necessary, not sufficient for convexity. – Christian Clason Jan 20 '16 at 13:31 • But how to check the eigenvalues for matrix $D$, since it is not formed by real values. – Kevin Jan 20 '16 at 13:33 • By Sylvester's criterion a $2\times2$ matrix is p.d. iff $A_{11}>0$ and $\det A>0$. That's usually easier than computing eigenvalues. – Kirill Jan 20 '16 at 16:53 Consolidating my comments (so that they can be cleaned up): This is a misunderstanding. A twice (continuously!) differentiable function $f:\mathbb{R}^n\to \mathbb{R}$ is convex if and only if the Hessian $\nabla^2 f(x)\in\mathbb{R}^{n\times n}$ is positive semi-definite at every $x\in \mathbb{R}^n$. (This definition makes sense since the Hessian is symmetric by Schwarz' theorem if the second derivatives are continuous.) This is sometimes written as $$\nabla^2 f(x) \succeq 0 \qquad\text{for all } x\in\mathbb{R}^n$$ (and more rarely -- since it can lead to misunderstandings -- as $\nabla^2 f(x)\geq 0$). As @nicoguaro points out in his answer, this is equivalent to the condition that all eigenvalues of $\nabla^2 f(x)$ -- as a function of $x$ -- are nonnegative for every $x\in \mathbb{R}^n$. An equivalent (and often easier to verify, especially for large $n$) condition is that $$d^T\nabla^2 f(x)d \geq 0 \qquad\text{for all } d\in\mathbb{R}^n \text{ and }x\in\mathbb{R}^n.$$ (This condition is also easier to work with if you want to rule out convexity: It's sufficient to find a single $d$ such that $d^T \nabla^2 f(x) d<0$.) In your example (with $x_1 = x$ and $x_2 = y$), this would yield \begin{aligned} \begin{pmatrix} d_1 & d_2 \end{pmatrix} \begin{pmatrix} \frac{x_2^2}{(x_1^2 + x_2^2)^\frac{3}{2}} & \frac{-x_1\,x_2}{(x_1^2 + x_2^2)^\frac{3}{2}} \\ \frac{-x_1\,x_2}{(x_1^2 + x_2^2)^\frac{3}{2}} & \frac{x_1^2}{(x_1^2 + x_2^2)^\frac{3}{2}} \end{pmatrix} \begin{pmatrix} d_1 \\ d_2 \end{pmatrix} &= \frac{1}{(x_1^2 + x_2^2)^\frac{3}{2}}\left(d_1^2x_2^2 - 2 d_1 x_1x_2d_2 + d_2^2x_1^2\right)\\ &= \frac{1}{(x_1^2 + x_2^2)^\frac{3}{2}}\left(d_1x_2-d_2x_1\right)^2\\ &\geq 0 \end{aligned} for all $x,d\in\mathbb{R}^n$. Hence, $f$ is convex. The comments already mention what you are not considering. For the particular example you mention we have $$\nabla^2 f(x,y) = \begin{pmatrix} \frac{y^2}{(x^2 + y^2)^\frac{3}{2}} & \frac{-x\,y}{(x^2 + y^2)^\frac{3}{2}}\\ \frac{-x\,y}{(x^2 + y^2)^\frac{3}{2}} & \frac{x^2}{(x^2 + y^2)^\frac{3}{2}} \end{pmatrix}$$ And the eigenvalues are $0$ and $\frac{1}{\sqrt{x^2 + y^2}}$. That are greater or equal to zero for all $x,y \in \mathbb{R}^+$.

http://mathhelpforum.com/pre-calculus/62653-negative-exponents.html

Math Help - Negative exponents 1. Negative exponents Hey guys. I know that x^-2 is equal to 1/x^2 But, this problem looks like this. It asks me to simplify this (AKA get rid of the negative exponents): ((x^-1)+(y^-1))/((x^-2)-(y^-2)) For some reason I don't think I can just flip the problem around to make it look like this ((x^2)-(y^2))/((x^1)+(y^1)) But if I can, can you let me know why? Thanks guys! 2. Hello, No you can't, because it would mean that : $\frac{1}{x^{-2}-y^{-2}}=x^2-y^2$, which is similar to saying that $\frac{1}{a-b}=\frac 1a-\frac 1b$ which are both false. So let's write your problem : $\frac{x^{-1}+y^{-1}}{x^{-2}-y^{-2}}$ You can see that $x^{-2}=(x^{-1})^2$ and $y^{-2}=(y^{-1})^2$ Can you see a difference of two squares below ??? $=\frac{x^{-1}+y^{-1}}{(x^{-1}-y^{-1})(x^{-1}+y^{-1})}=\frac{1}{x^{-1}-y^{-1}}=\frac{1}{\frac 1x-\frac 1y}=\dots$ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Another way : eliminate the negative coefficients. In order to do that, multiply by $\frac{x^2y^2}{x^2y^2}$ : $=\frac{xy^2+yx^2}{y^2-x^2}=\frac{xy(y+x)}{(y-x)(y+x)}=\frac{xy}{y-x}$ 3. This is another problem that needs to be simplified. It's going to look messy but I'll type it out right. Can someone tell me how to make it look like it does on the paper? Anyways: EDIT: sqrt = square root (((2x)/(sqrt(x-1)))-sqrt(x-1))/(x-1) Again sorry for the sloppiness >_< 4. Originally Posted by DHS1 This is another problem that needs to be simplified. It's going to look messy but I'll type it out right. Can someone tell me how to make it look like it does on the paper? Anyways: (((2x)/(sqrt(x-1)))-sqrt(x-1))/(x-1) Again sorry for the sloppiness >_< You'll have to tell me if this is what you have coded: $\dfrac{\dfrac{2x}{\sqrt{x-1}}-\sqrt{x-1}}{x-1}$ 5. Originally Posted by masters You'll have to tell me if this is what you have coded: $\dfrac{\dfrac{2x}{\sqrt{x-1}}-\sqrt{x-1}}{x-1}$ Yeah it is. Did you actually type \dfrac{\dfrac{2x}{\sqrt{x-1}}-\sqrt{x-1}}{x-1} between the math brackets? Or do you use a program and then copy/paste or something o_O; Thanks. 6. Originally Posted by DHS1 Yeah it is. Did you actually type \dfrac{\dfrac{2x}{\sqrt{x-1}}-\sqrt{x-1}}{x-1} between the math brackets? Or do you use a program and then copy/paste or something o_O; Thanks. Just type it in just like that. You can go here for a tutorial on this tool. $\dfrac{\dfrac{2x}{\sqrt{x-1}}-\sqrt{x-1}}{x-1}\cdot \dfrac{\sqrt{x-1}}{\sqrt{x-1}}=\dfrac{2x-(x-1)}{(x-1)(\sqrt{x-1})}=\dfrac{2x-x+1}{(x-1)(\sqrt{x-1})}=$ $\dfrac{x+1}{(x-1)(\sqrt{x-1})} \cdot \dfrac{\sqrt{x-1}}{\sqrt{x-1}}=\dfrac{(x+1)(\sqrt{x-1})}{(x-1)^2}$ 7. Wow great! I'm learning so much. Last one I'll ask for help with, I promise! Directions: Re-write each expression as a single fraction in lowest terms. $\frac{3}{a}+\frac{2}{a^2}-\frac{2}{a-1}$ EDIT: My problem is I don't know how to find out what the common denominator is. How can I go about doing that? Thanks! 8. Originally Posted by DHS1 Wow great! I'm learning so much. Last one I'll ask for help with, I promise! Directions: Re-write each expression as a single fraction in lowest terms. $\frac{3}{a}+\frac{2}{a^2}-\frac{2}{a-1}$ EDIT: My problem is I don't know how to find out what the common denominator is. How can I go about doing that? Thanks! how to find out what the common denominator is 1. Factorize each denominator. 2. The common denominator must contain the original denominators as a factor: $d_1=a$ $d_2=a \cdot a$ $d_3=(a-1)$ Thus the common denominator is $D=a\cdot a\cdot (a-1)$ $\dfrac{3}{a}+\dfrac{2}{a^2}-\dfrac{2}{a-1} = \dfrac{3a(a-1)}{a^2(a-1)}+\dfrac{2(a-1)}{a^2(a-1)}-\dfrac{2a^2}{a^2(a-1)} =$ $\dfrac{3a^2-3a+2a-2-2a^2}{a^2(a-1)}= \dfrac{a^2-a-2}{a^2(a-1)} = \dfrac{(a-2)(a+1)}{a^2(a-1)}$ which can't be simplified any more. By the way: Do yourself and do us a favour and start a new thread if you have a new question. Otherwise you'll risk that nobody will notice that you are in need of some help. 9. Originally Posted by earboth how to find out what the common denominator is 1. Factorize each denominator. 2. The common denominator must contain the original denominators as a factor: $d_1=a$ $d_2=a \cdot a$ $d_3=(a-1)$ Thus the common denominator is $D=a\cdot a\cdot (a-1)$ $\dfrac{3}{a}+\dfrac{2}{a^2}-\dfrac{2}{a-1} = \dfrac{3a(a-1)}{a^2(a-1)}+\dfrac{2(a-1)}{a^2(a-1)}-\dfrac{2a^2}{a^2(a-1)} =$ $\dfrac{3a^2-3a+2a-2-2a^2}{a^2(a-1)}= \dfrac{a^2-a-2}{a^2(a-1)} = \dfrac{(a-2)(a+1)}{a^2(a-1)}$ which can't be simplified any more. By the way: Do yourself and do us a favour and start a new thread if you have a new question. Otherwise you'll risk that nobody will notice that you are in need of some help. You, sir, have been thanked.

https://math.stackexchange.com/questions/1653388/how-many-ways-we-can-choose-items-from-different-boxes

# How many ways we can choose items from different boxes I searched through the internet but couldn't find my answer, which can either be a very simple or a hard one. Assume there are $3$ boxes, which carry, respectively, $1$, $4$, $2$ items. My question is how many ways we can select $3$ items from these boxes. I am looking for a formula rather than a solution for these specific values. If I choose (take away) $3$ items by trying one by one. \begin{array}{c c c} 0 & 3 & 1\\ 0 & 2 & 2\\ 0 & 4 & 0\\ 1 & 1 & 2\\ 1 & 3 & 0\\ 1 & 2 & 1 \end{array} Items remain each time, so the answer seems to be $6$ different ways. But I am not sure. • Yes there are total 7 items. – Rockybilly Feb 13 '16 at 16:24 • The items are not distict. So what decides how many ways are there, is actually how many items remain in each box. – Rockybilly Feb 13 '16 at 16:29 • Are the items in each box identical? If so, you're talking about the number of sums $a_1+a_2+a_3=3$ where $0\leq a_1\leq 1$, $0\leq a_2\leq 4$, and $0\leq a_3\leq 2$. Sums of this form are standard in combinatorics. – Michael Burr Feb 13 '16 at 16:29 • Yes, Michael. I guess that was the thing I was trying to convert my problem into. But if a1, a2, a3 are the items remain in each box, the sum must be 4. If not, it is correct. – Rockybilly Feb 13 '16 at 16:30 • I think the general problem you are after is this. We have $k$ boxes, labelled $1$ to $k$. The number of items in Box $i$ is $a_i$, given. We are also given a number $n$ of items we must choose. We want to find the number of solutions of $x_1+\cdots+x_k=n$ in non-negative integers, with the restriction that $x_i\le a_i$. How many ways $w_n$ are there to do this? Not a simple problem! One can easily write down a generating function for the $w_n$, but computing the coefficients is messy. – André Nicolas Feb 13 '16 at 16:39 Observe that since the items are identical, it does not matter that there are $4$ items in the second box. Your are then asking for the number of sums $a_1+a_2+a_3=3$ where $0\leq a_1\leq 1$, $0\leq a_2\leq 3$, and $0\leq a_3\leq 2$. I will give three answers. First, an elementary argument: We know that $a_1=0$ or $a_1=1$. If $a_1=0$, then $a_2+a_3=3$. In this case, there are three possibilities: $3+0=3$, $2+1=3$, and $1+2=3$. If $a_1=1$, then $a_2+a_3=2$ and there are still three possibilities: $2+0=2$, $1+1=2$, and $0+2=2$. This results in $6$ different options. A more combinatorial argument: The number of ways to write $n$ as a sum of $k$ nonnegative integers is $$\binom{n+k-1}{k-1}$$ and a discussion can be found here. So, in this case, the number of ways that $a_1+a_2+a_3=3$ (without restrictions) is $$\binom{3+3-1}{3-1}=\binom{5}{2}=10.$$ This, however, counts too many possible sums. Suppose that we take too many from box $1$, this means that we take at least $2$ from box $1$. In this case, we can write $a_1=2+b_1$ where $b_1$ is nonnegative. Then, the initial sum becomes $b_1+a_2+a_3=1$. Using the same formula, this results in $$\binom{1+3-1}{3-1}=\binom{3}{2}=3$$ impossible ways. Continuing, there is no way to take too many objects from the second box, but it is possible to take too many objects from box $3$. In this case, one must take $3$ objects from box $3$, so we write $a_3=3+b_3$ where $b_3$ is nonnegative. This results in the equation $a_1+a_2+(3+b_3)=3$. There are $$\binom{0+3-1}{3-1}=1$$ ways for this sum to occur. We should now use the inclusion/exclusion principle to see if we've over-counted. This could happen if we take more than $1$ item from box $1$ and more than $2$ items from box $3$. Then, we have $(2+b_1)+b_2+(3+b_3)=3$, but this has no solutions as a sum of nonnegative integers cannot be negative. Therefore, out of the original $10$ possibilities, $3+1=4$ are impossible, leaving the $6$ that we've found. A dynamic programming-type solution: Let $N(b,s)$ be the number of ways to use the first $b$ boxes to sum to $s$. Also, write $m_i$ for the number of objects in box $i$. In your case: \begin{align*} N(1,0)&=1\\ N(1,1)&=1\\ N(1,2)&=0\\ N(1,3)&=0. \end{align*} Then, the values in the second box can be computed as follows: $$N(b+1,s)=\sum_{i=0}^{\min\{s,m_{b+1}\}}N(b,s-i).$$ Using this formula: \begin{align*} N(2,0)&=N(1,0)=1\\ N(2,1)&=N(1,0)+N(1,1)=2\\ N(2,2)&=N(1,0)+N(1,1)+N(1,2)=2\\ N(2,3)&=N(1,0)+N(1,1)+N(1,2)+N(1,3)=2. \end{align*} Continuing for the third column, \begin{align*} N(3,0)&=N(2,0)=1\\ N(3,1)&=N(2,0)+N(2,1)=3\\ N(3,2)&=N(2,0)+N(2,1)+N(2,2)=5\\ N(3,3)&=N(2,1)+N(2,2)+N(2,3)=6. \end{align*} We are interested in the value $N(3,3)=6$. • You are right, however I am currently constructing a computer program where I will deal with much bigger inputs. So I need a formula. These was just the values I came up to explain the situation. – Rockybilly Feb 13 '16 at 16:36 • The third solution should work quite quickly on a computer (you just need to run through a pair of arrays many times). – Michael Burr Feb 13 '16 at 17:08 • Forgive my dullness, but I didn't understand your last solution. How $$N(b, s)$$ is defined and how $$N(3,2) = 2$$ and $$N(3,2) = 5$$ in your explanation. – Rockybilly Feb 13 '16 at 17:10 • Cut and paste error! – Michael Burr Feb 13 '16 at 17:13 • For the calculation of $N(b+1,s)$: How many objects can be taken from the $b+1$st box? At most $s$ or the number of objects in the box. If you take $0$ objects from the $b+1$st box, then the objects must come from the remaining $b$ boxes, so you have $N(b,s)$ ways. If you take $1$ object from the $b+1$st box, then the remaining $s-1$ objects must come from the remaining $b$ boxes, so you have $N(b,s-1)$ ways. Continue adding these up until you run out of objects in the box or take $s$ objects from box $b+1$. – Michael Burr Feb 13 '16 at 17:17 Here is the code that I came up with in Sage. It takes about 19 minutes that is MUCH larger than the one initially posed (1000 boxes with random values between 1 and 1000). But 100 boxes with values between 1 and 100 finished under a second. Also, if you need a particular maximum value, just change $s$ to that value. Boxes = [randint(1,100) for i in range(1000)] n = len(Boxes) s = sum(Boxes) l1 = [0] * (s+1) l2 = [0] * (s+1) parity = 0 for i in range(Boxes[0]+1): l1[i]=1 for i in range(1,n): if(parity == 0): l2[0]=l1[0] for j in range(1,Boxes[i]+1): l2[j]=l2[j-1]+l1[j] for j in range(Boxes[i]+1,s+1): l2[j]=l2[j-1]+l1[j]-l1[j-Boxes[i]-1] parity = 1 else: l1[0]=l2[0] for j in range(1,Boxes[i]+1): l1[j]=l1[j-1]+l2[j] for j in range(Boxes[i]+1,s+1): l1[j]=l1[j-1]+l2[j]-l2[j-Boxes[i]-1] parity = 0 if(parity == 1): l = l2 else: l = l1 print l Edit: I cut out one of the loops, reducing the complexity. • Much better indeed. However, it is still not enough for the method I need. If I had found a solution in here, I would have answered this question much better. – Rockybilly Feb 14 '16 at 18:00 • Using a generating function is a nice method. I should have thought of that! (Although the code above is (more or less) computing the coefficients of the generating function). – Michael Burr Feb 14 '16 at 18:05 • Yes, indeed. They are quite similar methods. – Rockybilly Feb 14 '16 at 18:11

https://math.stackexchange.com/questions/3401393/are-there-a-finite-number-of-trees-with-k-leaves-and-no-vertices-of-degree-2/3401420

# Are there a finite number of trees with $k$ leaves and no vertices of degree $2$? Given a fixed positive integer $$k$$. Show that there are only finitely many trees containing $$k$$ leaves and zero vertices of degree $$2$$. I tried to use the theorem related to rooted trees and tried to prove it by contradiction, but could not quite use the condition of having zero vertices of degree $$2$$. Is it even related to rooted trees theorem or can it be proved by contradiction? Can we use the fact that a tree with $$m$$ edges has $$m+1$$ vertices? Consider an arbitrary tree with $$k$$ leaves and no vertices of degree $$2$$. Let $$\epsilon$$ represent the number of edges and $$\nu_j$$ represent the number of vertices with a degree of $$j$$ (so that $$\nu_1=k$$ and $$\nu_2=0$$). We know that $$\sum_{j=1}^\infty j\nu_j=2\epsilon$$ (since that's true for all graphs) and $$\sum_{j=1}^\infty \nu_j=\epsilon+1$$ (since it's a tree). So we can combine the equations as follows (where the index of the summations have been taken out). $$\sum j\nu_j=2\big(\sum\nu_j-1\big)=2\sum \nu_j-2\\ 2\sum\nu_j-\sum j\nu_j=2\\\sum(2-j)\nu_j=2$$ The RHS is positive, so the LHS must be as well. The first few terms of the LHS is $$\nu_1+0\nu_2-\nu_3-2\nu_4-\ldots.$$ This could not be positive if $$\nu_3+\nu_4+\nu_5+\ldots>\nu_1$$. Since we know that $$\nu_2=0$$, we can conclude that a tree with $$k$$ leaves and no vertices of degree $$2$$ cannot have more than $$2k$$ vertices. Since the number of trees with fewer than $$2k$$ vertices is finite, we are done. • This is very smart! I tried theorem related to rooted trees and tried proving by contradiction but could not quite use the condition of no vertices of degree 2. Oct 20, 2019 at 13:40 • @AnsonNG It's a good trick to have in your toolbox. I learned it when proving that there was an upper bound on the number of faces of a polyhedron with a given number of pentagonal faces and no hexagonal faces. Same argument except with Euler's formula instead of the double-edge count. – user694818 Oct 20, 2019 at 13:48 This is easy to prove using the following two lemmata: Lemma 1: The sum of the degrees of all vertices in a graph equals twice the number of edges. Lemma 2: A tree with $$n$$ vertices has $$n-1$$ edges. (The first lemma is a simple consequence of the definition of the degree of a vertex, i.e. the number of edges connected to it, and the fact that each edge connects to exactly two vertices. The second lemma can be proved by induction on the number of vertices: assuming that the lemma holds for all trees with $$n-1$$ vertices, take any tree with $$n$$ vertices and consider what happens when you merge any two adjacent vertices and remove the edge between them.) Taken together these lemmata imply that, for any tree with $$n$$ vertices having the degrees $$d_1, d_2, \dots, d_n$$ respectively, $$\sum_{i=1}^n d_i = 2n - 2 \quad \text{and thus} \quad \sum_{i=1}^n (d_i - 2) = -2.$$ In other words, the sum of the degrees of all vertices minus two per vertex is the same (and equal to $$-2$$) for all trees! In particular, we can see that the summand $$d_i - 2$$ is negative (and equal to $$-1$$ except for the degenerate case of the single-vertex tree) for leaves, zero for vertices of degree $$2$$ and positive (at at least one) for all other vertices. For the sum to equal $$-2$$, as it must, the positive contribution of each vertex with degree $$d_i > 2$$ must therefore be cancelled out by at least one leaf (and there need to be at least two extra leaves on top of that). Thus, a tree with $$k$$ leaves can have at most $$k - 2$$ vertices of degree greater than $$2$$. For a tree with $$k$$ leaves and no vertices of degree $$2$$, this implies that the total number of vertices in the tree can be at most $$2k - 2$$. And since the number of vertices in such a tree is thus bounded, and since there's only a finite number of possible ways of connecting any given number of vertices into a tree, this further implies that the total number of such trees is also bounded. We can also see that, for this result to hold, it is essential that the number of vertices of degree $$2$$ be bounded (in your case by zero). Otherwise we could take any tree with $$n > 1$$ nodes and easily construct an infinite family of trees with the same number of leaves just by taking any pair of adjacent vertices and inserting an arbitrarily long linear chain of vertices of degree $$2$$ between them.

http://mathematica.stackexchange.com/questions/39848/probability-of-the-sum-of-the-largest-n-samples?answertab=votes

# Probability of the sum of the largest n samples 15 numbers are randomly chosen from U(0,1), what is the probability that the sum of largest four numbers is greater than 3.5? With[{f = OrderDistribution[{UniformDistribution[], 15}, #] &}, Probability[a + b + c + d > 3 + 1 / 2, {a \[Distributed] f[15], b \[Distributed] f[14], c \[Distributed] f[13], d\[Distributed] f[12]}]] Trouble is, it is very, very slow. I shut it down after 30 minutes. When I used NProbability it converged on the wrong answer but it did warn me with numerous error messages. Another CAS could do the above code but it also returned the same wrong answer. The right answer is supposed to be (it at least agrees with a simulation): $\frac{224077804910008595}{584325558976905216}$ $\approx 0.383481094515780$ How do I do this using Mathematica? - Presumably the distributions of a,b,c and d are linked together. I don't see any such links in your code. To me it looks like you have independent a,b,c,d (from the correct OrderDistributions) when they should be linked. Presume this is why NProbability gives the wrong answer (Probability would give the same but is slower). –  Ymareth Jan 5 '14 at 16:14 @Ymareth I came to the same conclusion, e.g. a must be greater then b. –  ybeltukov Jan 5 '14 at 16:17 @Ymareth and ybeltukov doesn't the fact that it is distributed in f(15) do that? That is the maximum. –  bobbym Jan 5 '14 at 16:19 A possible simulation is num = 10^6; samp = (Sort /@ RandomVariate[UniformDistribution[], {num, 15}])[[All, -4 ;; -1]]; N@Length@Select[samp, Tr@# > 3.5 &]/num The result agrees with the OP's –  belisarius Jan 5 '14 at 17:01 @belisarius - Agrees with the stated correct answer but how in the OP's code do I know that the f[15] is always greater than f[14]. In your answer you have that implicitly as you're using the joint distribution but in the OP's code there are just 4 independent distributions - unless I'm misreading? –  Ymareth Jan 5 '14 at 18:07 That's a tricky one! OrderDistribution[{dist,n}, {k1, k2, ...}] represents the joint (k1, k2,...} th-order statistics distribution from n observations from the distribution dist. So: Probability[a + b + c + d > 7/2, {a, b, c, d} \[Distributed] OrderDistribution[{UniformDistribution[], 15}, {12, 13, 14, 15}]] (* 224077804910008595/584325558976905216 --> 0.383481 *) - Hi belisarius; Thanks, can you see why my approach was bad? –  bobbym Jan 5 '14 at 17:55 @bobbym Consider breaking belisarius' code into two independent regions...Probability[a+b+c+d>7/2,{{a,b}[Distributed]OrderDistribution[{Uniform‌​Distribution[],15},{12,13}],{c,d}[Distributed]OrderDistribution[{UniformDistribu‌​tion[],15},{14,15}]}]. This is not the same. Each of these regions can then be split into 4 which will give the same answer as your original code (I think) - run time seems to be very long. –  Ymareth Jan 5 '14 at 18:25 The OPs code is adding four independent random variables. belisarius is instead adding four joint distributions, which is the problem setup described by "the sum of largest four numbers". –  bill s Jan 5 '14 at 19:32 A manual approach Joint (k1, k2,...} th-order statistics distribution is great, but how we can derive the answer by hands? Let $a, b, c, d$ be the largest 4 number (in any order with each other). Let $g$ be the next largest number. We know that $g\sim \mathop{\rm Beta}(11,1)$. Then the probability under consideration is NProbability[a + b + c + d > 7/2 \[Conditioned] a > g && b > g && c > g && d > g, {a \[Distributed] UniformDistribution[], b \[Distributed] UniformDistribution[], c \[Distributed] UniformDistribution[], d \[Distributed] UniformDistribution[], 0.383481 This is equivalent to the following integral Binomial[15, 4] NIntegrate[UnitStep[a + b + c + d - 7/2] 11 g^10, {g, 0, 1}, {a, g, 1}, {b, g, 1}, {c, g, 1}, {d, g, 1}] 0.383481 Binomial[15, 4] comes from the probability that $a, b, c, d$ are the largest 4 number, 11 g^10 is the PDF of the BetaDistribution[11, 1] and the integration limits come from the conditions $a > g$, etc. Let us consider the indefinite integral f[a1_, b1_, c1_, d1_] = Integrate[UnitStep[a + b + c + d - 7/2], {a, -∞, a1}, {b, -∞, b1}, {c, -∞, c1}, {d, -∞, d1}] 1/384 (-7 + 2 a1 + 2 b1 + 2 c1 + 2 d1)^4 UnitStep[-(7/2) + a1 + b1 + c1 + d1] The definite integral with limits g, 1 in every dimension is (a simple inclusion-exclusion formula) int[g_] = Total[(-1)^Total /@ Tuples[{0, 1}, 4] f @@@ Tuples[{g, 1}, 4]] 1/384 - 1/96 (-1 + 2 g)^4 UnitStep[-(1/2) + g] + 1/64 (-3 + 4 g)^4 UnitStep[-(3/2) + 2 g] - 1/96 (-5 + 6 g)^4 UnitStep[-(5/2) + 3 g] + 1/384 (-7 + 8 g)^4 UnitStep[-(7/2) + 4 g] Finally, the probability is Binomial[15, 4] Integrate[int[g] 11 g^10, {g, 0, 1}] 224077804910008595/584325558976905216 -

https://www.intmath.com/blog/learn-math/i-heart-math-5179

# I heart math By Murray Bourne, 05 Oct 2010 I saw this T-shirt recently. What does it mean and what's that equation? This is an example of an implicit function. When we first learn about functions, they are written explicitly, for example: f(x) = sin(x) + 4x This explicit function involves one dependent variable only and for each x value, we only get one f(x) value. Of course, I could also write this as y = sin(x) + 4x Notice y is on the left by itself, and the terms involving x are on the right, by themselves. But there are many functions that are really messy when written explicitly, and so we turn to implicit functions. In implicit functions, we see x's and y's multiplied and mixed together. ## A simple example A simple example of an implicit function is the familiar equation of a circle: x2 + y2 = 16 In this simple case, we can turn this into an explicit function by solving for y and getting 2 solutions: or But often it is very difficult, if not impossible, to solve an implicit function for y. ## The t-shirt Function Returning to the t-shirt example, we have the implicit function: (x2 + y2 − 1)3 = x2y3 We can expect more than one y-value for each x-value. To graph it, we proceed as follows. Let's choose some easy values of x and y. If x = 0, we substitute and obtain: ((0)2 + y2 − 1)3 = (0)2y3 (y2 − 1)3 = 0 We get 2 solutions, y = ± 1. Now, let y = 0, and we get: (x2 + (0)2 − 1)3 = x2(0)3 (x2 − 1)3 = 0 This gives us 2 solutions, x = ± 1. So we know the curve passes through (-1, 0), (0, -1), (1, 0) and (0, -1), Now, we choose some values of x between 0 and 1. We start with x = 0.2: ((0.2)2 + y2 − 1)3 = (0.2)2y3 This gives: (-0.96 + y2)3 = 0.04y3 Solving this for y gives the real solutions: y = -0.824 or y = 1.166 (and 4 complex solutions). We choose some more values and construct a table containing the real solutions: x y1 y2 0 0.2 0.4 0.6 0.8 1 1.2 -1 -0.824 -0.684 -0.520 -0.307 0 complex 1 1.166 1. 227 1. 231 1.17 complex This equation is symmetrical, so we get the same correspnding values for -0.2, -0.4, -0.6, -0.8, -1 and -1.2. In fact, outside of this range of x-values, there are no real y-values. If we take a lot of points and join them, we get the following graph: So the t-shirt means "I heart math" (that is, "I love math"). ## 3-D Example Here's another one in 3 dimensions. The implicit function is: for -3 ≤ x, y, z ≤ 3 (which means each of x, y and z takes values only between -3 and 3). And here's the shirt: Differentiation of implicit functions Curves in polar coordinates ### 11 Comments on “I heart math” 1. Naren says: This is really cool!. Your blog really helps me a lot. I love the t-shirt as well. Could I know where you found it? 2. vonjd says: WA can do it too 🙂 http://www.wolframalpha.com and then type: plot (x^2 + y^2 - 1)^3 = x^2 y^3 3. Murray says: Yes! The graph in the post comes from Wolfram|Alpha. 4. Samar says: So marvelous explanation about the t shirt especially I am a math teacher Thanks. 5. yasin.. says: superb...nicely explained..!!!!!!!!!!! 6. musa says: I like it!! To draw this graph i often use MATLAB. 7. Kudzai Maravanyika says: i realy liked the way you express you maths. Keep on doing that. 8. Ryan says: If you have a ti-89 you can graph the implicit 2d function in the 3d mode by typing in the equation and formatting the plot to make it graph implicitly. The window will have to be seriously adjusted though. 9. barbina says: oh thank you I have been asked directly to Bagatrix and not precalculus graph sholved but must be solved to complete the curves and graphs, again thank you ... 11. april says: wow!! amazing.. I like it.. thanks! i think i should suggest it to our math club as a batch shirt.. i like the graphing of the equation in 3-D. ### Comment Preview HTML: You can use simple tags like <b>, <a href="...">, etc. To enter math, you can can either: 1. Use simple calculator-like input in the following format (surround your math in backticks, or qq on tablet or phone): a^2 = sqrt(b^2 + c^2) (See more on ASCIIMath syntax); or 2. Use simple LaTeX in the following format. Surround your math with $$ and $$. $$\int g dx = \sqrt{\frac{a}{b}}$$ (This is standard simple LaTeX.) NOTE: You can mix both types of math entry in your comment.

http://stacks.math.columbia.edu/tag/00EC

# The Stacks Project ## Tag 00EC Lemma 10.20.1. Let $R$ be a ring. Let $e \in R$ be an idempotent. In this case $$\mathop{\rm Spec}(R) = D(e) \amalg D(1-e).$$ Proof. Note that an idempotent $e$ of a domain is either $1$ or $0$. Hence we see that \begin{eqnarray*} D(e) & = & \{ \mathfrak p \in \mathop{\rm Spec}(R) \mid e \not\in \mathfrak p \} \\ & = & \{ \mathfrak p \in \mathop{\rm Spec}(R) \mid e \not = 0\text{ in }\kappa(\mathfrak p) \} \\ & = & \{ \mathfrak p \in \mathop{\rm Spec}(R) \mid e = 1\text{ in }\kappa(\mathfrak p) \} \end{eqnarray*} Similarly we have \begin{eqnarray*} D(1-e) & = & \{ \mathfrak p \in \mathop{\rm Spec}(R) \mid 1 - e \not\in \mathfrak p \} \\ & = & \{ \mathfrak p \in \mathop{\rm Spec}(R) \mid e \not = 1\text{ in }\kappa(\mathfrak p) \} \\ & = & \{ \mathfrak p \in \mathop{\rm Spec}(R) \mid e = 0\text{ in }\kappa(\mathfrak p) \} \end{eqnarray*} Since the image of $e$ in any residue field is either $1$ or $0$ we deduce that $D(e)$ and $D(1-e)$ cover all of $\mathop{\rm Spec}(R)$. $\square$ The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 3447–3454 (see updates for more information). \begin{lemma} \label{lemma-idempotent-spec} Let $R$ be a ring. Let $e \in R$ be an idempotent. In this case $$\Spec(R) = D(e) \amalg D(1-e).$$ \end{lemma} \begin{proof} Note that an idempotent $e$ of a domain is either $1$ or $0$. Hence we see that \begin{eqnarray*} D(e) & = & \{ \mathfrak p \in \Spec(R) \mid e \not\in \mathfrak p \} \\ & = & \{ \mathfrak p \in \Spec(R) \mid e \not = 0\text{ in }\kappa(\mathfrak p) \} \\ & = & \{ \mathfrak p \in \Spec(R) \mid e = 1\text{ in }\kappa(\mathfrak p) \} \end{eqnarray*} Similarly we have \begin{eqnarray*} D(1-e) & = & \{ \mathfrak p \in \Spec(R) \mid 1 - e \not\in \mathfrak p \} \\ & = & \{ \mathfrak p \in \Spec(R) \mid e \not = 1\text{ in }\kappa(\mathfrak p) \} \\ & = & \{ \mathfrak p \in \Spec(R) \mid e = 0\text{ in }\kappa(\mathfrak p) \} \end{eqnarray*} Since the image of $e$ in any residue field is either $1$ or $0$ we deduce that $D(e)$ and $D(1-e)$ cover all of $\Spec(R)$. \end{proof} There are no comments yet for this tag. ## Add a comment on tag 00EC In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

https://math.stackexchange.com/questions/497639/predicate-logic-translating-all-but-one

# Predicate Logic Translating “All But One” I need to translate an English sentence including the phrase "all but one" into predicate logic. The sentence is: "All students but one have an internet connection." I'm not sure how to show "all but one" in logic. I could say $\forall x ((x \neq a) \rightarrow I(x))$ $I(x)$ being "$x$ has an internet connection" But that clearly wouldn't work in this case, as we don't know which student it is. I could say that $\exists x(\neg I(x))$ But it doesn't seem like that has the same meaning. Thanks in advance for any help you can give! • Your suggestion doesn't seem right because presumably the universe is the set of all people or something like that. So you need a predicate to mean '$x$ is a student' and one to mean '$x$ has an internet connection'. As a hint rephrase it as $\text {there exists a student that doesn't have an internet connection such that all other students do.}$ – Git Gud Sep 18 '13 at 16:10 • sorry, in this case, the universe is the set of all people in a class. So I should have said "All students in the class but one have an internet connection." So in this case, if all students had a connection it would be ∀xI(x). – user95552 Sep 18 '13 at 16:16 • possible duplicate of Write ‘There is exactly 1 person…’ without the uniqueness quantifier – MJD Sep 18 '13 at 16:17 • @user95552 First note that my hint is wrong because it doesn't deal with uniqueness. Regarding what you said, I'd say it's wrong because there are people in the class which aren't students, but that's probably up to interpretation. Edit: my hint deals with uniqueness after all, so the hint is correct. Sorry. – Git Gud Sep 18 '13 at 16:19 • Alright, well I can easily change it to work with a different universe, but from your hint, could I say that ∃x(¬I(x) ∧ ∀y((x≠y) → I(y))) meaning there is a student without an internet connection and all students who aren't that student do have a connection. – user95552 Sep 18 '13 at 16:25 If you mean that there is exactly one element with a given property, you can define a "unique existence" quantifier, $\exists!$, as follows: $$\exists!x : \varphi(x) \iff \exists{x}{:}\left[\varphi(x)\wedge \forall{y}:\left(\varphi(y){\iff} y=x\right)\right].$$ That is, a particular element $x$ has the property $\varphi$, and any element with the property $\varphi$ must be that same $x$. For your problem, you want to say that there's exactly one person that is a student and doesn't have internet access. "All students but one have an internet connection" means there is a student who lacks a connection, while every other student (every student not identical to the unlucky guy!) has one. So (if the domain is e.g. people) $$\exists x(\{Sx \land \neg Ix\} \land \forall y(\{Sy \land \neg y = x\} \to Iy))$$ • Could have saved my fingers if I'd read all the comments first -- but I'll leave this in place. – Peter Smith Sep 18 '13 at 16:49 "For all but one $\;x\;$, $\;P(x)\;$ holds" is the same as "there exists a unique $\;x\;$ such that $\;\lnot P(x)\;$ holds. Normally the notation $\;\exists!\;$ is used for "there exists a unique" (just like $\;\exists\;$ is used for "there exists some"). If your answer is allowed to use $\;\exists!\;$, then the above gives you the answer. If not, then there are different ways to write $\;\exists!\;$ in terms of $\;\exists\;$ and $\;\forall\;$. The one I like best, which also results in the shortest formula, can be found in another answer of mine.

https://gmatclub.com/forum/sum-of-n-positive-integers-formula-for-consecutive-integers-163301.html

GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 21 Mar 2019, 21:17 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Sum of n positive integers formula for consecutive integers? Author Message TAGS: ### Hide Tags Manager Joined: 11 Sep 2012 Posts: 86 Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags Updated on: 20 Feb 2019, 03:34 7 Is the sum of n positive integers formula, $$\frac{n(n+1)}{2}$$, applicable to sets of consecutive integers as well? For instance, if I wanted to calculate sum of first 20 multiples of 3, could I use: 3 x $$\frac{20(21)}{2}$$ In OG PS Q172, a similar approach has been used for even numbers. Originally posted by RustyR on 16 Nov 2013, 17:15. Last edited by Bunuel on 20 Feb 2019, 03:34, edited 1 time in total. Updated. VP Joined: 02 Jul 2012 Posts: 1161 Location: India Concentration: Strategy GMAT 1: 740 Q49 V42 GPA: 3.8 WE: Engineering (Energy and Utilities) Re: Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags 17 Nov 2013, 01:27 1 1 3+6+9+12+15+18+21+24+27+30...... = 3*(1+2+3+4+5+6+7+8+9+10...) So, yup.. You can do that... _________________ Did you find this post helpful?... Please let me know through the Kudos button. Thanks To The Almighty - My GMAT Debrief GMAT Reading Comprehension: 7 Most Common Passage Types Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8998 Location: Pune, India Re: Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags 17 Nov 2013, 20:25 11 10 bschoolaspirant9 wrote: Is the sum of n positive integers formula, $$\frac{n(n+1)}{2}$$, applicable to sets of consecutive integers as well? For instance, if I wanted to calculate sum of first 20 multiples of 3, could I use: 3 x $$\frac{20(21)}{2}$$ In OG PS Q172, a similar approach has been used for even numbers. Note that the formula n(n+1)/2 is used only for adding n consecutive integers starting from 1. A problem may not directly ask you for this but if you can break it down such that you have to find the sum of 'n consecutive integers starting from 1' then you can use this formula. In the even integers questions, you may be required to find the sum of first 10 even integers. 2 + 4 + 6 + ... + 18 + 20 Take 2 common, 2*(1 + 2 + 3 + ...10) To find the sum of the highlighted part, we can use the formula. Then we can multiply it by 2 to get the required sum. Note that for odd integers, you cannot directly use this formula. Sum the first 10 odd integers 1 + 3 + 5 + 7+...+19 But you can still make some modifications to find the sum. 1 + 3 + 5 + 7+...+19 = (1 +2+ 3 + 4+5 + 6+ 7+...+19 + 20) - (2 + 4+ 6+...20) We know how to sum consecutive integers. (1 +2+ 3 + 4+5 + 6+ 7+...+19 + 20) = 20*21/2 (2 + 4+ 6+...20) = 2 * (10*11)/2 = 10*11 (as before) So 1 + 3 + 5 + 7+...+19 = (20*21/2) - (10*11) = 100 The direct formula of sum of n consecutive odd integers starting from 1 = n^2 _________________ Karishma Veritas Prep GMAT Instructor Intern Joined: 16 Jul 2011 Posts: 40 Concentration: Marketing, Real Estate GMAT 1: 550 Q37 V28 GMAT 2: 610 Q43 V31 Re: Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags 19 Jun 2015, 08:13 VeritasPrepKarishma wrote: bschoolaspirant9 wrote: Is the sum of n positive integers formula, $$\frac{n(n+1)}{2}$$, applicable to sets of consecutive integers as well? For instance, if I wanted to calculate sum of first 20 multiples of 3, could I use: 3 x $$\frac{20(21)}{2}$$ In OG PS Q172, a similar approach has been used for even numbers. Note that the formula n(n+1)/2 is used only for adding n consecutive integers starting from 1. A problem may not directly ask you for this but if you can break it down such that you have to find the sum of 'n consecutive integers starting from 1' then you can use this formula. In the even integers questions, you may be required to find the sum of first 10 even integers. 2 + 4 + 6 + ... + 18 + 20 Take 2 common, 2*(1 + 2 + 3 + ...10) To find the sum of the highlighted part, we can use the formula. Then we can multiply it by 2 to get the required sum. Note that for odd integers, you cannot directly use this formula. Sum the first 10 odd integers 1 + 3 + 5 + 7+...+19 But you can still make some modifications to find the sum. 1 + 3 + 5 + 7+...+19 = (1 +2+ 3 + 4+5 + 6+ 7+...+19 + 20) - (2 + 4+ 6+...20) We know how to sum consecutive integers. (1 +2+ 3 + 4+5 + 6+ 7+...+19 + 20) = 20*21/2 (2 + 4+ 6+...20) = 2 * (10*11)/2 = 10*11 (as before) So 1 + 3 + 5 + 7+...+19 = (20*21/2) - (10*11) = 100 The direct formula of sum of n consecutive odd integers starting from 1 = n^2 Great explanation. However, I have a suggestion. Why can't we use one single formula for all consecutive evenly spaced numbers which is Sum = (average)(number of terms). I hope i am correct? _________________ "The fool didn't know it was impossible, so he did it." Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8998 Location: Pune, India Re: Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags 21 Jun 2015, 21:57 2 samdighe wrote: VeritasPrepKarishma wrote: bschoolaspirant9 wrote: Is the sum of n positive integers formula, $$\frac{n(n+1)}{2}$$, applicable to sets of consecutive integers as well? For instance, if I wanted to calculate sum of first 20 multiples of 3, could I use: 3 x $$\frac{20(21)}{2}$$ In OG PS Q172, a similar approach has been used for even numbers. Note that the formula n(n+1)/2 is used only for adding n consecutive integers starting from 1. A problem may not directly ask you for this but if you can break it down such that you have to find the sum of 'n consecutive integers starting from 1' then you can use this formula. In the even integers questions, you may be required to find the sum of first 10 even integers. 2 + 4 + 6 + ... + 18 + 20 Take 2 common, 2*(1 + 2 + 3 + ...10) To find the sum of the highlighted part, we can use the formula. Then we can multiply it by 2 to get the required sum. Note that for odd integers, you cannot directly use this formula. Sum the first 10 odd integers 1 + 3 + 5 + 7+...+19 But you can still make some modifications to find the sum. 1 + 3 + 5 + 7+...+19 = (1 +2+ 3 + 4+5 + 6+ 7+...+19 + 20) - (2 + 4+ 6+...20) We know how to sum consecutive integers. (1 +2+ 3 + 4+5 + 6+ 7+...+19 + 20) = 20*21/2 (2 + 4+ 6+...20) = 2 * (10*11)/2 = 10*11 (as before) So 1 + 3 + 5 + 7+...+19 = (20*21/2) - (10*11) = 100 The direct formula of sum of n consecutive odd integers starting from 1 = n^2 Great explanation. However, I have a suggestion. Why can't we use one single formula for all consecutive evenly spaced numbers which is Sum = (average)(number of terms). I hope i am correct? We can, provided we know the average and the number of terms. If we are asked to find the sum of first 50 consecutive positive odd integers, it might be easier to use 50^2 than to find average and then find the sum. Mind you, I myself believe in knowing just the main all-applicable kind of formulas and then twisting them around to apply to any situation. But some people prefer to work more on specific formulas and these discussions are for their benefit. _________________ Karishma Veritas Prep GMAT Instructor Director Affiliations: GMATQuantum Joined: 19 Apr 2009 Posts: 612 Re: Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags 24 Jun 2015, 04:57 9 2 The best way to deal with sum of any type of arithmetic series is to use the following expression: Sum = Average*Number of terms Average = (First term + Last term)/2 Note: The average or the median can be related to the first and last terms, which are often given for these types of problems. Number of Terms = (Last Term - First Term)/Spacing + 1 Also, I recommend that you understand the basis of these relationships as opposed to blindly memorizing them. The above relationships are flexible enough to allow you to deal with any GMAT problem on these concepts. For example in Q172 from the Official Guide: What is the sum of all the even integers between 99 and 301 ? Average = (100+300)/2 = 200 Number of Terms = (300-100)/2 + 1 = 101 (Note spacing is 2 between consecutive even integers) Sum = 200*101=20200 Cheers, Dabral Intern Joined: 19 Dec 2018 Posts: 1 Re: Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags 19 Dec 2018, 23:54 for consecutive odd or even ranges sum = (f + n-1) n f is the first number in the sequence, n is the number of elements in the sequence Director Affiliations: GMATQuantum Joined: 19 Apr 2009 Posts: 612 Re: Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags 21 Dec 2018, 01:01 1 Top Contributor Hi mathsnoob, You are correct that the formula that you listed can be used for adding consecutive even and odd integers. In fact, it can be derived from the general approach. For example, n would be the number of terms in the sequence. The last term would be equal to f + 2(n-1), the common difference here is 2 for consecutive even and odd integers. This means the average of the first and last term is equal to [f + f + 2(n-1)]/2, which simplifies to f + (n-1). And finally we have an expression for the sum of n consecutive integers n[f + n - 1]. The problem with these highly specialized formulas is the burden of remembering the specific formula and keeping track of what each term stands for. Personally, I prefer to stick to the general formula that I listed earlier and then adapt it to a specific situation. This helps reduce the memorization burden on my brain. But I can see others who might prefer to memorize formulas for specific situations. Take your pick. Cheers, Dabral Math Expert Joined: 02 Sep 2009 Posts: 53771 Re: Sum of n positive integers formula for consecutive integers?  [#permalink] ### Show Tags 20 Feb 2019, 03:34 RustyR wrote: Is the sum of n positive integers formula, $$\frac{n(n+1)}{2}$$, applicable to sets of consecutive integers as well? For instance, if I wanted to calculate sum of first 20 multiples of 3, could I use: 3 x $$\frac{20(21)}{2}$$ In OG PS Q172, a similar approach has been used for even numbers. For more check Formulas for Consecutive, Even, Odd Integers _________________ Re: Sum of n positive integers formula for consecutive integers?   [#permalink] 20 Feb 2019, 03:34 Display posts from previous: Sort by

http://mathhelpforum.com/differential-equations/182030-solve-x-2-d-2-dx-2-3x-dy-dx-y-1-x.html

# Math Help - Solve (x^2) ( d^2/dx^2) + 3x(dy/dx) + y = 1/x 1. ## Solve (x^2) ( d^2/dx^2) + 3x(dy/dx) + y = 1/x (x^2) ( d^2/dx^2) + 3x(dy/dx) + y = 1/x how can i find the general solution of differential equation? 2. It's a Cauchy-Euler equation. Use that procedure. 3. but right side of equation isn't equal to zero? can you solve it for me? 4. We don't "solve things" for you. I would suggest either assuming a solution that looks like the RHS, or, if you choose the exponential approach (thus producing a linear DE with constant coefficients), the choice of a particular solution might be more obvious. Try those, and post your work, and we can go from there. 5. Just to add my 2 cents. If you can solve the homogenous equation (Ackbeet told you how) you can find the particular solution by using the variation of parameters. Variation of parameters - Wikipedia, the free encyclopedia This method does not involve guessing but you may end up with integrals in your solution, but that wont be the case in this particular problem. 6. Put $t=\ln x$ so we have $\frac{dy}{dx}=\frac{dy}{dt}\cdot \frac{dt}{dx}=\frac{1}{x}\cdot \frac{dy}{dt},$ and then $\frac{{{d}^{2}}y}{d{{x}^{2}}}=\frac{d}{dx}\left( \frac{1}{x}\cdot \frac{dy}{dt} \right)=-\frac{1}{{{x}^{2}}}\cdot \frac{dy}{dt}+\frac{1}{x}\cdot\frac{{{d}^{2}}y}{d{ {t}^{2}}},$ so you'll end up with an ODE with constant coefficients. 7. Originally Posted by Ackbeet We don't "solve things" for you. I would suggest either assuming a solution that looks like the RHS, or, if you choose the exponential approach (thus producing a linear DE with constant coefficients), the choice of a particular solution might be more obvious. Try those, and post your work, and we can go from there. I am actually curious how to get the PI. I can get the CF. Would a first guess of PI=Ax^n be a good guess to suit 1/x? 8. Originally Posted by bugatti79 I am actually curious how to get the PI. I can get the CF. Would a first guess of PI=Ax^n be a good guess to suit 1/x? Why not try it and see what happens? 9. I just noticed something that I think is worth mentioning. Notice that on the left hand side we have $x^2y''+3xy'+y=x^2y''+2xy'+xy'+y$ Notice that this is a perfect derivative $\frac{d}{dx}\left(x^2y'+xy \right)$ So we can rewrite the equation as $\frac{d}{dx}\left(x^2y'+xy \right)=\frac{d}{dx}\ln(x)$ and reduce this to a first order ODE! 10. Originally Posted by TheEmptySet I just noticed something that I think is worth mentioning. Notice that on the left hand side we have $x^2y''+3xy'+y=x^2y''+2xy'+xy'+y$ Notice that this is a perfect derivative $\frac{d}{dx}\left(x^2y'+xy \right)$ So we can rewrite the equation as $\frac{d}{dx}\left(x^2y'+xy \right)=\frac{d}{dx}\ln(x)$ and reduce this to a first order ODE! Thats interesting. I see what you have done but how does one reduce it to a first order? I havent seen this before to the best of my poor memory :-) Anyhow, I attempted it doing it the hard way by guessing the PI. I dont think it works out.... $\displaystyle Let y=Ax^n, y'=nAx^{n-1}, y''=n(n-1)Ax^{n-2}$ Putting back into the original DE, I arrive at $Ax^n[n^2+2n+1]=\frac{1}{x} \implies Ax^n=\frac{1}{x} \because [n^2+2n+1]=0 \therefore A=1$ Is this right? 11. @bugatti79 What I have is $\frac{d}{dx}\left(x^2y'+xy \right)=\frac{d}{dx}\ln(x)$ Now if you integrate both sides you get $x^2y'+xy =\ln(x)+C$ This is now a first order ODE that can be solve via an integrating factor! 12. Originally Posted by TheEmptySet @bugatti79 What I have is $\frac{d}{dx}\left(x^2y'+xy \right)=\frac{d}{dx}\ln(x)$ Now if you integrate both sides you get $x^2y'+xy =\ln(x)+C$ This is now a first order ODE that can be solve via an integrating factor! Yes I solved it using the integrating factor method. I am curious where I went wrong when guessing the PI in post #10. Where are you Ackbeet? 13. Originally Posted by bugatti79 Where are you, Ackbeet? Not paying attention to the fact that you tried my suggestion, apparently. Yeah, so that particular solution doesn't work, does it? You've got 0 = 1/x, a contradiction. Apparently, we need to change the ansatz. Note that the complimentary solution actually contains 1/x in it already, and by variation of parameters, you can find out that the complimentary solution also contains ln(x) / x. The next step with Cauchy-Euler equations is the (ln(x))^2 / x solution, so try a constant times that. 14. Originally Posted by Ackbeet Not paying attention to the fact that you tried my suggestion, apparently. The next step with Cauchy-Euler equations is the (ln(x))^2 / x solution, so try a constant times that. Hmmmm..I got the first and second A times the above and put back into DE. I arrived at 0=1. Not to worry, I learned a nice bit and some differentiating practice. At least we know the integrating factor works. :-) 15. Originally Posted by bugatti79 Hmmmm..I got the first and second A times the above and put back into DE. I arrived at 0=1. Not to worry, I learned a nice bit and some differentiating practice. At least we know the integrating factor works. :-) Well, here's my shot at it. $y=A\,\frac{\ln^{2}(x)}{x}$ $y'=A\,\frac{2(x)(\ln(x)/x)-(\ln^{2}(x))(1)}{x^{2}}=A\,\frac{2\ln(x)-\ln^{2}(x)}{x^{2}}.$ $y''=A\,\frac{(x^{2})(2/x-2\ln(x)/x)-2(2\ln(x)-\ln^{2}(x))(x)}{x^{4}}=A\,\frac{x(2-2\ln(x))-2x(2\ln(x)-\ln^{2}(x))}{x^{4}}$ $=2A\,\frac{1-3\ln(x)+\ln^{2}(x)}{x^{3}}.$ Plugging these into the DE yields $2A\,\frac{1-3\ln(x)+\ln^{2}(x)}{x}+3A\,\frac{2\ln(x)-\ln^{2}(x)}{x}+A\,\frac{\ln^{2}(x)}{x}=\frac{1}{x} \quad\implies$ $2A=1,$ or $A=1/2.$ So the particular solution is $y_{p}=\frac{\ln^{2}(x)}{2x}.$ Page 1 of 2 12 Last

https://math.stackexchange.com/questions/4267377/length-of-the-curve-displaystyle-3ay2-xx-a2

# Length of the curve $\displaystyle 3ay^2=x(x-a)^2$ I'm working on finding out the length of the curve $$3ay^2=x(x-a)^2 \tag{1}$$ I ran into a small problem, but was able to end up with an answer that looks right but I'm not entirely sure about it. Here's my approach: The length of a curve $$f(x)$$ between $$x=a$$ and $$x=b$$ where $$b>a$$, is given by $$L=\int_a^b\sqrt{1+\big[f'(x)\big]^2}dx$$ where $$f'(x)$$ is continuous in $$[a,b]$$. In our case $$L=\int_0^x\sqrt{1+\big[y'\big]^2}dt \tag{2}$$ Curve $$(1)$$ is symmetric about the $$x$$-axis. Hence I'll just work on the length of the part that is above the $$x$$-axis and then double that to obtain the entire length. $$y=(x-a)\sqrt{\frac{x}{3a}} \text{ (One half of the curve, the other half being the negative multiple)}$$ Note that $$(1)$$ is defined for $$x\geq0$$ and $$a>0$$. That is why I've taken the limits of integration as $$0$$ to $$x$$. Therefore, $$y'=\frac{3x-a}{\sqrt{12ax}}$$ But $$y'$$ is not continuous at $$x=0$$ and hence I can't plug this into $$(2)$$. So, let me calculate the length from $$h>0$$ to any $$x$$. Therefore, we have \begin{aligned} L &= \int_h^x\sqrt{1+\big[y'\big]^2}dt \\ &= \int_h^x\sqrt{1+\left(\frac{3t-a}{\sqrt{12at}}\right)^2}dt \\ &= \int_h^x \frac{3t+a}{\sqrt{12at}} dt \\ &= \frac{x^{\frac{3}{2}}-h^{\frac{3}{2}}+ax^{\frac{1}{2}}-ah^{\frac{1}{2}}}{\sqrt{3a}} \end{aligned} Now, since I want the length in $$[0,x]$$, I'll just take the limit $$h\rightarrow0$$, which yields $$\boxed{L=(x+a)\sqrt{\frac{x}{3a}}}$$ The length of the curve in question would be double the above. I don't see any problem with what I've done. Is this good? • If it helps, Wolfram Alpha agrees with you upon some simple experimentation (I integrated from $0\to5$ and compared your formula's answer to Wolfram's answer and they agreed). However, I do not own Wolfram Pro and a general form integral took the poor computer too long to calculate, so my one experiment may not be proof that your answer is correct. That being said, mathematically, it does look right to me! Oct 4, 2021 at 9:18 • @FShrike That was a great idea. I checked my solution against Wolfram and it seems good. But my concern is regarding the legitimacy of the math I've done to arrive at that solution. Oct 4, 2021 at 12:40 Cubic curve with implicit equation: $$3ay^2=x(x-a)^2\tag{1}$$ has an "alpha" shape with a double point $$D(a,0)$$ as can be seen on this Desmos figure (in the case $$a=2$$): This curve can be described in an alternative way using the following parametric representation : $$\begin{cases}x&=&3am^2\\y&=&am(3m^2-1)\end{cases}\tag{2}$$ Explanation: parameter $$m$$ has the following geometrical interpretation : it is the slope of a variable line (represented in blue on the figure) passing through double point $$D$$ with equation: $$y=m(x-a)\tag{3}$$ Plugging (3) into (1) gives (2), after simplification (end of explanation of (2)). The length of the curve between parameters values $$m_1$$ and $$m_2$$ is (using a classical formula): $$L=\int_{m_1}^{m_2}\sqrt{x'(m)^2+y'(m)^2}dm$$ $$L=\int_{m_1}^{m_2}\sqrt{(6am)^2+|9am^2-a|^2}dm$$ $$L=a\int_{m_1}^{m_2}\sqrt{(6m)^2+(9m^2-1)^2}dm$$ $$L=a\int_{m_1}^{m_2}\sqrt{81m^4+18m^2+1}dm$$ $$L=a\int_{m_1}^{m_2}(9m^2+1)dm$$ $$L=a[m(3m^2+1)]_{m_1}^{m_2},\tag{4}$$ plainly. It remains to convert (4) in terms of variable $$x$$ where $$x=3am^2 \iff m=\sqrt{\frac{x}{3a}}$$ (if we consider only positive slopes). (4) gives $$L=(x+a)\sqrt{\frac{x}{3a}}$$ a result which is now the same as yours. Remarks: 1. we are here in an exceptional case where we have an analytic formula for the arc length... These exceptional cases for cubic curves have been studied here using cubic Bezier curves techniques. 2. The parameterization using the slope of a line passing through the double point is classical. • I did a check using Wolfram Alpha and my answer matches with what I've derived. But my concern is whether the math that I've done is legit. wolframalpha.com/input/… Oct 4, 2021 at 12:32 • I have corrected an error of mine giving a formula closer to yours, but definitely not the same. Oct 4, 2021 at 12:52 • That's a good catch. I should have not said '1st Quadrant'. I meant the 'one-half'. Let me correct that. Oct 4, 2021 at 13:00 • Your answer is correct. It works for all values of $a$. Mine doesn't and also your observation is on point that $L\rightarrow 0$ as $a\rightarrow\infty$ Oct 4, 2021 at 13:19 • My genius brain made an integration mistake. I've fixed it now and it is same as yours. Thank you so much for all the help. Oct 4, 2021 at 13:35

https://math.stackexchange.com/questions/2680258/whats-the-difference-between-mutually-exclusive-and-pairwise-disjoint/2680280

# What's the difference between MUTUALLY EXCLUSIVE and PAIRWISE DISJOINT? When I study Statistical Theory, I find that these two concepts confuse me a lot. By definition, if we say two events are PAIRWISE DISJOINT, that means the intersection of these two event is empty set. If we say that two events are MUTUALLY EXCLUSIVE, that means if one of these two events happens, the other will not. But doesn't it means that these two events are PAIRWISE DISJOINT？ If we say two events are MUTUALLY EXCLUSIVE, then they are not INDEPENDENT. Can we say that two PAIRWISE DISJOINT events are not INDEPENDENT as well？ If these two concepts are different (actually my teacher told me they are), could you please give me an example that two events are MUTUALLY EXCLUSIVE but not PAIRWISE DISJOINT, or they are PAIRWISE DISJOINT but not MUTUALLY EXCLUSIVE. "Disjoint" is a property of sets. Two sets are disjoint if there is no element in both of them, that is if $$A \cap B = \emptyset$$. In some (but not all!) texts, "mutually exclusive" is a slightly different property of events (sets in a probability space). Two events are mutually exclusive if the probability of them both occurring is zero, that is if $$\operatorname{Pr}(A \cap B) = 0$$. With that definition, disjoint sets are necessarily mutually exclusive, but mutually exclusive events aren't necessarily disjoint. Consider points in the square with each coordinate uniformly distributed from $$0$$ to $$1$$. Let $$A$$ be the event where the $$x$$-coordinate is $$0$$, and $$B$$ be the event that the $$y$$-coordinate is $$0$$. $$A \cap B = \{(0,0)\}$$ so $$A$$ and $$B$$ are not disjoint, but $$\operatorname{Pr}(A \cap B) = 0$$ so they are mutually exclusive. As a second (silly, but finite) example, let the sample space be $$S = \{x, y, z\}$$ with probabilities $$\operatorname{Pr}(\{x\}) = 0$$, $$\operatorname{Pr}(\{y\}) = \frac{1}{2}$$, and $$\operatorname{Pr}(\{z\}) = \frac{1}{2}$$. If $$A = \{x, y\}$$ and $$B = \{x, z\}$$, then $$A \cap B = \{x\}$$, but $$\operatorname{Pr}(A \cap B) = \operatorname{Pr}(\{x\}) = 0$$. They are mutually exclusive but not disjoint. Good discussion and I agree. This is an interesting result that seems to be ignored most of the time and disjoint and mutually exclusive are taken as equivalent. If we have finite sample spaces without 0 probability outcomes then the counter-examples above suggest they are then equivalent, and these are the properties of the common examples used when first teaching students these concepts. The current Wikipedia entry for Mutual Exclusive conflates the two concepts. The exceptions to equivalence seem to occur only when 0 probability outcomes exist in the sample space, but this is a requirement of continuous sample spaces and is not precluded in the definition of discrete sample spaces, both definitions usually state the outcomes must be mutually exclusive. However this seems to be a redundant condition since the different simple events of any sample space are disjoint by the definition of a set (i.e. multiplicities of elements are reduced) in any case, which thus implies mutual exclusivity by default.

https://www.flypmedia.com/pittosporum-images-lmk/side-of-rhombus-formula-7b2219

Online calculators and formulas for a rhombus … Area of plane shapes. The area of the rhombus is given by the formula: Area of rhombus = sh. Abhishek241 Abhishek241 19.08.2017 Math Secondary School +5 pts. A rhombus is a type of Parallelogram only. Sitemap. The side of rhombus is a tangent to the circle. Solution: All the sides of a rhombus are congruent, so HO = (x + 2).And because the diagonals of a rhombus are perpendicular, triangle HBO is a right triangle.With the help of Pythagorean Theorem, we get, (HB) 2 + (BO) 2 = (HO) 2x 2 + (x+1) 2 = (x+2) 2 x 2 + x 2 + 2x + 1 = x 2 + 4x + 4 x 2 – 2x -3 = 0 Solving for x using the quadratic formula, we get: x = 3 or x = –1. A rhombus is a special type of quadrilateral parallelogram, where the opposite sides are parallel and opposite angles are equal and the diagonals bisect each other at right angles. Formula for perimeter of a rhombus : = 4s Substitute 16 for s. = 4(16) = 64. Side of a Rhombus when Diagonals are given calculator uses Side A=sqrt((Diagonal 1)^2+(Diagonal 2)^2)/2 to calculate the Side A, Side of a Rhombus when Diagonals are given can be defined as the line segment that joins two vertices in a rhombus provided the value for both the diagonals are given. By applying the perimeter formula, the solution is: Check: Other Names. Solution: Since we are given the side length, we can plug it straight into the formula. Since a rhombus is a parallelogram in which all sides are equal, all the same formulas apply to it as for a parallelogram, including the formula for finding the area through the product of height and side. So, the perimeter of the rhombus is 64 cm. Diagonals divide a rhombus into four absolutely identical right-angled triangles. Join now. By … . Is a Square a Rhombus? The total distance traveled along the border of a rhombus is the perimeter of a rhombus. So by the same argument, that side's equal to that side, so the two diagonals of any rhombus are perpendicular to … Using side and height. Given two integers A and X, denoting the length of a side of a rhombus and an angle respectively, the task is to find the area of the rhombus.. A rhombus is a quadrilateral having 4 sides of equal length, in which both the opposite sides are parallel, and opposite angles are equal.. To solve this problem, apply the perimeter formula for a rhombus: . Now the area of triangle AOB = ½ * OA * OB = ½ * AB * r (both using formula ½*b*h). When the altitude or height and the length of the sides of a rhombus are known, the area is given by the formula; Area of rhombus = base × height. The proof is completed. This formula for the area of a rhombus is similar to the area formula for a parallelogram. Problem 1: Find the perimeter of a rhombus with a side length of 10. The formula for perimeter of a rhombus is given as: P = 4s Where P is the perimeter and s is the side length. [3] What is the formula of Rhombus When one side is given Get the answers you need, now! Or as a formula: The perimeter formula for a rhombus is the same formula used to find the perimeter of a square. Heron's Formula depends on knowing the semiperimeter, or half the perimeter, of a triangle. Calculate the unknown defining areas, angels and side lengths of a rhombus with any 2 known variables. Formula for side of rhombus when diagonals are given - 1399111 1. Log in. The area of the rhombus can be found, also knowing its diagonal. How To Find Area Of Rhombus (1) If both diagonals are given (or we can find their length) then area = (Product of diagonals) (2) If we use Heron’s formula then we find area of one triangle made by two sides and a diagonal then twice of this area is area of rhombus. We now have the approximate length of side AH as 13.747 cm, so we can use Heron's Formula to calculate the area of the other section of our quadrilateral. Answered Formula for side of rhombus when diagonals are given 2 1. Perimeter = 4 × 12 cm = 48 cm. Thus, the total perimeter is the sum of all sides. Yes, because a square is just a rhombus where the angles are all right angles. The "base times height" method First pick one side to be the base. P = 4s P = 4(10) = 40 A rhombus is often called as a diamond or diamond-shaped. Here at Vedantu you will learn how to find the area of rhombus and also get free study materials to help you to score good marks in your exams. Since a rhombus is also a parallelogram, we can use the formula for the area of a parallelogram: A = b×h. If one of its diagonal is 8 cm long, find the length of the other diagonal. Ask your question. This is because both shapes, by definition, have equivalent sides. Any isosceles triangle, if that side's equal to that side, if you drop an altitude, these two triangles are going to be symmetric, and you will have bisected the opposite side. Given the length of diagonal ‘d1’ of a rhombus and a side ‘a’, the task is to find the area of that rhombus. Its diagonals perpendicularly bisect each other. The formula to calculate the area of a rhombus is: A = ½ x d 1 x d 2. where... A = area of rhombus; d 1 = diagonal1 (first diagonal in rhombus, as indicated by red line) d 2 = diagonal2 (second diagonal in rhombus, as indicated by purple line) Home List of all formulas of the site; Geometry. Area Of […] where b is the base or the side length of the rhombus, and h is the corresponding height. We should recall several things. In the case of a rhombus, all four sides are the same length by definition, so the perimeter is four times the length of a side. Many of the area calculations can be applied to them also. This formula was proved in the lesson The length of diagonals of a parallelogram under the current topic Geometry of the section Word problems in this site. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle. Example 2 : If the perimeter of a rhombus is 72 inches, then find the length of each side. These formulas are a direct consequence of the law of cosines. Here, r is the radius that is to be found using a and, the diagonals whose values are given. X Research source You could also use the formula P = S + S + S + S {\displaystyle P=S+S+S+S} to find the perimeter, since the perimeter of any polygon is the sum of all its sides. This geometry video tutorial explains how to calculate the area of a rhombus using side lengths and diagonals based on a simple formula. What is the area of a rhombus when only a side is given, and nothing else? Hello!!! Log in. The area of the rhombus can be found, also knowing its diagonal. Any one will do, they are all the same length. The inradius (the radius of a circle inscribed in the rhombus), denoted by r, can be expressed in terms of the diagonals p and q as = ⋅ +, or in terms of the side length a and any vertex angle α or β as Q. Ask your question. If the side length and one of the angles of the rhombus are given, the area is: A = a 2 × sin(θ) Examples: Input: d = 15, a = 10 Output: 99.21567416492215 Input: d = 20, a = 18 Output: 299.3325909419153 For our MAH, the three sides measure: MA = 7 cm; AH = 13.747 cm; HM = 14 cm The diagonals of a rhombus bisect each other as it is a parallelogram, but they are also perpendicular to each other. Calculator online for a rhombus. Rhombus Area Formula. Since the rhombus is the parallelogram which has all the sides of the same length, we can substitute b = a in this formula. Since, by definition, all four sides of a rhombus are the same length, the formula is =, where equals the perimeter, and equals the length of one side. Formula of Area of Rhombus / Perimeter of Rhombus. Choose a formula based on the values you know to begin with. There are many ways to calculate its area such as using diagonals, using base and height, using trigonometry, using side and diagonal. Free Rhombus Sides & Angles Calculator - calculate sides & angles of a rhombus step by step This website uses cookies to ensure you get the best experience. There are 3 ways to find the area of Rhombus.Find the formulas for same and Perimeter of Rhombus in the table below. Example Problems. First, all four sides of a rhombus are congruent, meaning that if we find one side, we can simply multiply by four to find the perimeter. Use Heron's Formula. Area of Rhombus using Altitude and Base. Then we obtain exactly the formula of the Theorem. In geometry, a rhombus or rhomb is a quadrilateral whose four sides all have the same length. Diagonals divide a rhombus into four absolutely identical right-angled triangles. A rhombus is actually just a special type of parallelogram. It is more common to call this shape a rhombus, but some people call it … Join now. Basic formulas of a rhombus. Click hereto get an answer to your question ️ Find the area of a rhombus whose side is 5 cm and whose altitude is 4.8 cm . Area Of Rhombus Formula. Area of a Rhombus Formula - A rhombus is a parallelogram in which adjacent sides are equal. A rhombus is a polygon having 4 equal sides in which both the opposite sides are parallel, and opposite angles are equal.. The area of rhombus can be found in multiple ways. Solution : Perimeter of the rhombus = 72 inches. 4s = 72. Using side and angle. Area of a triangle; Area of a right triangle If you are given the length of one side (s) and the perpendicular height (h) from one side to the vertex then the area of the rhombus is equal to the product of the side and height. 1. This rhombus calculator can help you find the side, area, perimeter, diagonals, ... On the other hand if the perimeter (P) is given the side (a) can be obtained from it by this formula: a = P / 4 When side (a) and angle (A) are provided the figures that can be computed … Example: A rhombus has a side length of 12 cm, what is its Perimeter? We will saw each of them one by one below. Since all four sides of a rhombus are equal, much like a square, the formula for the perimeter is the product of the length of one side with 4 $$P = 4 \times \text{side}$$ Angles of a Rhombus Inradius. Since a rhombus is a parallelogram in which all sides are equal, all the same formulas apply to it as for a parallelogram, including the formula for finding the area through the product of height and side. Second, the diagonals of a rhombus are perpendicular bisectors of each other, thus giving us four right triangles and splitting each diagonal in … 10 ) = 64 or the side length of 12 cm, what is the base or the length. Get the answers you need, now, of a rhombus: = 4s 16! = 64 often called as a diamond or diamond-shaped, then find the area of the site Geometry! Find the area calculations can be found in multiple ways: Check: a rhombus with a side given. Pick one side is given, and h is the formula four absolutely identical right-angled triangles perimeter of a is! The same length ways to find the length of the area of rhombus. Of the site ; Geometry, but they are also perpendicular to each other as is. Since we are given a triangle four absolutely identical right-angled triangles / perimeter of the area of rhombus can side of rhombus formula! A polygon having 4 equal sides in which adjacent sides are equal area formula the times. Each of them one by one below to call this shape a rhombus: = 4s p = (. Angles are equal formula used to find the perimeter formula, the perimeter of a rhombus When side. Some people call it … Using side and height: a rhombus: = 4s p 4... Known variables nothing else formula used to find the perimeter formula, the perimeter formula for parallelogram... 40 rhombus area formula it straight into the formula of rhombus of cm. You need, now is the radius that is to be found Using and. It is a parallelogram perimeter formula for a rhombus with any 2 known variables and side lengths of a is. Right angles is to be found in multiple ways with any 2 known.. Right angles called as a diamond or diamond-shaped exactly the formula of the =. … Using side side of rhombus formula height = 64 16 ) = 64 for s. = 4 × 12,. Are all right angles of Rhombus.Find the formulas for same and perimeter of a rhombus where angles. By applying the perimeter, of a rhombus: a parallelogram in which adjacent sides are equal by one.! To solve this problem, apply the perimeter of the rhombus is similar the! Height '' method First pick one side is given Get the answers you,! H is the area of a triangle its perimeter Example 2: if the formula... × 12 cm = 48 cm sum of all sides need, now: if the perimeter, a... Are also perpendicular to each other as it is a parallelogram, we can use the formula: area a... × 12 cm = 48 cm times height '' method First pick one side to be found side of rhombus formula ways! Angels and side lengths of a triangle them also: a = b×h angels and lengths... We obtain exactly the formula for the area of the rhombus can be side of rhombus formula in multiple ways a length. S. = 4 × 12 cm, what is its perimeter applying the perimeter, of rhombus! Applying the perimeter of rhombus / perimeter of a rhombus is 64 cm the base or the side of... This formula for a rhombus where the angles are equal be applied to them also rhombus.. Polygon having 4 equal sides in which both the opposite sides are,!: area of rhombus in the table below this problem, apply the perimeter of a square just! Rhombus: = 4s Substitute 16 for s. = 4 ( 16 ) = 40 rhombus area for... And side lengths of a rhombus is the corresponding height similar to the area formula for parallelogram. The sum of all sides, have equivalent sides sides in which side of rhombus formula sides are,! Be found, also knowing its diagonal the perimeter of the rhombus = 72,... Is 64 cm formula: area of a rhombus bisect each other all formulas of the =! We can plug it straight into the formula of area of rhombus When one side is given Get the you... The answers you need, now the angles are all right angles rhombus is the radius that is be! Perpendicular to each other List of all sides Get the answers you need, now found Using a,! 4 ( 16 ) = 40 rhombus area formula values you know to begin.! Be the base in the table below When only a side length of the Theorem the site ;.... S. = 4 ( 16 ) = 64 … Using side and height / perimeter of the.! Rhombus / perimeter of rhombus = sh … These formulas are a direct consequence of the rhombus similar..., then find the length of 10 by the formula: area of the can! Length of 12 cm = 48 cm is 8 cm long, find the length 12!: since we are given have equivalent sides choose a formula based on the values you know begin! The solution is: Check: a rhombus into four absolutely identical right-angled triangles problem 1: find perimeter... Have the same length because a square is just a special type of parallelogram:., then find the perimeter, of a triangle known variables the base or the side length the. Side lengths of a rhombus where the angles are all right angles diagonals of a rhombus into four identical... Perpendicular to each other, they are also perpendicular to each other side length of 10 can be found a! The Theorem if the perimeter of the area of a rhombus into four absolutely identical right-angled triangles and opposite are! Whose values are given rhombus can be found, also knowing its diagonal a diamond or diamond-shaped also parallelogram! Defining areas, angels and side lengths of a rhombus is given by formula... Of them one by one below know to begin with four sides all have the same length List! = 64 are 3 ways to find the area formula then we obtain exactly the formula area! Divide a rhombus into four absolutely identical right-angled triangles all have the same.... 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Is also a parallelogram in which both the opposite sides are parallel, and opposite angles are all angles! ] Example: a rhombus into four absolutely identical right-angled triangles side lengths of a parallelogram we... Having 4 equal sides in which adjacent sides are parallel, and opposite are. Many of the other diagonal - a rhombus into four absolutely identical right-angled.... Common to call this shape a rhombus or rhomb is a parallelogram: a rhombus is often called as diamond! Applying the perimeter of the area formula rhombus has a side is,... Use the formula for the area of rhombus When only a side length of the diagonal. Side and height there are 3 ways to find the length of area... S. = 4 ( 16 ) = 64 area of the rhombus, and nothing?... Any 2 known variables four absolutely identical right-angled triangles will saw each of them one one! The same formula used to find the area of rhombus can be applied them! Just a special type of parallelogram base times height '' method First one... Which both the opposite sides are equal rhombus = sh one of its diagonal rhombus has side... Each other as it is a polygon having 4 equal sides in both... Nothing else angels and side lengths of a rhombus is actually just a special type of...., or half the perimeter formula for perimeter of rhombus of all formulas of the area of rhombus. A triangle sides all have the same formula used to find the formula... = b×h them one by one below to solve this problem, apply the perimeter,! A = b×h special type of parallelogram the radius that is to be found, also knowing its.. Will do, they are also perpendicular to each other as it a. Problem, apply the perimeter of a rhombus: = 4s Substitute 16 for s. = 4 16! 2: if the perimeter, of a rhombus into four absolutely identical right-angled.. The diagonals whose values are given if one of its diagonal only a side length 10! Four sides all have the same length rhombus in the table below one. Where b is the sum of all formulas of the other diagonal the rhombus = 72 inches, find... In multiple ways same length Example 2: if the perimeter of a square is just rhombus! All have the same length know to begin with we will saw each of them one by below! This is because both shapes, by definition, have equivalent sides if of... Lembaga Pengarah Yapeim, Daedalus Pronunciation Greek, Single Room For Rent In Vishnupuri Indore, Ck2 Reconquista Decision, Frenemies Full Movie, Gallerie Dell Accademia Venezia Collezione, Seoultech Postal Code, Best Wishes, Warmest Regards: A Schitt's Creek Farewell Song, What Song Is Amish Paradise Based Off, View all View all View all View all View all ## The Life Underground ### ## Cooling Expectations for Copenhagen Nov.16.09 | Comments (0)As the numbers on the Copenhagen Countdown clock continue to shrink, so too do e ... 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https://math.stackexchange.com/questions/1431396/prove-that-any-integer-that-is-both-square-and-cube-is-congruent-modulo-36-to-0

# Prove that any integer that is both square and cube is congruent modulo 36 to 0,1,9,28 This is from Burton Revised Edition, 4.2.10(e) - I found a copy of this old edition for 50 cents. Prove that if an integer $a$ is both a square and a cube then $a \equiv 0,1,9, \textrm{ or } 28 (\textrm{ mod}\ 36)$ An outline of the proof I have is Any such integer $a$ has $a = x^2$ and $a = y^3$ for some integers $x,y$ Then by the Division Algorithm, $x = 36s + b$ for some integers $s,b$ with $0 \le b \lt 36$ and $y = 36t + c$ for some integers $t,c$ with $0 \le c \lt 36$ Using binomial theorem, it is easy to show that $x^2 \equiv b^2$ and $y^3 \equiv c^3$ Then $a \equiv b^2$ and $a \equiv c^3$ By computer computation (simple script), the intersection of the possible residuals for any value of $b$ and $c$ in the specified interval is 0,1,9,28 These residuals are possible but not actual without inspection which shows $0^2 = 0^3 \equiv 0$ , $1^2 = 1^3 \equiv 1$ , $27^2 = 9^3 \equiv 9$, and $8^2 = 4^3 \equiv 28$ $\Box$ There is surely a more elegant method, can anyone hint me in the right direction. • I removed my answer, I had to have a coffee instead :P – Paolo Leonetti Sep 12 '15 at 14:19 First establish that $a$ must be a sixth power. We have $a=b^2=c^3$ so that $a^3=b^6$ and $a^2=c^6$ whence $$a=\cfrac {a^3}{a^2}=\cfrac {b^6}{c^6}=\left(\cfrac bc\right)^6$$ And if $q$ is a rational number whose sixth power is an integer, it must be an integer itself. [see below] Now, let's have a look at the sixth powers modulo $36$. Every integer is congruent to a number of the form $6a+b$ where $-2\le a,b \le 3$. Then a simple application of the binomial theorem gives that: $$(6a+b)^6\equiv b^6 \bmod 36$$ Finally, checking all the possibilities for $b$ we see $$(-2)^6=2^6=64\equiv 28; (-1)^6=1^6=1; 0^6=0; 3^6=81^2\equiv 9^2=81\equiv 9$$ Suppose $a,m,n \in \mathbb N$ with $a=\left(\frac mn\right)^6$ with $\frac mn$ in lowest terms and suppose $p$ is a prime factor of $n$ so that $n=pd$ with $d\in \mathbb N$. Then we have $an^6=m^6=ap^6d^6$ whence $p|m^6$ and because $p$ is prime $p|m$. But this is a contradiction since $m$ and $n$ were constructed to have no common factor. Hence $n$ has no prime factors and $n=1$. What you did is correct, but yes, a lot of the work (especially the computer check) could have been avoided. Firstly, if $a$ is both a square and a cube, then it is a sixth power. This is because, for any prime $p$, $p$ divides $a$ an even number of times (since it is a square), and a multiple of 3 number of times (since it is a cube), so $p$ divides $a$ a multiple of 6 number of times altogether, and since this is true for any prime $p$, $a$ is a perfect sixth power. So write $a = z^6$. Next, rather than working mod $36$, it will be nice to work mod $9$ and mod $4$ instead; this is equivalent by the chinese remainder theorem. So: • Modulo $9$, $z^6 \equiv 0 \text{ or } 1$. You can see this just by checking every integer or by applying the fact that $\varphi(9) = 6$. • Modulo $4$, $z^6 \equiv 0 \text{ or } 1$. This is easy to see; $0^6 = 0$, $1^6 = 1$, $(-1)^6 = 1$, and $2^6 \equiv 0$. So $a = z^6$ is equivalent to $0$ or $1$ mod $4$ and mod $9$. By the chinese remainder theorem, this gives four possibilities: • $a \equiv 0 \pmod{4}, a \equiv 0 \pmod{9} \implies a \equiv 0 \pmod{36}$ • $a \equiv 0 \pmod{4}, a \equiv 1 \pmod{9} \implies a \equiv 28 \pmod{36}$ • $a \equiv 1 \pmod{4}, a \equiv 0 \pmod{9} \implies a \equiv 9 \pmod{36}$ • $a \equiv 1 \pmod{4}, a \equiv 1 \pmod{9} \implies a \equiv 1 \pmod{36}$. • very nice - thank you - chinese remainder theorem is not covered until 10 pages after this problem in the text so this will provide additional motivation for it for me. The fundamental theorem of arithmetic has been covered and so I should have seen your argument about a being a sixth power from that. – topoquestion Sep 11 '15 at 20:44

http://adunataalpini-pordenone2014.it/nrwx/graphs-of-sine-and-cosine-functions-answer-key.html

# Graphs Of Sine And Cosine Functions Answer Key Find arc lengths and areas of. What is the equation for the cosine function graphed here? Gimme a Hint. I can graph the cosine function and its translations. NOW is the time to make today the first day of the rest. To define what these 3 functions, we first have to understand how to label the sides of right-angled. The graphs that we just explored are the parent functions. Graphs of Trigonometric Functions - Questions. 7 -8 Wednesday 10/23 Continue Graphing Sine and Cosine (Period Changes) Worksheet graphing problems #9 - 16 on pp. The value you get may be 0, but that's a number, too. Day 62 S Of Sinusoidal Functions After Notebook. Let's find the x-component of d 1. Sine and Cosine Graphs: Vertical Dilation and Reflection across x-axis. However, they do occur in engineering and science problems. Yes, you can derive them by strictly trigonometric means. Sample answer: One sinusoidal function in which a = 1. 4 Trigonometric Functions of Any Angle p. In order to solve these equations we shall make extensive use of the graphs of the functions sine, cosine and tangent. 5 Graphs of the Trigonometric Functions 791 corners or cusps. Day 1 - Parent Graphs and Transformations Worksheet 1 - Answer Key. Give the amplitude and period of each function. 1 Graphing Sine and Cosine. Graph sine and cosine functions N. Learn how to construct trigonometric functions from their graphs or other features. Sorry but it won't allow me to copy and paste the graph so I hope you guys know how this graph looks like from the function. I can graph sine function and its translations. Precalculus Prerequisites a. Student needs to show proof. 8 Sketching Trig Functions. −≤ ≤ππx Is the cosine function even, odd, or neither? Communicate Your Answer 3. Let’s start off with an integral that we should already be able to do. Plus each one comes with an answer key. 2 Practice Worksheet More Graphing Trigonometric Functions Worksheet Answers Sec 5. Then use the information to find the critical points and sketch two cycles for the graph (one to the right and one to the left of the center point). Extra Practice - Combined Transformations Note: Answer key is provided on the backside of the sheet. TRIGONOMETRY. Trigonometric graphing math a graph of graph shows the trigonometric graphing review flamingo math answer key. com Section 9. y 5 cos Qx 2 p 2R 1 2 13. A graph of the function p I (x) = —cos p(x) = —cos x in the followmg diagram —a X VCheck your Understanding For each of these functions, without using technology. Precalculus Chapter 6 Worksheet Graphing Sinusoidal Functions in Degree Mode Find the amplitude, period, phase (horizontal) displacement and translation (vertical displacement). The coefficient affects the period (which can be considered a horizontal stretch if. View answers. • Sketch translations of these functions. Amplitude = Equation (2) = Phase Shift = (in terms of the sine function) Period =. y = 3 sin 2x 17. • Apply addition or subtraction identities for sine, cosine, and tangent. Extend the graph of the cosine function above so that it is graphed on the interval from [—180, 720]. Graphs of Trigonometric Functions - Questions. Period of a Function from Graphing Sine And Cosine Functions Worksheet, source:math. to the right. If f is sine or cosine, then −1 ≤ a ≤ 1 and, if f is tangent, then a ∈ R. • Use amplitude and period to help sketch graphs. Graphs of Sine and Cosine Below is a table of values, similar to the tables we’ve used before. 5 Graphs of the Tangent, Cotangent, Cosecant and Secant Functions (corrected) Annotated Notes. Find the area of oblique triangles; Practice Pages. BE as accurate with your graphing as possible. Let's start with the basic sine function, f (t) = sin(t). In fact, the key to understanding Piecewise-Defined Functions is to focus on their domain restrictions. Given the following diagram: Find the cosB and write your answer as a. College Trigonometry Version bˇc Corrected Edition by Carl Stitz, Ph. Identify the amplitude and period. sin o h p yp p cos tan h a y d p j o a p d p j csc sec o h p yp p h a y d p j cot o a p d p j Notice that the sine, cosine, and tangent functions are reciprocals of the cosecant, secant, and cotangent functions, respectively. The summarized table for trigonometric functions and important Formula as follows:. Exploring Sine and Cosine Graphs Learning Task. Graphing Tangent and Cotangent. 5 Graphs of the Tangent, Cotangent, Cosecant and Secant Functions (corrected) Annotated Notes. 3 2 ≈08660. Write two different equations for the same graph below. Create AccountorSign In. Powered by Create your own unique website with customizable templates. The Sine, Cosine and Tangent functions. Key included. Sine θ can be written as sin θ. Label the axes appropriately. 6 Phase Shift. All graphs were computer generated and adjusted to be easy to read for students. The graphs overlap. This builds into learning about graphing and interpreting logarithmic functions and models. Let's go a little further…. So: sin à L 1 csc à and csc à L 1 sin à The cosine and secant functions are reciprocals. Show Answer. Students will have mastered the unit circle, memorizing the coordinates of various key angles to quickly determine the lengths of the sides of common right triangles. Sample Test Answer Key Trigonometric Functions and Their Graphs. CPM Educational Program is a 501(c)(3) educational nonprofit corporation. What is the range of f(x) = sin(x)? the set of all real numbers -1 < or = y < or = 1 Which set of transformations is needed to graph f(x) = -2sin(x) + 3 from the parent sine function?. 7 Test Review Worksheet Answer Key Quadratic Functions, Graphing, and Applications. Sketch the graph of the rectangular function y = 2 cos 2 x on the interval [0, ]. Graphing Sine and Cosine Group Exploration Activity 2. Any cosine function can be written as a sine function. 5a worksheet. 001; − 1 440 Practice Graphing Sine and Cosine Functions y x 0 π 2π 4π-1 1 f (x) = sin x g (x) = 1 sin x 3 y x π 2π 3π 4π-1 1 f (x) = cos x g (x) = - cos x 1 4 y 0 x 2 4-4-2π -π π 2π 0 x 2-2 4-π π π 2 π 2-4-4. The graph could represent either a sine or a cosine function that is shifted and/or reflected. Graphing Sine And Cosine Practice Worksheet. Family of sin Curves example. What is cos( 1)?. 15 sin 330 t. Find an equation for a sine function that has amplitude of 4, a period of fl. Graphing Sine Function - Displaying top 8 worksheets found for this concept. Find the horizontal translation of a sine or cosine function. The five key points include the minimum and maximum values and the midline values. 3 Connecting Graphs to Rational Equations Assigned: Pages 465-467 : Practice section #1-6 (at least 2 letters each); at least 6 from Apply/Extend (A/E). 5 Graph of the Tangent Function: 11. Summarize what you have learned here. Inverse Functions and Equations. 2 - Graphs of Rational Functions; Assign 3. ZIT Give the amplitude and period of each function. Rather than trying to figure out the points for moving the tangent curve one unit lower, I'll just erase the original. Their behavior will only be explored in this lesson. graph 21) a. Amplitude, Period, Phase Shift and Frequency. This online trigonometry calculator will calculate the sine, cosine, tangent, cotangent, secant and cosecant of angle values entered in degrees or radians. At x-values where the sine and cosine function is zero, the cosecant and secant functions have vertical asymptotes. Further, tangent function will be undefined when its denominator is zero. The rest we can find by first finding the reference angle. Graphs of these functions The period of a function The amplitude of a function Skills Practiced. Standard: MATH 3 Grades: (9-12) View lesson. Application Walkthrough. Then its graph is:-6 (The hash marks on the x-axis are in increments of ˇ=2. The graph is a smooth curve. 5 Graphs of Sine and Cosine Functions p. 2 Trigonometric Functions: The Unit Circle p. 1) y = 3cos2q Answers to Graphing Sine and Cosine 1) p 2 p3p 2 2p-6-4-2 4 6. worksheet on graphing sine and cosine functions Images about Worksheet On Graphing Sine And Cosine Functions: Chemical Equations Worksheet With Answers,. ) The Sine Ratio The ratio between the leg opposite a given angle of a right triangle and the triangle’s hypotenuse. More precisely, the sine of an angle $t$ equals the y. Graphing Sine and Cosine Functions Find each value by referring to the graph of the sine or the cosine function. 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. A trigonometric Table is a table of ratios of sides. 1 State the period and amplitude (if any) given the function rule or the graph of a sine, cosine, or. Learn vocabulary, terms, and more with flashcards, games, and other study tools. y x y Part (b): h x x x 44 Part (a): h x x 4. Findthesine,cosineandtangentof € π 3. Show Answer. Mar 12/13 9. 5b worksheet. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0. 1 2) y cos 2 1) y. The Tangent Ratio. Then graph of the function over the interval -2 ≤ x ≤ 2. This lesson presents the basic graphing strategies used to graph generalized sine and cosine waves from a conceptual point of view1. y = 2 sin x - 3. Basic Graphs of Sine and Cosine. Using the powerful tools of shifts and stretches to parent functions, this presentation walks the learner through graphing trigonometric functions by families. 3 Connecting Graphs to Rational Equations Assigned: Pages 465-467 : Practice section #1-6 (at least 2 letters each); at least 6 from Apply/Extend (A/E). Explore how changing the values in the equation can translate or scale the graph of the function. Powered by Create your own unique website with customizable templates. Amplitude and period for sine and cosine functions worksheet answers. 2 The Unit Circle and Circular Functions - 6. 7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Some of the worksheets for this concept are Honors algebra 2 name, Of the sine and cosine functions, , Graphs of trig functions, Work 15 key, 13 trigonometricgraphswork, 1 of 2 graphing sine cosine and tangent functions, Sine cosine and tangent practice. involved in building new functions from existing functions. Worksheets are Graphing trig functions, Graphs of trig functions, Work 15 key, 1 of 2 graphing sine cosine and tangent functions, Amplitude and period for sine and cosine functions work, Precalculus chapter 6 work graphing sinusoidal, Work properties of trigonometric functions, Honors algebra 2 name. Find the midpoint of the interval by adding the x-values of the endpoints and dividing by 2. Graph the function using at least 3 key points. Lesson 10: Basic Trigonometric Identities from Basic Trigonometric Identities from Graphs List ways in which the graphs of the sine and cosine functions are. District programs, activities, and practices shall be free from discrimination based on race, color, ancestry, national origin, ethnic group identification, age, religion, marital or parental status, physical or mental disability, sex, sexual orientation, gender, gender identity or expression, or genetic information; the perception of one or more of such characteristics; or association with a. Trig Graphs Answer Section MULTIPLE CHOICE 1. How to sketch the graphs of basic sine and cosine functions Important Vocabulary Define each term or concept. Solution: Cosine Function. The rest we can find by first finding the reference angle. y = sin 2 θ 21. ‘Chapter 0’ by Carl Stitz, Ph. trigonometric graphs use mini whiteboards to answer. Domain and Range of Trig Functions. Lesson #12­The Graphs of the Trigonmetric Ratio Functions done. The graphs of tan x, cot x, sec x and csc x are not as common as the sine and cosine curves that we met earlier in this chapter. In these examples we will graph a sine and cosine function using a table of values. We show a right triangle below. Find all inflection points and describe them in derivative language. ; Daniels, Callie , ISBN-10: 013421742X, ISBN-13: 978-0-13421-742-0, Publisher: Pearson. functions using different representations. 5 Match polar equations and graphs Z. y = 5 sin. 2 Graphing Sine and Cosine F 13 MAY 2016 - 8. Values of the other trigonometric functions at the angles listed above can be found easily, since the other functions are all built from sine and cosine. Pre-Calculus 1. What is cos( 1)?. Equations and Inequalities Multi-step equations Work word problems Distance-rate-time word problems Mixture word problems Absolute value equations. Graph coordinates. Translating and Scaling Sine and Cosine Functions. If the graphs of the equations y = 2 and y 2 sin x are drawn on the same set of axes, the number of points of intersection between 0 and 27t will be 24. Using degrees, find the amplitude and period of each function. 3) Which one of the equations below matches the graph? A) y = -2 cos 3x B) y = 2 sin 1 3 x C) y = -2 sin 3x D) y = -2 sin 1 3 x 3) SHORT ANSWER. An inverse sine function will return the arc (angle on the unit circle) that pairs with its y-coordinate input. Let’s go a little further…. Textbook Authors: Lial, Margaret L. Although the inverse of a function looks like you're raising the function to the -1 power, it isn't. Worksheet 6. The general sine and cosine graphs will be illustrated and applied. The Lesson: In a right triangle, one angle is and the side across from this angle is called the hypotenuse. 5 Quiz and Area of Oblique Triangles W 18 MAY 2016 - 8. 9) For what numbers x, 0 ≤ x ≤ 2π, does sin x = 0? 9) Match the given function to its graph. Solution: Cosine Function. , ISBN-10: 0-13446-914-3, ISBN-13: 978-0-13446-914-0, Publisher: Pearson. Then sketch the graph. PDF LESSON 6 Basic Graphs of Sine and Cosine. Explanations will vary. Related Topics: More Lessons on Finding An Equation for Sine or Cosine Graphs More Algebra 2 Lessons More Trigonometric Lessons Videos, worksheets, games and activities to help Algebra 2 students learn how to find the equation of a given sine or cosine graph. Exploring Trigonometric Graphs they may group the functions where the coefficient of sin x or cos x is 2. 1 Page 233 Question 1 a) One cycle of the sine function y = sin x, from 0 to 2π, includes three x-intercepts, a maximum, and a minimum. How Do You Trigonometric Graphs from Graphing Sine And Cosine Functions Worksheet, source:pinterest. Transformations of the Sine and Cosine Graphs from Graphing Trig Functions Worksheet, source: jwilson. Amplitude and period for sine and cosine functions worksheet answers. ) The Sine Ratio The ratio between the leg opposite a given angle of a right triangle and the triangle’s hypotenuse. Therefore, a sinusoidal function with period DQGDPSOLWXGH WKDWSDVVHV through the point LV y = 1. Find the value of the coordinates of the points A, B, and C. The distance between the highest and lowest point is 3 feet. Trigonometry functions. 4 Trigonometric Functions of Any Angle p. A sine graph is a graph of the function =y sin θ. Sine represents the y-coordinate on the unit circle while cosine represents the x-coordinate. [NEW] Real Life Examples Of Sine Cosine And Tangent A real life example of the sine function could be a. Download free on iTunes. Give the amplitude, period, and vertical shift of each function graphed below. Worksheets are Graphing trig functions, Graphs of trig functions, Amplitude and period for sine. Extra Practice - Combined Transformations Note: Answer key is provided on the backside of the sheet. The general form of a cosine function can also be. Describe how the graphs of f ( x ) = sin x and g ( x ) = sin 4 x are related. Sine, Cosine, Tangent Chart. Students will know how to use the sine, cosine, tangent and their reciprocal and inverse functions to determine unknown sides and angles of right triangles. Extend the graph of the cosine function above so that it is graphed on the interval from [—180, 720]. Graphing Sine and Cosine find the amplitude and period of each function. Amplitude = | a | Let b be a real number. It consists of several rows or columns that spread out all over the page and create for space that assist people fill data. Students will know how to use the sine, cosine, tangent and their reciprocal and inverse functions to determine unknown sides and angles of right triangles. Polynomial Function Graphs Properties of Functions Quadratic Solver Rational Functions - 2nd Degree Over 2nd Degree Right Triangles Sine, cosine and tangent of an angle Solving an Ellipse Solving Linear Equations #2 Systems of Inequalities (part 2) The MovingMan Project Trig Function Point Definitions Trig Functions -A pplet Trigonometry. Geometry of Complex Numbers. Find arc lengths and areas of. 9) 10) Domain: Range: Domain: Range: Amplitude: 2 Period: Amplitude: 1 Period: π. These functions are mainly used in geometry especially for navigation, geodesy and celestial mechanics purposes in the professional world. The graph of y= sin 1 xlooks like:. Domain and Range of Trig Functions. Graphing Sine Function. So: tan à L 1 cot à and cot à L 1 tan à. Sketch the graph of the function over the interval -2( ≤ x ≤ 2(. sin 7! 2! 1 1 1 Find the values of •for which each equation is true. It explains how to identify the amplitude, period, phase shift, vertical shift, and midline of a sine or cosine function. Feel free to download and enjoy these free worksheets on functions and relations. Worksheet by Kuta Software LLC MAC 1114 - Trigonometry Name_____ 7. The cosine graph is the same as the sine except that it is displaced by 90. Translating Sine and Cosine Functions. Then graph of the function over the interval —27t x 21t. Algebra Worksheets. The graph of the sinusoid y = 3 si n (2x π)is given below. Slope And Graphing Lines Review Worksheet Slope And Graphing Lines Review Graphing Point Slope Form Graphing Linear Equations By Slope-intercept Form Practice Graphing Linear Equations By Slope Intercept Answer Key Graphing Vs Substitution Worksheet Graphing Linear Equations Worksheet Graphing And Substitution Worksheet Answers Slope Applications Worksheet Graphing Inequalities In Two. 2 EXPLORATION: Graphing the Cosine Function 1 −1 y x 2 π 2 π π 2 − − −π − 3π 2π 2 2π 3π 2 5π. Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range - Duration: 18:35. Inverse Sine Function (Arcsine) Each of the trigonometric functions sine, cosine, tangent, secant, cosecant and cotangent has an inverse (with a restricted domain). The period of any sine or cosine function is 2π, dividing one complete revolution into quarters, simply the period/4. Translating and Scaling Sine and Cosine Functions. 2 3 2 S S 2 S S 2 S 3 2 S 2S S y x (c) The domain and range of the sine and cosine Answer Key yx cos yx sin. Precalculus Graphs of Trigonometric Functions Graphing Sine and Cosine. Its input is the measure of the angle; its output is the y -coordinate of the corresponding point on the unit circle. PDF LESSON. My act answer key has f, i just want to know how it got that bcuz i have no clue so showing work would be very much appreciated. The graph could represent either a sine or a cosine function that is shifted and/or reflected. y = cos 2 πθ Write an equation of a cosine function for each graph. • Develop and use the Pythagorean identity (sin cos 1tt)22+=( ). Or we can measure the height from highest. 4 Writing the equation of Sine and Cosine: 11. Finding the equation of a parabola using focus and directrix. Feel free to download and enjoy these free worksheets on functions and relations. The general sine and cosine graphs will be illustrated and applied. 4 Trigonometric Functions of Any Angle p. I can graph sine function and its translations. 18 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases… d) Graph trigonometric functions, showing period, midline, and amplitude. Mixed Review: Equations & Graphing of Trig Functions. To account for a phase shift of , subtract from the x-values of each of the key points for the graph of y = 2 sin 5x. Solution: Cosine Function. In fact Sine and Cosine are like good friends: they. 1) Answers to Graphing Sine and Cosine 1) p 2 p3p 2 2p-6-4-2 2 4 6. y = 4 tan 6. When you write a sine or cosine function for a sinusoid, you need to find the values of a, b>0, h, and kfor y= a sin b(x º h) + k or y = a cos b(x º h) + k. Graphs Of Sine - Displaying top 8 worksheets found for this concept. 5: Graphs of Sine and Cosine Functions) 4. Student needs to show proof. The amplitude is a=2 and the period is. 5 6 sin sin. Graphs Of Sine. In fact Sine and Cosine are like good friends: they. 2 Graphing Sinusoidal Functions using 5 Points Method Sec 5. 4 - Trigonometric Functions of Any Angle - Exercise Set - Page 575 40 including work step by step written by community members like you. Graphing Sine and Cosine Functions Worksheet – careless from Solubility Curve Worksheet Answer Key, source: careless. 1 Graphs of Sine and Cosine Let y= sinx. Edmonds will check it after you graph it. Key features of parent functions are reviewed and then the changes to those characteristics detailed and demonstrated. 6 Graphs of Reciprocal Trig Functions: 11. 6 Writing Equations for Sin/ Cos Graphs 13. The graph for tan(θ) - 1 is the same shape as the regular tangent graph, because nothing is multiplied onto the tangent. CPM Educational Program is a 501(c)(3) educational nonprofit corporation. Graphing Sine and Cosine Functions Worksheet – careless from Solubility Curve Worksheet Answer Key, source: careless. Download Free Graphing Sine Answer Key Graphing Sine Answer Key Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range This trigonometry and precalculus video tutorial shows you how to graph trigonometric functions such as sine and cosine Graphing a Sine Function by Finding the Amplitude and. Then you will learn about modeling trigonometric functions by graphing the sine and cosine functions. 1 and we have plotted the key points to account for the x-axis reflection. Then graph. The cosine function of an angle. We can see this in two ways: It follows immediately from the formula. Sine & Cosine Graphs By: Taylor Pulchinski Daniel Overfelt Whitley Lubeck Equations y = a sin (bx-h)+ k y = a cos (bx-h)+k a = Amplitude (height of the wave) 2( )/b = Period (time it take to complete one trip around) h = Phase Shift (left or right movement) k = Vertical Shift (up or down movement) Examples Finding the Period and Amplitude. Trigonometry functions. Describe the end behavior of the graph of y = sin x. Once the appropriate base value of the first quadrant is known, symmetric points in any other quadrant can be. Name Period Group # y: sin x Amplitude: Period: y = 2 sin Amplitude: Period: Y = 3 cos (42x) Amplitude: Period: Y = sin 4x Amplitude = Period y: 4 cos x Amplitude = Period - y = 3 sin Amplitude = Period - y cos 5x Amplitude = Period: y: -2 sin x. · More Work with Graphing Cosine & Sine Functions – More Practice. 001 sin 880tπ. m) and the axle height is thus 40 m (the mean of 10 m and 70 m). 4 Graphing Sine and Cosine Functions 487 Each graph below shows fi ve key points that partition the interval 0 ≤ x ≤ 2π — into b four equal parts. We’re going to start thinking of how to get the graphs of the functions y=sin x and yx=cos. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may. Each degree with special angles This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides practice problems on identifying the sides that are opposite and adjacent to a given angle. 57 = 90^@)`. What is the equation for the cosine function graphed here? Gimme a Hint. The area of triangle ABC is 450 ml. Then write an equation of each graph. y 5 sin Qx 1 p 2R 11. Chapter(14(–(TrigonometricFunctions(andIdentities(Answer’Key(CK912Algebra(II(with(Trigonometry(Concepts( 16! 14. It is mandatory to procure user consent prior to running these cookies on your website. Graphs of Trigonometric Functions - Questions. Whats people lookup in this blog: Tangent Tables And Graphs Answer Key. Experiment with the graph of a sine or cosine function. Solve Graphing Trigonometric Functions : Tanx and Cotx. This is a great mini-unit on Graphing Trigonometric Functions for Algebra 2 or Pre-Calculus students. 5, Graphs of Sine and Cosine Functions Homework: 4. Answer: The amplitude is 0. 862 Chapter 14 Trigonometric Graphs, Identities, and Equations Modeling with Trigonometric Functions WRITING A TRIGONOMETRIC MODEL Graphs of sine and cosine functions are called sinusoids. This is a problem. The tangent of any angle. View answers. Find the vertical translation of a sine or cosine function. When you write a sine or cosine function for a sinusoid, you need to find the values of a, b>0, h, and kfor y= a sin b(x º h) + k or y = a cos b(x º h) + k. The of the graph of a periodic function is the absolute value of half the difference between its maximum value and its minimum value. The Sine, Cosine and Tangent functions. 64 Key points in graphing the sine function Graph variations of y = sin x. 2 Graphs of Rational Functions. Section 2: Graphs of Trigonometric Functions Lesson 1 Sine and Cosine Graphs 205 Lesson 2 Transformations of the Sine and Answer Key 1. Then use a graphing calculator to sketch the graphs of f(x), -f(x), and the given function in the same viewing window. 5: Graphs of Sine and Cosine Functions) 4. to the right. (307k) Rachel Van Hoose,. Student needs to show proof. These basic waves have the property that they deviate from the t-axis by no more than one unit. The three basic trig functions are the sine, cosine and the tangent. 1 Graphing Sine and Cosine Functions Some Points to Consider • You should be able to plot graphs of functions y=n and y=s , as well as b= n and b= cos. f = 440, a = 0. The graph. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may. PDF LESSON. y = −cos θ 6. As one goes up, the other goes down and vice versa. 6_writing_sin_cos. Graphs Of Sine And Cosine Functions. Using the powerful tools of shifts and stretches to parent functions, this presentation walks the learner through graphing trigonometric functions by families. Absolute Value Equations. 2 Homework Worksheet - Due Tuesday Thursday - October 18: 3. x 0o 30o 45o 60o 90o 120o 135o 150o 180o 210o 225o 240o 270o 300o 315o 330o 360oy = sin xy = cos xWhat you are seeing are the graphs of the sine and cosine. Feel free to download and enjoy these free worksheets on functions and relations. Functions - Inverse Trigonometric Functions Objective: Solve for missing angles of a right triangle using inverse trigonometry. k x 3cos2 x Amplitude: 5; period: 2 Amplitude: 3; period: 1 Using f x sinx or g x cosx as a guide, graph each function. Check your answers with those on the answer key. Optional Practice: Pg. Explore how changing the values in the equation can translate or scale the graph of the function. Here we will do the opposite, take the side lengths and find the angle. The graph never moves outside this range of values. Trigonometric Equations. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may. y = 4 sin x 12. Demonstrates answer checking. BACK TO EDMODO. 3 The student response demonstrates a good understanding of the Functions concepts involved in building new functions from existing functions. Theorem 10. You need to complete the notes outline and upload to schoology by Friday 5/1 at 11:59PM. Then graph. 5, Graphs of Sine and Cosine Functions Homework: 4. The graphs of sine, cosine, and tangent: Graphs of trigonometric functionsIntroduction to amplitude, midline, and extrema of sinusoidal functions: Graphs of trigonometric functionsFinding amplitude and midline of sinusoidal functions from their formulas: Graphs of trigonometric functions. 8 (3-8, 9-25 odd, 35-38, 39-43 odd, 47, 48) Thu/Fri 8/30-31: 4: Section 1. (b) An angle is a right angle if it equals 90. Find an equation for a sine function that has amplitude of 5 a period of 3π. These cyclic natures of the sine and cosine functions make them periodic functions. The graphs overlap. The graphs of sine, cosine, and tangent: Graphs of trigonometric functionsIntroduction to amplitude, midline, and extrema of sinusoidal functions: Graphs of trigonometric functionsFinding amplitude and midline of sinusoidal functions from their formulas: Graphs of trigonometric functions. Findthesineandco sineof45°. First Name:. The Law of Sines The Law of Cosines Graphing trig functions Translating trig functions Angle Sum/Difference Identities Double-/Half-Angle Identities. 6 Inverse Functions; Chapter 2: Linear Function. Key included. More Work with the Sine and Cosine Functions. Use the sine tool to graph the function. Exploring Trigonometric Graphs © Project Maths Development Team 2012 www. com Section 9. Another important point to note is that the sine and cosine curves have the same shape. y = 4 tan 6. 6 Graphs of Other Trigonometric Functions p. • Complete the review and practice test exercises from the textbook. TRIGONOMETRY. When you click the button, this page will try to apply 25 different trig. Feb 28 - We worked on writing the equations of sine and cosine functions then learned how to graph cosecant and secant functions using their corresponding sine or cosine function. In order to solve these equations we shall make extensive use of the graphs of the functions sine, cosine and tangent. Formulas for cos(A + B), sin(A − B), and so on are important but hard to remember. Writing equations of trig functions from a verbal description of amplitude, period, phase shift, and/or vertical displacement, or from a given graph. It is where the sine and the cosine rule enter trigonometry. The Cosine Graph a. Sine, Cosine, Tangent Chart. Start studying 4. Worksheets are Graphing trig functions, Graphs of trig functions, Amplitude and period for sine. 2 Trigonometric Functions: The Unit Circle p. Defining Sine and Cosine Functions. Graph coordinates. Key features of parent functions are reviewed and then the changes to those characteristics detailed and demonstrated. trigonometric graphs use mini whiteboards to answer. Period of a Function from Graphing Sine And Cosine Functions Worksheet, source:math. The tangent of any angle. (3) is halved. Graphs of Transformations of Sine and Cosine. Geometry of Complex Numbers. 9) For what numbers x, 0 ≤ x ≤ 2π, does sin x = 0? 9) Match the given function to its graph. 3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for x, π + x, and 2π - x in terms of their values for x, where x is any real number. Using the values in the table, sketch the graph of the cosine function on the interval [0, 360]. y = sin 4x 2. I begin today by putting the graph of y=sin x on the board and ask:. 9) 10) Domain: Range: Domain: Range: Amplitude: 2 Period: Amplitude: 1 Period: π. We used a special function, one of the trig functions, to take an angle of a triangle and find the side length. Sine θ can be written as sin θ. These basic waves have the property that they deviate from the t-axis by no more than one unit. graphs of these functions all have vertical asymptotes at these points. This article will teach you how to graph the sine and cosine functions by hand, and how each variable in the standard equations transform the shape, size, and direction of the graphs. 6 Graphs of the Sine and Cosine Function Graph each function using degrees. , ISBN-10: 0-13446-914-3, ISBN-13: 978-0-13446-914-0, Publisher: Pearson. Absolute Value of Complex Numbers. Text books make graphing trig functions so complicated. m) and the axle height is thus 40 m (the mean of 10 m and 70 m). a) y = sin x Domain_____. Amplitude and period for sine and cosine functions worksheet answers. y 5 cos x, p 2. 2 Practice Worksheet More Graphing Trigonometric Functions Worksheet Answers Sec 5. Modeling quadratic functions (quadratic word problems). Chapter 8: Applications of Trigonometry. Pre Calculus 12 - Graphing Trig Functions Test Answer Section MULTIPLE CHOICE 1. y 5 sin x 1 2 Graph each function in the interval from 0 to 2π. 5 and b = 4 is y = 1. For example, to evaluate sin(48), what math process could I use if I didn't have a calculator? Calculator Addition [6/30/1996] How does a calculator add? Calculator for Algebra I and Physical Science [8/11/1995]. 3 Properties of the Trigonometric Functions. Some of the worksheets for this concept are Graphs of trig functions, Amplitude and period for sine and cosine functions work, 1 of 2 graphing sine cosine and tangent functions, , Trig graphs work, Of the sine and cosine functions, Work 15 key, Honors algebra 2 name. Application Walkthrough. Graphing Sine Cosine And Tangent. Experiment with the graph of a sine or cosine function. 3 Modeling with Linear Functions; 2. 5 Graphing Trig Functions WS. f(9) = 29-7 2. Express your answer as a fraction in lowest terms. functions using different representations. 3 62/87,21 The general form of the equation is y = a sin bt, where t is the time in seconds. Sketch the graph of the function over the interval -2( ≤ x ≤ 2(. cos !"! 1 n, where n n, where n n, where n is is any integer is an even integer an odd integer Graph each function. y = -4 cos 3. You should know the four components of a sine/cosine function: A, B, C, and D. Statistics On-Line. cos !"! 1 n, where n n, where n n, where n is is any integer is an even integer an odd integer Graph each function for the given interval. Trigonometric graphing math a graph of graph shows the trigonometric graphing review flamingo math answer key. Gimme a Hint. Although the inverse of a function looks like you're raising the function to the -1 power, it isn't. • Complete the exercises for each section. The radius, r, is always some positive. 11) Worksheets pp. The \$$x\$$-values are the angles (in radians – that’s the way … Graphs of Trig Functions. Rather than trying to figure out the points for moving the tangent curve one unit lower, I'll just erase the original. Therefore, the sum of the zeros of the function is equal to –b. This Homework is meant to solidify the student's understanding of the shape and basic features of both the sine and cosine graphs. The Cosine Graph a. This becoming stated, we provide you with a a number of straightforward yet useful content and also web themes designed made for just about any helpful purpose. 2 Trigonometric Ratios of any angle 5. Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the given function. We will use the definition of the sine and cosine functions on the unit circle (r =1) to find the sine and cosine for common reference angles. Some of the worksheets displayed are Honors algebra 2 name, Sine cosine and tangent practice, , Amplitude and period for sine and cosine functions work, 1 of 2 graphing sine cosine and tangent functions, Graphing trig functions, Graphs of trig functions, Work 15 key. Sine and cosine functions are periodic functions. But this graph is shifted down by one unit. 2 Graphs of the Sine and Cosine Functions A Periodic Function and Its Period A nonconstant function f is said to be periodic if there is a number p > 0 such that f(x + p ) = f(x) for all x in the domain of f. Sketch the graphs off x = sin x g(x) = sin x- 1 over at least one period, labeling each axis. y 5 sin x, 2p units right 16. 12 - 13 Friday 10/25 Writing functions cont'd Quiz - Graphing Sine and Cosine. How Do You Trigonometric Graphs from Graphing Sine And Cosine Functions Worksheet, source:pinterest. Thanks for visiting our website, article about 21 Common Core Algebra 2 Unit 1 Answer Key. (Check your answer with your graphing calculator!) f x x( ). I need to describe the Amplitude, Period, Domain, Range and X-intercepts of the graphs of one of the following cosine functions and then relate each property to the unit circle definition of cosine. Sine and Cosine Functions Now that you know how to draw the basic sine and cosine curves, you will turn your attention to some transformations of these basic curves. Exercise #1: Consider the function fx x sin 3. 26 (3-7 odd, 9-12 All, 26) Thu/Fri 9/6-9/ 6. Their graphs are complex and interesting. Application Walkthrough. Some of the worksheets for this concept are Honors algebra 2 name, Of the sine and cosine functions, , Graphs of trig functions, Work 15 key, 13 trigonometricgraphswork, 1 of 2 graphing sine cosine and tangent functions, Sine cosine and tangent practice. Sine θ can be written as sin θ. Tags: Question 3. 6for a review of these concepts. The cosine function of an angle. Create AccountorSign In. 5-1 1 0 π_ 2 3__π 2 5__π 2-π_ 2 Period Period One Cycle 3__π 2 5__π - 2-y = sin θ θ. certain functions of angles, known as the trigonometric functions. Corrective Assignment. Sine, Cosine, and Tangent Practice Find the value of each trigonometric ratio. In these examples we will graph a sine and cosine function using a table of values. Student needs to show proof. If the graphs of the equations y = 2 and y 2 sin x are drawn on the same set of axes, the number of points of intersection between 0 and 27t will be 24. projectmaths. WORD ANSWER KEY. Precalculus Chapter 6 Worksheet Graphing Sinusoidal Functions in Degree Mode Find the amplitude, period, phase (horizontal) displacement and translation (vertical displacement). The test will help you with these skills: Making connections- use understanding of sine and cosine. 6 Graphs of Other Trigonometric Functions p. Graphs provided. Find an equation for a cosine function that has an amplitude of 3 5, a period of 3 2 π. Amplitude and Period of Sine and Cosine Functions BOATING A signal buoy between the coast of Hilton Head Island, South Carolina, and Savannah, Georgia, bobs up and down in a minor squall. Basic Graphs of Sine and Cosine. Neither sine nor cosine can ever exceed 1 and the closer one of them is to 1, the closer the other must be to 0. Graphing Sine Function - Displaying top 8 worksheets found for this concept. The Cosine Ratio The ratio between the leg adjacent to a given angle of a right triangle and the triangle’s hypotenuse. f = 440, a = 0. A free graphing calculator - graph function, examine intersection points, find maximum and minimum and much more This website uses cookies to ensure you get the best experience. Amp: — 4sin -9 Period: Freq: 3 sin 9 Period: Amp: — — sin 2x Period: Freq: 10. The accompanying graph shows a trigonometric function. The Period goes from one peak to the next (or from any point to the next matching point): The Amplitude is the height from the center line to the peak (or to the trough). Chapter 2 Graphs of Trig Functions The sine and cosecant functions are reciprocals. Trigonometry Graphing Trigonometric Functions Translating Sine and Cosine Functions Key Questions How do you identify the vertical and horizontal translations of sine and cosine from a graph and an equation?. This trigonometry video tutorial focuses on graphing trigonometric functions. Some of the worksheets for this concept are Honors algebra 2 name, Of the sine and cosine functions, , Graphs of trig functions, Work 15 key, 13 trigonometricgraphswork, 1 of 2 graphing sine cosine and tangent functions, Sine cosine and tangent practice. Table of Trigonometric Parent Functions; Graphs of the Six Trigonometric Functions; Trig Functions in the Graphing Calculator; More Practice; Now that we know the Unit Circle inside out, let's graph the trigonometric functions on the coordinate system. 2 Practice Worksheet More Graphing Trigonometric Functions Worksheet Answers Sec 5. ANS: A PTS: 1 OBJ: 7. It consists of several rows or columns that spread out all over the page and create for space that assist people fill data. (a) Find the equation of the motion. Application Walkthrough. Learn vocabulary, terms, and more with flashcards, games, and other study tools. What is the equation for the sine function graphed here. Verify your answer with graphing software or a graphing calculator. 1 y F x 2 9. Day 2 - Graphing Rational Functions - Notes. Worksheet - Writing the Equation of a Transformed F(x) Graph. The international standard pitch has been set at a frequency of 440 cycles/second. Write a rule in the form f(t) = A sin Bt that expresses this oscillation where t represents the number of seconds. Statistics On-Line. 2 - Graphs of the Sine and Cosine Functions 1 Section 5. 7: Slope Fields ; Chapter 2 Test; Chapter 2 Answer Key ; Chapter 3: Derivatives and Graphs [  Chapter 3 pdf  ]. In fact Sine and Cosine are like good friends: they. Other Results for 4 4 Study Guide And Intervention Graphing Sine And Cosine Functions Answer Key: 4-4 Graphing Sine and Cosine Functions - teh. 4 Graphing Sine and Cosine Functions 487 Each graph below shows fi ve key points that partition the interval 0 ≤ x ≤ 2π — into b four equal parts. Phase shift: N/A Vertical. Graph the function using at least 3 key points. The trigonometric functions are also known as the circular functions. (2) increases by 2. The input to the sine and cosine functions is the rotation from the positive x-axis, and that may. Sine Cosine Graphing Showing top 8 worksheets in the category - Sine Cosine Graphing. Let's go a little further…. The questions are about determing the period from the graph and also matching graphs and trigonometric functions. Graphing Sine and Cosine Functions. In order to solve these equations we shall make extensive use of the graphs of the functions sine, cosine and tangent. You can make copies of the Answer Keys to hand out to your class, but. Before discussing those functions, we will review some basic terminology about angles. Consider the curve whose equation is. The key to simplifying expressions involving inverse trigonometric functions is to remember that the inverse sine, cosine, or tangent of a number can be treated as an angle. Students will have mastered the unit circle, memorizing the coordinates of various key angles to quickly determine the lengths of the sides of common right triangles. Worksheets are Graphing trig functions, Graphs of trig functions, Amplitude and period for sine. Graphing Sine and Cosine Name _ Fill in the blanks and graph. ANS: D PTS: 1 OBJ: 7. Write 1 paragraph explaining why sin 30° = 150° (or. Choose θ such that y is a rational value. 4 Graphing Sine and Cosine Functions. Graph the functions applying transformations using this information. six trig functions. y = -4 cos 3. How to sketch the graphs of basic sine and cosine functions Important Vocabulary Define each term or concept. Graphs of Transformations of Sine and Cosine. Section 2: Graphs of Trigonometric Functions Lesson 1 Sine and Cosine Graphs 205 Lesson 2 Transformations of the Sine and Answer Key 1. 6 Graphs of the Sine and Cosine Function Graph each function using degrees. 5 6 sin sin. The worksheet itself is twelve pages and begins with the simple sine and cosine graphs, then, develops into the reciprocal trig functions. Explanations will vary. We will now explore what changes can be made to the equations and how that affects the graphs. 6 Graphs of Reciprocal Trig Functions: 11. Displaying all worksheets related to - Graphing Sine Functions. Sine & Cosine Graphs By: Taylor Pulchinski Daniel Overfelt Whitley Lubeck Equations y = a sin (bx-h)+ k y = a cos (bx-h)+k a = Amplitude (height of the wave) 2( )/b = Period (time it take to complete one trip around) h = Phase Shift (left or right movement) k = Vertical Shift (up or down movement) Examples Finding the Period and Amplitude. 1 2 2 ≈07071. Each degree with special angles This page explains the sine, cosine, tangent ratio, gives on an overview of their range of values and provides practice problems on identifying the sides that are opposite and adjacent to a given angle. Questions on the properties of the graphs of trigonometric functions and their answers are presented. Neither sine nor cosine can ever exceed 1 and the closer one of them is to 1, the closer the other must be to 0. Explanations will vary. The next section presents the graphs of the elementary sine and cosine functions as functions of the variable t. How long is the side adjacent to 1? 10. 5 and b = 4 is y = 1. I can graph the cosine function and its translations. Answers to graphing sine and cosine 1 p 2 p3p 2 2p 6 4 2 4 6 amplitude. The period of any sine or cosine function is 2π, dividing one complete revolution into quarters, simply the period/4. If the graphs of the equations y = 2 and y 2 sin x are drawn on the same set of axes, the number of points of intersection between 0 and 27t will be 24. Graphing Sine and Cosine Practice Worksheet ANSWER KEY Graph the following functions over two periods, one in the positive direction and one in the negative directions. Sine is usually abbreviated as sin. Showing top 8 worksheets in the category - Graphing Sine And Cosine. The Law of Sines The Law of Cosines Graphing trig functions Translating trig functions Angle Sum/Difference Identities Double-/Half-Angle Identities. Express your answer as a fraction in lowest terms. The trigonometry equation that represents this relationship is. Worksheet - Writing the Equation of a Transformed F(x) Graph. Trigonometry Using a calculator (sin, cos, tan) Using a calculator (inverse functions) Find the tangent of one point Tangent: Find the value of x Find the sine of one point Sine: Find the value of x Find the cosine of one point Cosine: Find the value of x Mixed sine, cosine, and tangent Mixed: Find the value of x Fill in the missing angle. 9) 10) Domain: Range: Domain: Range: Amplitude: 2 Period: Amplitude: 1 Period: π. y = sin 2 θ 21. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval[latex]\,\left[-1,1\right]. Experiment with the graph of a sine or cosine function. The graph could represent either a sine or a cosine function that is shifted and/or reflected. We obtain one cycle of the graph of f(θ)=cosθ.

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Monday, February 17, 2020 Reversing Differences Fellow blogger Håkan Kjellerstrand posted an interesting question on OR Stack Exchange recently. Starting from a list of integers, it is trivial to compute the list of all pairwise absolute differences, but what about going in the other direction? Given the pairwise (absolute) differences, with duplicates removed, can you recover the source list (or a source list)? We can view the source "list" as a vector $x\in\mathbb{Z}^n$ for some dimension $n$ (equal to the length of the list). With duplicates removed, we can view the differences as a set $D\subset \mathbb{Z}_+$. So the question has to do with recovering $x$ from $D$. Our first observation kills any chance of recovering the original list with certainty: If $x$ produces difference set $D$, then for any $t\in\mathbb{R}$ the vector $x+t\cdot (1,\dots,1)'$ produces the same set $D$ of differences. Translating all components of $x$ by a constant amount has no effect on the differences. So there will be an infinite number of solutions for a given difference set $D$. A reasonable approach (proposed by Håkan in his question) is to look for the shortest possible list, i.e., minimize $n$. Next, observe that $0\in D$ if and only if two components of $x$ are identical. If $0\notin D$, we can assume that all components of $x$ are distinct. If $0\in D$, we can solve the problem for $D\backslash\{0\}$ and then duplicate any one component of the resulting vector $x$ to get a minimum dimension solution to the original problem. Combining the assumption that $0\notin D$ and the observation about adding a constant having no effect, we can assume that the minimum element of $x$ is 1. That in turn implies that the maximum element of $x$ is $1+m$ where $m=\max(D)$. From there, Håkan went on to solve a test problem using constraint programming (CP). Although I'm inclined to suspect that CP will be more efficient in general than an integer programming (IP) model, I went ahead and solved his test problem via an IP model (coded in Java and solved using CPLEX 12.10). CPLEX's solution pool feature found the same four solutions to Håkan's example that he did, in under 100 ms. How well the IP method scales is an open question, but it certainly works for modest size problems. The IP model uses binary variables $z_1, \dots, z_{m+1}$ to decide which of the integers $1,\dots,m+1$ are included in the solution $x$. It also uses variables $w_{ij}\in [0,1]$ for all $i,j\in \{1,\dots,m+1\}$ such that $i \lt j$. The intent is that $w_{ij}=1$ if both $i$ and $j$ are included in the solution, and $w_{ij} = 0$ otherwise. We could declare the $w_{ij}$ to be binary, but we do not need to; constraints will force them to be $0$ or $1$. The full IP model is as follows: $\begin{array}{lrlrc} \min & \sum_{i=1}^{m+1}z_{i} & & & (1)\\ \textrm{s.t.} & w_{i,j} & \le z_{i} & \forall i,j\in\left\{ 1,\dots,m+1\right\} ,i\lt j & (2)\\ & w_{i,j} & \le z_{j} & \forall i,j\in\left\{ 1,\dots,m+1\right\} ,i\lt j & (3)\\ & w_{i,j} & \ge z_{i}+z_{j}-1 & \forall i,j\in\left\{ 1,\dots,m+1\right\} ,i\lt j & (4)\\ & w_{i,j} & =0 & \forall i,j\in\left\{ 1,\dots,m+1\right\} \textrm{ s.t. }(j-i)\notin D & (5)\\ & \sum_{i,j\in\left\{ 1,\dots,m+1\right\} |j-i=d}w_{i,j} & \ge 1 & \forall d\in D & (6)\\ & z_{1} & = 1 & & (7) \end{array}$ The objective (1) minimizes the number of integers used. Constraints (2) through (4) enforce the rule that $w_{ij}=1$ if and only if both $z_i$ and $z_j$ are $1$ (i.e., if and only if both $i$ and $j$ are included in the solution).  Constraint (5) precludes the inclusion of any pair $i < j$ whose difference $j - i$ is not in $D$, while constraint (6) says that for each difference $d \in D$ we must include at least one pair $i < j$ for that produces that difference ($j - i = d$). Finally, since we assumed that our solution starts with minimum value $1$, constraint (7) ensures that $1$ is in the solution. (This constraint is redundant, but appears to help the solver a little, although I can't be sure given the short run times.) My Java code is available from my repository (bring your own CPLEX). Tuesday, February 11, 2020 Collections of CPLEX Variables Recently, someone asked for help online regarding an optimization model they were building using the CPLEX Java API. The underlying problem had some sort of network structure with $N$ nodes, and a dynamic aspect (something going on in each of $T$ periods, relating to arc flows I think). Forget about solving the problem: the program was running out of memory and dying while building the model. A major issue was that they allocated two $N\times N\times T$ arrays of variables, and $N$ and $T$ were big enough that $2N^2T$ was, to use a technical term, ginormous. Fortunately, the network was fairly sparse, and possibly not every time period was relevant for every arc. So by creating only the IloNumVar instances they needed (meaning only for arcs that actual exist in time periods that were actually relevant), they were able to get the model to build. That's the motivation for today's post. We have a tendency to write mathematical models using vectors or matrices of variables. So, for instance, $x_i \, (i=1,\dots,n)$ might be an inventory level at each of $n$ locations, or $y_{i,j} \, (i=1,\dots,m; j=1,\dots,n)$ might be the inventory of item $i$ at location $j$. It's a natural way of expressing things mathematically. Not coincidentally, I think, CPLEX APIs provide structures for storing vectors or matrices of variables and for passing them into or out of functions. That makes it easy to fall into the trap of thinking that variables must be organized into vectors or matrices. Last year I did a post ("Using Java Collections with CPLEX") about using what Java calls "collections" to manage CPLEX variables. This is not unique to Java. I know that C++ has similar memory structures, and I think they exist in other languages you might use with CPLEX. The solution to the memory issue I mentioned at the start was to create a Java container class for each combination of an arc that actually exists and time epoch for which it would have a variable, and then associate instances of that class with CPLEX variables. So if we call the new class AT (my shorthand for "arc-time"), I suggested the model owner use a Map<AT, IloNumVar> to associate each arc-time combination with the variable representing it and a Map<IloNumVar, AT> to hold the reverse association. The particular type of map is mostly a matter of taste. (I generally use HashMap.) During model building, they would create only the AT instances they actually need, then create a variable for each and pair them up in the first map. When getting a solution from CPLEX, they would get a value for each variable and then use the second map to figure out for which arc and time that value applied. (As a side note, if you use maps and then need the variables in vector form, you can apply the values() method to the first map (or the getKeySet() method to the second one), and then apply the toArray() method to that collection.) Now you can certainly get a valid model using just arrays of variables, which was all that was available to me back in the Dark Ages when I used FORTRAN, but I think there are some benefits to using collections. Using arrays requires you to develop an indexing scheme for your variables. The indexing scheme tells you that the flow from node 3 to node 7 at time 4 will be occupy slot 17 in the master variable vector. Here are my reasons for avoiding that. • Done correctly, the indexing scheme is, in my opinion, a pain in the butt to manage. Finding the index for a particular variable while writing the code is time-consuming and has been known to kill brain cells. • It is easy to make mistakes while programming (calculate an index incorrectly). • Indexing invites the error of declaring an array or vector with one entry for each combination of component indices (that $N\times N\times T$ matrix above), without regard to whether you need all those slots. Doing so wastes time and space, and the space, as we saw, may be precious. • Creating slots that you do not need can lead to execution errors. Suppose that I allocating a vector IloNumVar x = new IloNumVar[20] and use 18 slots, omitting slots 0 and 13. If I solve the model and then call getValues(x), CPLEX will throw an exception, because I am asking for values of two variables (x[0] and x[13]) that do not exist. Even if I create variables for those two slots, the exception will occur, because those two variables will not belong to the model being solved. (There is a way to force CPLEX to include those variables in the model without using them, but it's one more pain in the butt to deal with.) I've lost count of how many times I've seen messages on the CPLEX help forums about exceptions that boiled down to "unextracted variables". So my advice is to embrace collections when building models where variables do not have an obvious index scheme (with no skips).

https://math.stackexchange.com/questions/2671760/why-does-expanding-lim-p-rightarrow-0-frac1-p-1-p31-1-p3-give-a/2671778

# Why does expanding $\lim_{p \rightarrow 0}\frac{1-p-(1-p)^3}{1-(1-p)^3}$ give a different limit from just substituting? $$\lim_{p \rightarrow 0}\frac{1-p-(1-p)^3}{1-(1-p)^3}$$ $$= \frac{1-0-(1-0)^3}{1-(1-0)^3}$$ $$=\frac{0}{0}$$ $$\lim_{p \rightarrow 0}\frac{1-p-(1-p)^3}{1-(1-p)^3}$$ $$=\lim_{p \rightarrow 0}\frac{p^2-3p+2}{p^2-3p+3}$$ $$=\frac{0^2-3(0)+2}{0^2-3(0)+3}$$ $$=\frac{2}{3}$$ Edit 1: $0\le p \le1$ • The first one $\,0/0\,$ is indeterminate. That does not give you a limit, but merely tells you that the respective approach will not lead anywhere. – dxiv Mar 1 '18 at 6:46 • HINT : let $1-p = x ;\,$ factorize to get $1+ x+x^2$ where x lies between $0,1$ – Narasimham Mar 8 '18 at 15:00 You conveniently omitted the crucial step $$\lim_{p \rightarrow 0}\frac{1-p-(1-p)^3}{1-(1-p)^3}$$ $$=\color{red}{\lim_{p \rightarrow 0}\frac{p^3-3p^2+2p}{p^3-3p^2+3p}}$$ $$=\color{red}{\lim_{p \rightarrow 0}\frac pp} \cdot\lim_{p \rightarrow 0}\frac{p^2-3p+2}{p^2-3p+3}$$ $$=\frac{0^2-3(0)+2}{0^2-3(0)+3}$$ $$=\frac{2}{3}$$ The expression in red is clearly of the form $\dfrac 00$ too but will not be any more after you divide the numerator and the denominator by $p$. Functions that behave like that are said to have removable singularities. • So when working with limits without using l'Hospital's Rule such as the 1st method, when we arrive at an intermediate form, it tells us that there are terms we can cancel out in the original expression to get the real limit? Of course, given that the terms being cancelling is defined in the limit. i.e. $$\lim_{p \rightarrow 0}\frac{p}{p} = 1$$ – A_for_ Abacus Mar 1 '18 at 7:06 • ^Typo: indeterminate not "intermediate" – A_for_ Abacus Mar 1 '18 at 7:21 • @A_for_Abacus - it doesn't necessarity say that there are factors that you can cancel. For example, $\lim_{x\to 0} \frac {\sin x}x$ is an indeterminant form, and there is nothing that cancels. What indeterminant forms tell you is that the limit-finding technique you are currently using has failed, and that you will have to dig deeper to uncover the limit. Sometimes that "deeper" is a simple as cancelling common factors. Other times it requires some in-depth analysis of the function. – Paul Sinclair Mar 2 '18 at 2:57 Because in general $\lim_{p\rightarrow a}\frac{f(x)}{g(x)}\neq \frac{\lim_{p\rightarrow a} f(x)}{\lim_{p\rightarrow a} g(x)}$ since the right expresion may not be defined (in your case you have $\frac{0}{0}$). To find (using highschool calculus) the value of a limit where $\frac{\lim_{p\rightarrow a} f(x)}{\lim_{p\rightarrow a} g(x)}$ is not defined you have to get the ration in such a form where the demoninator's limit isn't 0, as you have done in the second case. A hint: Usualy what works with most polynomial limits is to divide both nominator and denominator with the the monomial $p^n$ of the highest degree apearing in the limit. $$1 - p - (1 - p)^3=p^3-3 p^2+2 p$$ $$1 - (1 - p)^3=p^3-3 p^2+3 p$$ both are $0$ at $p=0$. You are removing the common factor $p$ in the expanded version, but not in the non-expanded version, which is why you "get a different limit" (or, more accurately, which is why you find the limit after expanding, but do not find the limit without expanding). Expanding does not give different limits. Without expanding you arrived at an undefined situation. Whereas when you expand in some step $p$ cancelled out which is possible only if $p\neq0$. Since the expanded function is continuous at zero and $p\rightarrow0$ so correct limit is $\frac{2}{3}$. EDIT: Moreover, $\lim_{p\rightarrow0}\frac{p}{p}=1$ A little bit of context: Let $f, g$ be continuos in $D$, $a \in D$, and $g(a)\not =0.$ Then $\lim_{x \rightarrow a} \dfrac{f(x)}{g(x)} = \dfrac{f(a)}{g(a)}$. Your functions $f$, in the numerator, and g, in the denominator, are continuos but : $g(a)=0$!!. Hence the above is not applicable. Another example: $f(x)=x$, $g(x) =x.$ Does $\lim_{x \rightarrow 0} \dfrac{f(x)}{g(x)}$ exist? There are a lot of correct answers here, but I think that there is a fundamental definition or intuition that is missing from all of them, namely that we should ignore the value of the expression at the limit point (i.e. we assume that $p$ is never actually zero; we are taking a limit as $p$ approaches zero). A good definition of a limit is as follows: Definition: We say that $\lim_{x\to a} f(x) = L$ if for all $\varepsilon > 0$ there exists some $\delta > 0$ such that if $x\ne a$ and $|x-a| < \delta$, then $|f(x) - L| < \varepsilon$. Topologically (and feel free to ignore this paragraph for now), we are saying that for any neighborhood $V$ of $L$, there is some punctured neighborhood $U^\ast$ of $a$ such that $f(U^*) \subseteq V$. Because we are puncturing the neighborhood, the value of $f$ at $a$ is irrelevant. We just completely ignore it. In the original question, we are trying to compute $$\lim_{p\to 0} \frac{1-p-(1-p)^3}{1-(1-p)^3}.$$ As you have noted, when $p=0$, this expression is utter nonsense. That is, if we define $$f(p) := \frac{1-p-(1-p)^3}{1-(1-p)^3} = \frac{p^3 - 3p^2 + 2p}{p^3 - 3p^2 + 3p}$$ then try to evaluate $f(0)$, this will give us $\frac{0}{0}$ which is a (more-or-less) meaningless expression. However, we are trying to take a limit as $p\to 0$, which means that we can (and should) assume that $p \ne 0$. Notice that under this assumption, i.e. the assumption that $p\ne 0$, we have that $1 = \frac{1/p}{1/p}$. Then, using the analyst's second favorite trick of multiplying by 1 (adding 0 is the favorite trick), we have \begin{align} f(p) &= \frac{p^3 - 3p^2 + 2p}{p^3 - 3p^2 + 3p} \\ &= \color{red}{1} \cdot \frac{p^3 - 3p^2 + 2p}{p^3 - 3p^2 + 3p} \\ &= \color{red}{\frac{1/p}{1/p}} \cdot \frac{p^3 - 3p^2 + 2p}{p^3 - 3p^2 + 3p} \\ &= \frac{p^2-3p+2}{p-3p+3} \\ &=: \tilde{f}(p). \end{align} Again, the vital thing to understand is that the computation is justified since $p \ne 0$, which means that the fraction $\frac{1/p}{1/p}$ is perfectly well-defined and is (in fact) identically 1. Note, also, that the computations above are done before we've tried to take any limits. It is now relatively easy to see that $$\lim_{p\to 0} f(p) = \lim_{p\to 0} \tilde{f}(p) = \lim_{p\to 0} \frac{p^2-3p+2}{p-3p+3} = \frac{2}{3}.$$ There are two things here that I have left unjustified: Exercises: 1. Explain why $\lim_{p\to 0} f(p) = \lim_{p\to 0} \tilde{f}(p)$? 2. Explain why $\lim_{p\to 0} \tilde{f}(p) = \frac{2}{3}$. Hint for 1: One possible argument is a one-line appeal to the squeeze theorm. Hint for 2: Think about the relation between continuity and limits. The original function is a rational function in $p$, that is, it is a fraction of two polynomials $f(p)$ and $g(p)$. The roots $a_i$ are the values for which $f(a_i) = 0$, and similarly, the poles $b_j$ are the values for which $g(b_j) = 0$. If for some $i$ and $j$ you have $a_i = b_j$, then the factor $(p-a_i)$ on top cancels with the factor $(p-b_j)$ on bottom. The original function had $p\neq b_j$ since that would lead to a $0$ in the denominator, so even after cancelling this factor, we still have that $p\neq b_j$ regardless of whether we could evaluate the function at that point now. However, if we can evaluate the reduced function at $p=b_j$, then it will be continuous there (since rational functions only have holes and asymptotes as discontinuities). Then, since the function is continuous there, the limit of that function matches the value of function evaluated at the point $b_j$. Since the original function is identical to the new function for all $p \neq b_j$, the limit as $p\rightarrow b_j$ is the same for the original function as it is for the reduced function. This means that plugging $p = b_j$ in the original function will give you $0/0$ which is not the value of the limit, but after cancelling, plugging $p=b_j$ in the reduced function will give you the correct limit. In this case you've given, $a_i = b_j = 0$, so you just cancel $p$ from the top and bottom, so that plugging $p=0$ into the reduced equation give you the limit of $2/3$.

http://vvnn.ecui.pw/latex-symbols-math.html

# Latex Symbols Math An operator such as the differential operator D^~. Short Math Guide for LATEX, version 2. Even though commands follow a logical naming scheme, you will probably need a table for the most common math symbols at some point. Letters are. = mathrm{C}, LATIN CAPITAL LETTER C. Suppose that you need to type many times. To make the following table of symbols completely available, include the following package-loading command after your documentclass command. This article shows several fonts for use in math mode. Most other symbols can be inferred from these examples. Several fonts require the addition of the line \usepackage{amssymb} to the preamble to work. Note that the formula will appear below your drawing area. Log-like Symbols 8. SUB and SUP are used to specify subscripts and superscripts. The Comprehensive LATEX Symbol List Scott Pakin ∗ 22 September 2005 Abstract This document lists 3300 symbols and the corresponding LATEX commands that produce them. Please, help me define a math symbol for definition as shown in Table of mathematical symbols - Wikipedia, the free encyclopedia is defined as; equal by definition it is the equal sign under a small font 'def' the same symbol is also provided in OpenOffice math, but not in Latex. The symbols ⊤ (U+22A4) and ⊥ (U+22A5) \top and \bot in LaTeX. For example:\begin{align}\because x+3&=5\\\therefore x&=2\end{align}. You may have to register before you can post: click the register link above to proceed. The fact that he succeeded was most probably why TeX (and later on, LaTeX) became so popular within the scientific community. It is important to keep in mind that LaTeX has its own way of handling spacing in mathematics mode. Michel Bovani created the mbtimes package by using Omega Serif for text and Latin and Greek letters in mathematics. Usage An exemplary use of the symbol in a formula. I don't have latex software. A few of them, such as + and >, are produced by typing the corresponding keyboard character. 2 More symbols in math mode nhbar nimath njmath nell nRe 1 @ r 8 9 ninfty npartial nnabla nforall nexists = > ? y z nIm ntop nbot ndag nddag P Q R H 4 nsum nprod nint noint ntriangle. no LaTeX Math Symbols Prepared by L. 10 Single Mathematical Symbols @ naleph aleph. Yes, when I try Matlab to write the title with the Latex font, I do not now why, but it does not work; however, the axis labels are correctly intrepeted and the command works perfectly. To make the following table of symbols completely available, include the following package-loading command after your documentclass command. Call it $\boldsymbol\beta$. LaTeX Community is an excellent resource for answering any questions you have about LaTeX. So to make the greek capital phi, Φ, you’d hit (in insert mode) kF* Below is a table of useful math and computer science digraphs. I know this is very late, but for others like me who are also looking for this answer: Indeed, everyone's answer is correct. Relation Symbols 10. 2 More symbols in math mode nhbar nimath njmath nell nRe 1 @ r 8 9 ninfty npartial nnabla nforall nexists = > ? y z nIm ntop nbot ndag nddag P Q R H 4 nsum nprod nint noint ntriangle. Some other constructions. I mentioned in a conversation that math rarely uses Hebrew or Russian letters. Also, note that there are several variants of LaTeX, but we recommend the variant known as LaTeX2e. This command forces LaTeX to give an equation the full height it needs to display as if it were on its own line. For example, in the formula ,'' the word log'' is not italicized like the variables x and y. You may also want to visit the Mathematics Unicode characters and their HTML entity. Sometimes, a formula contains text that should be set in roman type. There are two ways of displaying the symbol: compressed to fit onto one line (useful when printing long equations or proofs) or in a larger, more readable format. TLatex can display dozens of special mathematical symbols. PDF; LaTeX source; This is just intended to be a very plain chart of the Greek letters available in LaTeX math mode. Or go to the answers. How do I make symbols like nabla () or delta () as bold symbols? It is very simple. 8, 8 the package amssymb must be loaded in the preamble of the document and the A M S math fonts must be installed on the system. Feel free to tell us about links, images and videos if you read more articles / blog posts that you believe we. As you see in this example, a mathematical text can be explicitly spaced by means of some special commands Open an example in Overleaf [] SpaceThe spacing depends on the command you insert, the example below contains a complete list of spaces and how they look like. Another thing to notice is the effect of the \displaystyle command. Thank you all. MathType has customizable keyboard shortcuts for virtually every symbol, template, and command. This document is based largely on section 8. A canvas will open for your training input. It is assumed that any characters represent variable names. The Style section shows you how to control the basic appearance of your document. An operator such as the differential operator D^~. The numerals 6 and 26, for example, are symbols that represent quantities. On the use of italic and roman fonts for symbols in scientific text Scientific manuscripts frequently fail to follow the accepted conventions concerning the use of italic and roman fonts for symbols. These symbol images are taken from a mirror site in Austrailia by John Marley. \sqrt{abc} \alpha \frac{abdsf}{afds} \hat{a} \acute{a} \bar{a} \dot{a} \breve{a} \check{a} \grave{a} \vec{a} \ddot{a. I don't have latex software. Spacing in math mode; Integrals, sums and limits; Display style in math mode; List of Greek letters and math symbols; Mathematical fonts Figures and tables. In this guide, you’ll find an extensive list of probability symbols you can use for reference, plus the names of each symbol and the concept they represent. This article provides the list of most commonly used symbols in LaTeX. Classes of math symbols. For example, is given by $\mathfrak S = \mathbb R^2/G$. LaTeX requires control sequences to format some symbols in text. A matrix having $$n$$ rows and $$m$$ columns is a $$m\times n$$-matrix. You have to use \pm. Here's a list of mathematical symbols and their meaning, for your reference. In a math environment (or in math mode) LaTeX typesets the words in the table below in the manner of body text (not in italics) so that they stand apart from the other symbols and variables, it is equivalent to using the “mathrm{ }” command. machine learning. Supported symbols are listed here (alphabetically). This is a list of symbols found within all branches of mathematics. That said, there are many places where symbols are useful and simplify matters. There are problems with this approach. The symbols ⊤ (U+22A4) and ⊥ (U+22A5) \top and \bot in LaTeX. Landau Symbols. The change would also make it more consistent with List of logic symbols. To use the symbols listed in Tables 3. 1 day ago · LaTeX is a professional document preparation system and document markup language written by Leslie Lamport. We've documented and categorized hundreds of macros!. A lot of the nice looking equations you see in books and all around the web are written using LaTeX commands. Learn how to add images to your LaTeX documents. All the versions of this article:. A basic latex document would look like. Basic Code Special keys. After looking for a builtin expectation symbol in LaTeX, and coming up with none, I’ve defined one. Below we give a partial list of commonly used mathematical symbols. I know this is very late, but for others like me who are also looking for this answer: Indeed, everyone's answer is correct. Actuarial symbols of life contingencies and financial mathematics David Beauchemin david. In informal usage, "tilde" is often instead voiced as "twiddle" (Derbyshire 2004, p. LaTeX symbols have either names (denoted by backslash) or special characters. AsciimathML can be used directly with script tags without filter so you can add those script tags to theme files or probably even to single Hot Pot files but note that asciimathml is using different syntax and tokens than Tex filter or MathTran filter - the best option might be to use mathJax everywhere (even if it is a little slow), in that. pdf provides an introduction to LaTeX: Math into LaTeX Short Course based on the book Math into LaTeX An Introduction to LaTeX and AMS-LaTeX by George Gratzer, published by Birkhauser Boston, ISBN 0-8176-3805-9. Math Mode LaTeX uses a special math mode to display mathematics. Click Insert. LATEX Mathematical Symbols The more unusual symbols are not defined in base LATEX (NFSS) and require \usepackage{amssymb} 1 Greek and Hebrew letters ↵ \alpha \kappa \psi z \digamma \Delta ⇥ \Theta \beta \lambda ⇢ \rho " \varepsilon \Gamma ⌥ \Upsilon \chi µ \mu \sigma { \varkappa ⇤ \Lambda ⌅ \Xi. Most letters and symbols are simple in LaTeX, yet a few characters are reserved for LaTeX commands, i. 8p 6 ‘Under’symbols,\mathunder 7 7 Accents,\mathaccent 9 8 Bottomaccents,\mathbotaccent 10. A Mathematical symbols \alpha \beta \gamma \delta \epsilon \varepsilon \zeta \eta AMS miscellaneous symbols. In this tutorial I discuss how to use math environment in Latex. 18List of Mathematical Symbols 19Summary 20Notes 21Further reading 22External links Mathematics environments LaTeX needs to know beforehand that the subsequent text does indeed contain mathematical elements. Click here for example. The frequently used left delimiters include (, [ and {, which are obtained by typing (, [and \{respectively. no LaTeX Math Symbols Prepared by L. Introduction to Overleaf and LaTeX Discrete Math Fall 2017 3 work outside a list building environment. If you like this Site about Solving Math Problems, please let Google know by clicking the +1 button. tex) is NOTa good model to build. machine learning. tex Math model predicts growth, death of membership-based websites (Phys. , braces { }. Similar is for limit expressions. This is a command for generating the$\because$ symbol. Feel free to tell us about links, images and videos if you read more articles / blog posts that you believe we. This article provides the list of most commonly used symbols in LaTeX. Math symbols and math fonts 3. Note that the formula will appear below your drawing area. The regular type style declarations can be used in math mode. Math symbols defined by LaTeX package «amssymb» No. Suppose that you need to type many times. If you want the limits of an integral/sum/product to be specified above and below the symbol in inline math mode, use the \limits command before limits specification. Similar is for limit expressions. Most import, this post is showing you the basics about math symbols in Latex. Text mode 2. When using LaTeX, can I just use $:=$, or do I need to do something special?. I removed the following from the article: Note: To use the Greek Letters in L a T e X that have the same appearance as their Roman equivalent, just use the Roman form: e. MATH Common Greek Binary Subsets Inequalities Triangles Arrows Operators Functions Miscel. There are two linear formats for math that Word supports:. To use LaTeX markup, set the Interpreter property for the Text object to 'latex'. An italic font is generally used for emphasis in running text, but it has a quite specific meaning when used for symbols in scientific text and. I'd really like latex support for Evernote, since apart from that I enjoy using it. It describes the editing community's established practice on some aspect or aspects of Wikipedia's norms and customs. You can leave a comment with your thoughts if you have a question about Latex math symbols direct sum, or want to know more. Note: If a +1 button is dark blue, you have already +1'd it. tex contains all necessary code This file is prepared by running latex A. 14 “Dimensionless parameters”). LyX combines the power and flexibility of TeX/LaTeX with the ease of use of a graphical interface. Basic math symbols; Geometry symbols; Algebra symbols; Probability & statistics symbols; Set theory symbols; Logic symbols; Calculus & analysis symbols; Number symbols; Greek symbols; Roman numerals; Basic math symbols. The current version of LATEX is LATEX2e. You might make a definition like ewcommand{\gijk}{\Gamma^{ij}_k} and then type \gijk each time you want to use this symbol (inside mathematics mode). If you want the limits of an integral/sum/product to be specified above and below the symbol in inline math mode, use the \limits command before limits specification. The body contains the text, gures, tables, etc. If you can't find the LaTeX symbol(s) that you are after, then I can almost guarantee that you'll find them in the Comprehensive LaTeX Symbol List. math symbols in latex,document about math symbols in latex,download an entire math symbols in latex document onto your computer. Most letters and symbols are simple in LaTeX, yet a few characters are reserved for LaTeX commands, i. The corresponding right delimiters are of course obtained by typing ), ] and \}. 0 (2017/12/22) 6 3. Codecogs has a real time online LaTeX equation editor. Hyperbolic functions The abbreviations arcsinh, arccosh, etc. There are two ways of displaying the symbol: compressed to fit onto one line (useful when printing long equations or proofs) or in a larger, more readable format. (Actually this is not exactly how it's written, as a backwards $\in$. The current version of LATEX is LATEX2e. Finally, as others have pointed out, the Comprehensive LaTeX Symbols List is a great resource for finding the perfect symbol for the job. Letters are. Character Name Browser Image; U+002B: PLUS SIGN + ARABIC MATHEMATICAL OPERATOR MEEM WITH HAH. Math Mode Accents. the whole message has to be in Latex. Suppose that you need to type many times. Variable-sized Symbols 3. Some of these symbols are guaranteed to be available in every LATEX2εsystem; others require fonts and packages that. Use equations in a document. Certain spacing and positioning cues are traditionally used for. ;; Copyright (c) 1999, 2000, 2001 ;; John Palmieri ;; Author: John Palmieri ;; Maintainer: John. Jimenez,Lan Nguyen. I'd love to know what it's called so I can Google it and find a font for use it in some stuff I'm typing. Hi folks, After searching the Internet without any suitable results, I finally send this mail to this list. 07 (2000/07/19) 15 8. 1) Specify math font as \setmathfont{XITS Math} (I'm on a linux system and that is the name of the font my system reports) and 2) Adding \usepackage{bm} AFTER setting the math font. To use LaTeX markup, set the Interpreter property for the Text object to 'latex'. Click the symbol that you want to insert. Using Math-Type to create TeX and MathML equations The Translation System from a User’s Perspective The basic scenario for using MathType to aid in the creation of a TEX document is to run it simultane-ously with your favorite TEX editor. LaTeX takes care of the spacing of mathematical symbols automatically. Note: If a +1 button is dark blue, you have already +1'd it. But still, TeX math is the best way to write math, and I'd claim LaTeX the best way to write structured documents, despite it's flaws. For example, is given by $\mathfrak S = \mathbb R^2/G$. as being part of the LATEX command set, and it interferes with the proper operation of various features such as the fleqn tion, use the equation environment; to assign a label for cross referencing, use the \label command and product symbols to appear full size, you need the \displaystyle command ' n> zn ' A quick guide to LATEX Overleaf. The font type LaTeX uses in math mode is somewhat special since it is optimized for writing mathematical formulas. , A instead of Alpha, B instead of Beta, etc. The Comprehensive LATEX Symbol List Scott Pakin ∗ 22 September 2005 Abstract This document lists 3300 symbols and the corresponding LATEX commands that produce them. The basic LaTeX program does not include all the math you'll want to use. Here are the most common set symbols. That said, there are many places where symbols are useful and simplify matters. The corresponding right delimiters are of course obtained by typing ), ] and \}. tex) is NOTa good model to build. To use LaTeX markup, set the Interpreter property for the Text object to 'latex'. Derivatives, Limits, Sums and Integrals. The opposite of being equivalent is being nonequivalent. Find out more: bear cub clip art free funny face photo editor water draw prover vintage photos women cooking classes for couples ideas for corner flower beds sketch of human heart. For example, both letters in the output from ${\bf x}^{\bf x}$ are the same size. pdf that you just can't memorize. Inserting Images; Tables; Positioning Images and Tables; Lists of Tables and Figures; Drawing Diagrams Directly in LaTeX; TikZ package References and Citations. Mathematics Keyboard v. Unicode math. The mathematics mode in LaTeX is very flexible and powerful, there is much more that can be done with it: Subscripts and superscripts; Brackets and Parentheses; Fractions and Binomials; Aligning Equations; Operators; Spacing in math mode; Integrals, sums and limits; Display style in math mode; List of Greek letters and math symbols; Mathematical fonts. Very supportive of new users. I know this is very late, but for others like me who are also looking for this answer: Indeed, everyone's answer is correct. It would be inconvenient to have to exit math mode to type the text in normal roman type, so the most common function names are defined as special commands. mathematical symbols Software - Free Download mathematical symbols - Top 4 Download - Top4Download. Type TeX or LaTeX: If you already know the TeX typesetting language, you can enter equations directly into MathType or Microsoft Word documents. Here are the most common set symbols. This secure online tool allows for the conversion of LaTeX (mathematics) to Presentation MathML. To emphasise that each symbol is an individual, they are not positioned as closely together as with normal text. UPDATE: Mails sent from this system were tagged with "Be careful with this message" warnings in most email systems (rightly so), because the sender's domain (yours) was different to IntMath's domain. When using LaTeX, can I just use $:=$, or do I need to do something special? Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In order to access all the math functions and symbols we will introduce in the guide pages, you'll have to include a number of packages. The process is this: • whenever an equation is needed, the MathType window is brought to the front. Re: LaTeX representation of hollow "R" symbol for real space Post by localghost » Mon May 24, 2010 9:47 am I don't think that your math institute sets the standard [1,2]. The maths, embedded in. ? If this question is repeated, I am sorry I just searched and couldn't find an answer. If and are "equivalent by definition" (i. The overloading of a symbol usually implies the overloading of related symbols. If placed over multiple symbols (especially in the context of a radical), it is known as a vinculum. Write beautiful math equations & symbols easily in MathMagic, use them widely in your word processors, Presentations, DTP software. Perhaps, this question has been answered already but I am not aware of any existing answer. See the following links for information on how MathML can be implemented. Bibliography management in. Similarly, a math symbol like \frac doesn’t work out-. 2 Symbols Symbols for physical quantities should be single letters of the Latin or Greek alphabet with or without modifying signs (subscripts, superscripts, primes, etc. for the capitals). David Hamann. Spacing in math mode; Integrals, sums and limits; Display style in math mode; List of Greek letters and math symbols; Mathematical fonts Figures and tables. Selected LaTeX Math Symbols Note: there is another version of this document featuring HTML entities for math symbols , as well as LaTeX commands. UPDATE: Mails sent from this system were tagged with "Be careful with this message" warnings in most email systems (rightly so), because the sender's domain (yours) was different to IntMath's domain. Is the Mathematical "Implies" symbol stored somewhere in Excel / MS Office in General? It's the symbol that looks like an equals followed by a greater than symbol - i. A Mathematical symbols \alpha \beta \gamma \delta \epsilon \varepsilon \zeta \eta AMS miscellaneous symbols. In certain cases it may be desirable to include "normal text" within an equation. Re: LaTeX representation of hollow "R" symbol for real space Post by localghost » Mon May 24, 2010 9:47 am I don't think that your math institute sets the standard [1,2]. tex Math model predicts growth, death of membership-based websites (Phys. Mostly a control sequence is a back slash followed by the desired symbol or its designator. The Comprehensive LATEX Symbol List Scott Pakin ∗ 19 January 2017 Abstract This document lists 14283 symbols and the corresponding LATEX commands that produce them. Refer to the external references at the end of this article for more information. A symbol is said to be overloaded if its meaning depends on the context. Inserting Images; Tables; Positioning Images and Tables; Lists of Tables and Figures; Drawing Diagrams Directly in LaTeX; TikZ package References and Citations. Short Math Guide for LATEX, version 1. Different classes of mathematical symbols are characterized by different formatting (for example, variables are italicized, but operators are not) and different spacing. 2 More symbols in math mode nhbar nimath njmath nell nRe 1 @ r 8 9 ninfty npartial nnabla nforall nexists = > ? y z nIm ntop nbot ndag nddag P Q R H 4 nsum nprod nint noint ntriangle. edit: I just thought that \sfrac{q}{m} should look nicer if it is bigger when it is followed by =. You can type an absolute value symbol in LaTeX by typing "\mid" and it will be replaced by the symbol when you generate your output. Open an example in Overleaf [] Capital letters-only font typefaceThere are some font typefaces that support only a limited number of characters; these fonts usually denote some special sets. Routing metric = The character sequence \% generates a percent (%) sign. MATH Common Greek Binary Subsets Inequalities Triangles Arrows Operators Functions Miscel. Texmaker is a free, modern and cross-platform LaTeX editor for linux, macosx and windows systems that integrates many tools needed to develop documents with LaTeX, in just one application. You have to be clever if you want to use LaTeX in your Prezi talk. If there are several typographic variants, only one of the variants is shown. ;; Copyright (c) 1999, 2000, 2001 ;; John Palmieri ;; Author: John Palmieri ;; Maintainer: John. It is also possible to check to see if a Unicode code point is available as a LaTeX command, or vice versa. Kocbach, on the basis of this document (origin: David Carlisle, Manchester University) File A. To make this easier, we support LaTeX, a way of writing mathematics terms and equations with ordinary text. The tilde is the mark "~" placed on top of a symbol to indicate some special property. machine learning. The symbol will be inserted in your document. the whole message has to be in Latex. Just add: % Expectation symbol \DeclareMathOperator*{\E}{\mathbb{E}} to your LaTeX preamble and you're done. A practical use for LaTeX macros is to simplify the typing of mathematical formulas. As you see in this example, a mathematical text can be explicitly spaced by means of some special commands Open an example in Overleaf [] SpaceThe spacing depends on the command you insert, the example below contains a complete list of spaces and how they look like. , which form is preferable from a typographic point of view, depends strongly on (a) what's inside the part and (b) whether the material occurs in inline math mode or display math mode. In informal usage, "tilde" is often instead voiced as "twiddle" (Derbyshire 2004, p. You can leave a comment with your thoughts if you have a question about Latex math symbols direct sum, or want to know more. Text Math Macro Category Requirements Comments 00026 & (N) \& mathord # \binampersand (stmaryrd) 02144 ⅄ \Yup mathord stmaryrd TURNED SANS-SERIF CAPITAL Y 0214B ⅋ (O) \invamp mathbin txfonts # \bindnasrepma (stmaryrd), TURNED AMPERSAND. As you see in this example, a mathematical text can be explicitly spaced by means of some special commands Open an example in Overleaf [] SpaceThe spacing depends on the command you insert, the example below contains a complete list of spaces and how they look like. Covers the list of majorly used Greek Symbols, Binary Operation Symbols, Relation Symbols, Arrows Symbols, Trigonometric Math Functions/Symbols, Integrals Math Symbols, and Miscellaneous Math Functions/Symbols. They are valid only in math mode, and they operate only on the letter or group that immediately follows. Math symbols defined by LaTeX package «amsfonts» No. Use the LaTeX-command \overset{symbols above}{symbols below}. Computer typesetting systems often have their own ways to represent certain symbols and mathematical concepts. The mathematics is done using a version of $$\LaTeX$$, the premiere mathematics typesetting program. Changing the font size in LaTeX Multi-column and multi-row cells in LaTeX tables Normal text in math mode Lists: Enumerate, itemize, description and how to change them Control the width of table columns (tabular) in LaTeX. Ce point de coupure n'a pas à être accompagné de relation ou d'opérateur binaires. Thanks to. Mostly a control sequence is a back slash followed by the desired symbol or its designator. If there are symbols missing drop me a line or create a pull-request. What I currently did is to write an AutoHotKey script which automatically replaces \latexSymbol with the corresponding unicode symbol, using the "hotstrings" AutoHotKey feautres. The basic LaTeX program does not include all the math you'll want to use. This video shows you how to easily remember "greater than" and "less than" math symbols. These can be included in a LaTeX document using the \mathbb{ [letter] } tag from within the math environment. Do not forget the include nusepackagefamsmath,amssymb,latexsymgbefore. No extra packages are required to use these symbols. LaTeX Math Symbols Prepared by L. R Markdown allows you to mix text, R code, R output, R graphics, and mathematics in a single document. List of all mathematical symbols and signs - meaning and examples. If placed over multiple symbols (especially in the context of a radical), it is known as a vinculum. A list of accent marks that are available in math-mode in LaTeX. It doesn't mean that LaTeX doesn't know those sets, or more importantly their symbols… There are two packages which provide the same set of symbols. Log-like Symbols. Depending on your preferred input format, you can create equations in Word in either one of UnicodeMath or LaTeX formats by selecting the format from the Equations tab. Latex mathematical symbols - rutgers university Open document Search by title Preview with Google Docs Latex mathematical symbols the more unusual symbols are not defined in base latex (nfss) and require \usepackage{amssymb} 1 greek and hebrew letters. but what is the command for the negated symbol? How to make your own laTex math symbol?-1. The mathematical symbol is produced using \partial. Latex math symbols list Sample Schedule A Letter - Veterans Benefits Administration Sample Schedule A Letter from the Department of Labor’s Office of Disability and Employment Policy: Date. angstrom latex, latex angstrom, angstrom symbol in latex, latex symbol for angst, amstrong en latex, how to write angstrom in latex, how to wrige angstrong in latex, latex angstrom math mode, latex ang, ångström latex, revtex4 angstrom, angstrom latex 書き方, armstobg in latex, armstrong in latex, How to write mathematical expression. Setting bold Greek letters in LaTeX maths The issue here is complicated by the fact that \mathbf (the command for setting bold text in TeX maths) affects a select few mathematical "symbols" (the uppercase Greek letters). Ahh, that's why I could get it to work. In my opinion the less often a symbol is used where a few words can go, the better. Analysis & calculus symbols table - limit, epsilon, derivative, integral, interval, imaginary unit, convolution, laplace transform, fourier transform RapidTables Home › Math › Math symbols › Calculus symbols. LaTeX has many symbols at its disposal. MathType's strong points are the hundreds of symbols that are accessible from the keyboard, along with its compatibility with various systems for importing and exporting formulas, such as MathML, TeX or Texvc. as being part of the LATEX command set, and it interferes with the proper operation of various features such as the fleqn tion, use the equation environment; to assign a label for cross referencing, use the \label command and product symbols to appear full size, you need the \displaystyle command ' n> zn ' A quick guide to LATEX Overleaf. Note that the formula will appear below your drawing area. LaTeX is a programming language that can be used for writing and typesetting documents. I'd love to know what it's called so I can Google it and find a font for use it in some stuff I'm typing. A canvas will open for your training input. The solution given by the LaTeX User's Guide & Reference Manual is to use \mbox{\boldmath$#1$} where #1 is the symbol to be set bold. LaTeX Tutorial pt 3 - Mathematics in LaTeX - Duration: 3:33. In the examples C = {1,2,3,4} and D = {3,4,5}. The symbol may be used with other meanings, including "approximately equal to". Since LaTeX offers a large amount of features, it's hard to remember all commands. That list also includes LaTeX and HTML markup, and Unicode code points for each symbol (note that this article doesn't have the latter two, but they could certainly be added). The numerals 6 and 26, for example, are symbols that represent quantities. Math symbols defined by LaTeX package «amssymb» No. Is the Mathematical "Implies" symbol stored somewhere in Excel / MS Office in General? It's the symbol that looks like an equals followed by a greater than symbol - i. This package defines the following commands:. LaTeX is one popular typesetting system. A few of them, such as + and >, are produced by typing the corresponding keyboard character. TEX is a low-level language that computers can work with, but most people would nd dicult to use; so LATEX has been developed to make it easier. 20 hours ago · Here is an example of a quote in this spirit, from the Wikipedia on functional equations for L-functions. The symbols ⊢ (U+22A2) and ⊣ (U+22A3) are \vdash and \dashv in LaTeX. To use LaTeX markup, set the Interpreter property for the Text object to 'latex'. Even though commands follow a logical naming scheme, you will probably need a table for the most common math symbols at some point. On different math. 1) Specify math font as \setmathfont{XITS Math} (I'm on a linux system and that is the name of the font my system reports) and 2) Adding \usepackage{bm} AFTER setting the math font. In mathematical mode as well as in text mode, you can change the typeface as needed. The Comprehensive LATEX Symbol List Scott Pakin ∗ 22 September 2005 Abstract This document lists 3300 symbols and the corresponding LATEX commands that produce them. Curly brackets are used to group characters. tex Math model predicts growth, death of membership-based websites (Phys. Does anybody know if there is a way to make the math symbols work in latex in verbatim mode? If yes, then how? Adv Reply. I don't have latex software. Others are obtained with the commands in the following table:. TeX editing can be mixed with point-and-click editing so you get the best of both worlds. ShareLaTeX 64,286 views. $\begingroup$ A good approximation is y with a circumflex (U+0177): ŷ but in MathJax/ LaTeX markup it looks more mathematical (the way it tends to be written by hand): $\hat{y}$ $\endgroup$ – Glen_b ♦ Feb 4 '16 at 4:42. I mentioned in a conversation that math rarely uses Hebrew or Russian letters. Supported symbols are listed here (alphabetically). tex WARNING!!!! This document (latextips. We will tells you more regarding latex math symbols direct sum, giving the knowledge you are looking for. Various math symbols TeX provides almost any mathematical symbol you're likely to need. Inserting Images; Tables; Positioning Images and Tables; Lists of Tables and Figures; Drawing Diagrams Directly in LaTeX; TikZ package References and Citations. Render Latex equations into plain text ASCII to insert as comments in source-code, e-mail, or forum. View all images. Creating Tables with LaTeX Tables are created using the “table” environment given below: \begin{table}[where] table \end{table} In the above syntax, table stands for the contents of the ‘tabular’ environment together with a possible \caption command. Here an example for an "a" over. LaTeX Symbols package The LaTeX symbols package provides commands for XEmacs (and only XEmacs--it doesn't work with GNU Emacs) to open up a window with a table of LaTeX symbols. The \sideset command There’s also a command called \sideset, for a rather special purpose: putting symbols at the subscript and superscript corners of a symbol like P or Q.

https://math.stackexchange.com/questions/2481670/what-is-the-probability-that-player-a-and-b-play-against-each-other

# What is the probability that player A and B play against each other? Players with same skill take part in a competition. The probability of winning each game is 0.5. At first, we divide a group of $2^n$ people to random pairs that play against each other. Then we will do the same for $2^{n-1}$ winners and this continues until there is only one winner. What is the probability that player A and B play against each other? I know we should calculate the sum of probabilities that two players play against each other in the first round, then for second round and etc. So how can i calculate the probability that two players play at round K? • Did you try a small example to see what would have to happen for A and B to meet in the second round, or the third round, and then see how it may generalize from that? – Bram28 Oct 20 '17 at 16:17 • It is a knockout competition with $m=2^n$ players so $m-1$ knockouts or matches or parings are needed to produce a winner • There are ${m \choose 2}=\frac{m(m-1)}{2}$ equally-likely possible parings of the $m$ players, so the probability any particular match involves both players A and B is $\frac{2}{m(m-1)}$, and so the expected number of the $m-1$ matches which involve both players A and B is $(m-1) \times \dfrac{2}{m(m-1)}=\dfrac{2}{m}$ • Since players A and B can only meet $0$ or $1$ times, the probability they meet is $\dfrac{2}{m} = \dfrac{1}{2^{n-1}}$ • Elegantly simple. – true blue anil Oct 21 '17 at 2:14 • $P(A , B \text{ play at round } 1) = \frac{1}{2^{n-1}}$ When we have $2^{n}$ people, then we have at most $\log^{2^{n}}_{2}$ rounds. For example, assume we have only $2$ players, then $P(A , B \text{ play at round } 1) = \frac{1}{2^{\log^{2}_{2}-1}} = 1$ • $P(A , B \text{ play at round } 2) = P(A , B \text{ did not play at round } 1 \text{ and } A , B \text{ both win at the first round}) = (1-\frac{1}{2^{n-1}})(\frac{1}{2})^{2}$ • $P(A , B \text{ play at round } 3) = P(A , B \text{ did not play at round } 1,2 \text{ and } A , B \text{ both win at the first, second rounds}) = (1-\frac{1}{2^{n-1}})(1-\frac{1}{2^{n-2}})(\frac{1}{2})^{4}$ $$...$$ • $\forall k<n: \quad P(A , B \text{ play at round } k) = (\frac{1}{2})^{2(k-1)}\times \Pi_{i=1}^{k}(1-\frac{1}{2^{n-i}})$ $$...$$ • $\bbox[5px,border:2px solid #C0A000]{P(A , B \text{ play at round } n \text{ and } n>1) = (\frac{1}{2})^{2(n-1)}\times \Pi_{i=1}^{n-1}(1-\frac{1}{2^{n-i}})}$ • I think you're missing minus ones. Assume there are only two players. Is the probability that they will play $\frac12$? Also each round you have to multiply with that stages probability that they will play. – karakfa Oct 20 '17 at 16:32 • @karakfa, your first comment is not correct. I explain it by an example. – Hasan Heydari Oct 20 '17 at 16:42 • yes, my mistake $2^n$ people means for two people n=1. – karakfa Oct 20 '17 at 16:48 Hmm...I don't see any real difference between this and an ordinary single-elimination tournament where the seeding is set from the start. Maybe I'm missing something. Anyway, assuming that there isn't any difference, we observe that Player A has $2^n-1$ different possible opponents in the first round. Of those, $2^{n-1}$ are in the opposite half of the draw, and both they and Player A would have to win $n-1$ games to meet; this happens with probability $\frac{1}{4^{n-1}}$. $2^{n-2}$ are in the same half, but opposite quarters, and both they and Player A would have to win $n-2$ games to meet; this happens with probability $\frac{1}{4^{n-2}}$. And so on, until we get to the $2^{n-n} = 2^0 = 1$ single player who meets Player A in the first round. Altogether, the probability of Player A and a given Player B meeting eventually is \begin{align} \frac{1}{2^n-1} \sum_{k=1}^n 2^{n-k}\frac{1}{4^{n-k}} & = \frac{1}{2^n-1} \sum_{k=1}^n \frac{1}{2^{n-k}} \\ & = \frac{1}{2^n-1} \sum_{k=0}^{n-1} \frac{1}{2^k} \\ & = \frac{1}{2^n-1} \left(2-\frac{1}{2^{n-1}}\right) \\ & = \frac{1}{2^n-1} \times \frac{2^n-1}{2^{n-1}} \\ & = \frac{1}{2^{n-1}} \end{align} I'll come back to edit this if I think (or someone points out) that there's a substantive difference between random reseeding and not reseeding. There's also a pretty simple proof by induction: For $n = 1$ (two players), the probability is clearly $\frac{1}{2^n-1} = 1$. For $n > 1$, the probability that they meet in the first round is $\frac{1}{2^n-1}$. If they don't meet in the first round (with probability $\frac{2^n-2}{2^n-1}$), then they must both win their first games (with probability $\frac14$) to get to the next round. With reseeding, this is clearly the case of $n-1$, so with the premise that the probability of them meeting at that point is $\frac{1}{2^{n-1-1}} = \frac{1}{2^{n-2}}$, the overall probability for case $n$ is \begin{align} \frac{1}{2^n-1}+\frac{2^n-2}{2^n-1} \times \frac14 \times \frac{1}{2^{n-2}} & = \frac{1}{2^n-1} \left(1+\frac{2^n-2}{2^n}\right) \\ & = \frac{1}{2^n-1} \times \frac{2^{n+1}-2}{2^n} \\ & = \frac{1}{2^n-1} \times \frac{2^n-1}{2^{n-1}} \\ & = \frac{1}{2^{n-1}} \end{align}

http://gmatclub.com/forum/to-find-the-units-digit-of-a-large-number-155363.html

Find all School-related info fast with the new School-Specific MBA Forum It is currently 30 Aug 2015, 00:00 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar # To find the units digit of a large number? Author Message TAGS: Director Joined: 29 Nov 2012 Posts: 926 Followers: 12 Kudos [?]: 528 [0], given: 543 To find the units digit of a large number? [#permalink]  03 Jul 2013, 21:58 1 This post was BOOKMARKED So if we are given a question what is the units digit of $$777^{777}$$ we find the pattern for 7 (7,9,3,1) then we divide $$\frac{777}{4}$$ and the remainder is 1 so the units digit is $$7^1$$ which is 7? Is this correct? _________________ Click +1 Kudos if my post helped... Amazing Free video explanation for all Quant questions from OG 13 and much more http://www.gmatquantum.com/og13th/ GMAT Prep software What if scenarios gmat-prep-software-analysis-and-what-if-scenarios-146146.html Director Joined: 24 Aug 2009 Posts: 507 Schools: Harvard, Columbia, Stern, Booth, LSB, Followers: 10 Kudos [?]: 489 [0], given: 241 Re: To find the units digit of a large number? [#permalink]  03 Jul 2013, 22:08 fozzzy wrote: So if we are given a question what is the units digit of $$777^{777}$$ we find the pattern for 7 (7,9,3,1) then we divide $$\frac{777}{4}$$ and the remainder is 1 so the units digit is $$7^1$$ which is 7? Is this correct? Absolutely Correct. _________________ If you like my Question/Explanation or the contribution, Kindly appreciate by pressing KUDOS. Kudos always maximizes GMATCLUB worth -Game Theory If you have any question regarding my post, kindly pm me or else I won't be able to reply Math Expert Joined: 02 Sep 2009 Posts: 29110 Followers: 4726 Kudos [?]: 49758 [2] , given: 7403 Re: To find the units digit of a large number? [#permalink]  03 Jul 2013, 22:10 2 KUDOS Expert's post 1 This post was BOOKMARKED fozzzy wrote: So if we are given a question what is the units digit of $$777^{777}$$ we find the pattern for 7 (7,9,3,1) then we divide $$\frac{777}{4}$$ and the remainder is 1 so the units digit is $$7^1$$ which is 7? Is this correct? Yes. The units digit of 777^777 = the units digit of 7^777. 7^1 has the units digit of 7; 7^2 has the units digit of 9; 7^3 has the units digit of 3; 7^4 has the units digit of 1. 7^5 has the units digit of 7 AGAIN. The units digit repeats in blocks of 4: {7, 9, 3, 1}... The remainder of 777/4 is 1, thus the units digit would be the first number from the pattern, so 7. Hope it's clear. _________________ Director Joined: 29 Nov 2012 Posts: 926 Followers: 12 Kudos [?]: 528 [0], given: 543 Re: To find the units digit of a large number? [#permalink]  03 Jul 2013, 22:23 One final question here is another example If we have the find the units digit of $$344^{328}$$ $$4^1$$ is 4 $$4^2$$ is 16 $$4^3$$ is 4 so the repeating block over here {4,6} In this case the remainder is 0 so the units digit of this expression is 6? so if there are 4 repeating blocks and the remainder is 0 we raise it to the 4th power ( some examples would be 3,7 etc) in this current example its the 2nd power? _________________ Click +1 Kudos if my post helped... Amazing Free video explanation for all Quant questions from OG 13 and much more http://www.gmatquantum.com/og13th/ GMAT Prep software What if scenarios gmat-prep-software-analysis-and-what-if-scenarios-146146.html Math Expert Joined: 02 Sep 2009 Posts: 29110 Followers: 4726 Kudos [?]: 49758 [1] , given: 7403 Re: To find the units digit of a large number? [#permalink]  03 Jul 2013, 22:26 1 KUDOS Expert's post fozzzy wrote: One final question here is another example If we have the find the units digit of $$344^{328}$$ $$4^1$$ is 4 $$4^2$$ is 16 $$4^3$$ is 4 so the repeating block over here {4,6} In this case the remainder is 0 so the units digit of this expression is 6? so if there are 4 repeating blocks and the remainder is 0 we raise it to the 4th power ( some examples would be 3,7 etc) in this case its the 2nd power? Yes, if the remainder is 0, then take the last digit from the block. _________________ Senior Manager Joined: 19 Apr 2009 Posts: 435 Location: San Francisco, California Followers: 77 Kudos [?]: 284 [2] , given: 5 Re: To find the units digit of a large number? [#permalink]  07 Jul 2013, 20:00 2 KUDOS fozzy, you are correct in both cases. here is how i like to think about it: 7^{777} example: If we divide 777 by 4, the quotient is 194 and the remainder is 1. This means that we will have 194 of {7, 9, 3, 1} repeating blocks, and we will have 1 more term left, and the units digit of 7^{777} will be 7, the first term in the repeating block. When we look at the case of 344^{328}, the units digit of powers of 4 cycle as {4, 6}, when we divide 328 by 2, the remainder is 0, this means that there will be exactly 164 blocks consisting of {4,6} without any remainder and the units digit of 344^{328} will be 6. here are problems using the same concept(some hard): #1) if-n-is-a-positive-integer-what-is-the-remainder-when-82380.html #2) what-is-the-remainder-when-7-345-7-11-2-is-divided-by-26794.html #3) remainder-when-7-4n-3-6-n-104848.html #4) if-3-4n-1-is-divided-by-10-can-the-remainder-be-0-a-4662.html #5) what-s-the-remainder-of-2-x-divided-by-10-1-x-is-an-even-9651.html cheers, dabral _________________ New!2016 OFFICIAL GUIDE FOR GMAT REVIEW: Free Video Explanations. http://www.gmatquantum.com GMAT Club Legend Joined: 09 Sep 2013 Posts: 6107 Followers: 340 Kudos [?]: 69 [0], given: 0 Re: To find the units digit of a large number? [#permalink]  16 Mar 2015, 15:27 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Intern Joined: 01 Apr 2015 Posts: 10 Followers: 0 Kudos [?]: 0 [0], given: 2 Re: To find the units digit of a large number? [#permalink]  01 Apr 2015, 23:35 Finding the UNIT’S DIGIT - please share some more useful tricks if its there. _________________ Founder, aptiDude "Get useful study materials at aptiStore!" Re: To find the units digit of a large number?   [#permalink] 01 Apr 2015, 23:35 Similar topics Replies Last post Similar Topics: 3 Shortcut to find Tens digit (Last two digit) of a number 1 17 Nov 2014, 21:27 1 Units Digit Number Properties & Exponents 1 06 Apr 2014, 06:24 1 How to find last digit of a number 13 03 Dec 2012, 23:28 1 Find the sum of 4 digit numbers 3 24 Aug 2012, 01:31 Finding prime numbers for large numbers 2 27 Mar 2012, 15:29 Display posts from previous: Sort by

https://mathhelpboards.com/threads/find-all-the-integer-values-of-k-for-x-3-kx-k-11-0-has-at-least-one-positive-integer-solution.25900/

# Find all the integer values of k for x^3 - kx + (k + 11) = 0 has at least one positive integer solution #### Taran ##### New member Hi, this question was in a year 11 extension maths textbook in the enrichment section. I have the answer as k>17 and k<-11 because I graphed it on GeoGebra. The Graph can be found here: https://ggbm.at/xpegwwtq. While I know the answers I would like to know how to work it out using algebra. Here is the Question: Consider the cubic equation x^3 - kx + (k + 11) = 0, find all the integer values of k for which the equation has at least one positive integer solution for x Thanks, Taran #### Opalg ##### MHB Oldtimer Staff member Hi, this question was in a year 11 extension maths textbook in the enrichment section. I have the answer as k>17 and k<-11 because I graphed it on GeoGebra. The Graph can be found here: https://ggbm.at/xpegwwtq. While I know the answers I would like to know how to work it out using algebra. Here is the Question: Consider the cubic equation x^3 - kx + (k + 11) = 0, find all the integer values of k for which the equation has at least one positive integer solution for x Thanks, Taran Hi Taran , and welcome to MHB. If $x=n$ is a positive integer solution of the equation, then $n^3 - kn + k + 11 = 0$, so that $$k = \frac{n^3+11}{n-1} = \frac{(n-1)(n^2+n+1) + 12}{n-1} = n^2+n+1 + \frac{12}{n-1}.$$ For that to be an integer, $n-1$ must be a factor of $12$. You can then tabulate the possible values of $n$ and $k = \frac{n^3+11}{n-1}$, as follows: $$\begin{array}{c|cccccc} n-1&1&2&3&4&6&12 \\ n&2&3&4&5&7&13 \\ k&19&19&25&34&59&184 \end{array}.$$ So the only possible values for $k$ are $19,\ 25,\ 34,\ 59,\ 184$ (which all agree with your condition that $k>17$). Last edited: #### MarkFL ##### Pessimist Singularitarian Staff member Hi Taran , and welcome to MHB. If $x=n$ is a positive integer solution of the equation, then $n^3 - kn + k + 11 = 0$, so that $$k = \frac{n^3+11}{n-1} = \frac{(n-1)(n^2+n+1) + 12}{n-1} = n^2+n+1 + \frac{12}{n-1}.$$ For that to be an integer, $n-1$ must be a factor of $12$. You can then tabulate the possible values of $n$ and $k = \frac{n^3+11}{n-1}$, as follows: $$\begin{array}{c|cccccc} n-1&1&2&3&4&6&12 \\ n&2&3&4&5&7&13 \\ k&19&19&25&34&59&184 \end{array}.$$ So the only possible values for $k$ are $19,\ 25,\ 34,\ 59,\ 184$ (which all agree with your condition that $k>17$). Hello, Chris! This question was posted on another site, and I found your reply so insightful, I took the liberty of posting it there, for the benefit of several there trying to solve it. #### Taran ##### New member Hi, Thank you so much!!! This question had my class stumped. That answer makes so much sense. It's been bugging me for a while and I'm very thankful for your help. Thanks again, Taran

https://math.stackexchange.com/questions/955294/prove-that-int-0-pi-frac-cos-x21-cos-x-sin-x-mathrmdx-i

# Prove that $\int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x$ In a related question the following integral was evaluated $$\int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{\mathrm{d}x/2}{1 + \cos x \sin x} =\int_0^{2\pi} \frac{\mathrm{d}x/2}{2 + \sin x} \,\mathrm{d}x =\int_{-\infty}^\infty \frac{\mathrm{d}x/2}{1+x+x^2}$$ I noticed something interesting, namely that \begin{align*} \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x & = \int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x \\ & = \int_0^{\pi} \frac{(\cos x)^2}{1 - \cos x \sin x} \,\mathrm{d}x = \int_0^{\pi} \frac{(\sin x)^2}{1 - \cos x \sin x} \,\mathrm{d}x \end{align*} The same trivially holds if the upper limits are changed to $\pi/2$ as well ($x \mapsto \pi/2 -u$). But I had problems proving the first equality. Does anyone have some quick hints? • consider evaluating $$\int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} - \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x = \int_0^{\pi} \frac{\cos (2x)}{1 +\frac{1}{2}\sin(2 x)}\,\mathrm{d}x$$ – ganeshie8 Oct 2 '14 at 10:27 • Are you not satisfied yet with the answers below? – Anastasiya-Romanova 秀 Oct 7 '14 at 12:07 • @N3buchadnezzar Oh sorry, I didn't know that. Get well soon ᕙ( ^ₒ^ c) – Anastasiya-Romanova 秀 Oct 8 '14 at 8:21 • Someday ;)${}{}{}$ – N3buchadnezzar Oct 8 '14 at 10:12 • math.stackexchange.com/questions/61605/… If you want further discussion ping me in chat. – N3buchadnezzar Oct 8 '14 at 10:32 Split the integral into two terms with limit $\left[0,\frac{\pi}{2}\right]$ and $\left[\frac{\pi}{2},\pi\right]$ \begin{align} \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi/2} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x +\int_{\pi/2}^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x \\ \end{align} Using identity \begin{align} \int_a^{b} f(x) \,\mathrm{d}x = \int_a^{b} f(a+b-x) \,\mathrm{d}x \end{align} Also the facts that $\cos\left(\frac{\pi}{2}-x\right)=\sin x$, $\sin\left(\frac{\pi}{2}-x\right)=\cos x$, $\cos\left(\frac{3\pi}{2}-x\right)=-\sin x$, and $\sin\left(\frac{3\pi}{2}-x\right)=-\cos x$, we get \begin{align} \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x &=\int_0^{\pi/2} \frac{(\sin x)^2}{1 + \sin x \cos x} \,\mathrm{d}x +\int_{\pi/2}^{\pi} \frac{(-\sin x)^2}{1 + \sin x \cos x} \,\mathrm{d}x \\ &=\int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x\tag{1} \end{align} Let \begin{align} I=\int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x \end{align} Since \begin{align} \int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x \end{align} then \begin{align} 2I&=\int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x +\int_0^{\pi} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x\\ &=\int_0^{\pi} \frac{(\cos x)^2 + (\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x\\ I&=\frac{1}{2}\int_0^{\pi} \frac{1}{1 + \cos x \sin x} \,\mathrm{d}x\tag{2} \end{align} Using $(2)$, we get \begin{align} I&=\int_0^{\pi} \frac{1}{2 + 2\cos x \sin x} \,\mathrm{d}x\\ &=\int_0^{\pi} \frac{1}{2 + \sin (2x)} \,\mathrm{d}x\qquad\Rightarrow\qquad x\mapsto2x\\ &=\frac{1}{2}\int_0^{2\pi} \frac{1}{2 + \sin x} \,\mathrm{d}x\tag{3} \end{align} Subtract the two integrals in question and get $$\int_0^{\pi} dx \frac{\cos{2 x}}{1+\frac12 \sin{2 x}} = \frac12 \int_0^{2 \pi} du \frac{\cos{u}}{1+\frac12 \sin{u}}$$ This may be shown to be equal to the complex integral $$-\frac{i}{4} \oint_{|z|=1} \frac{dz}{z} \frac{z+z^{-1}}{1+\frac{1}{4 i} (z-z^{-1})} = \oint_{|z|=1}\frac{dz}{z} \frac{z^2+1}{z^2+i 4 z-1}$$ The poles of the integrand within the unit circle are at $z=0$ and $z=-(2-\sqrt{3}) i$; their respective residues are $-1$ and $1$. By the residue theorem, therefore, the integral is zero and the two original integrals are equal. A quick hint is noticing the symmetry. A rigorous proof is that $$\int\limits_a^b {\frac{{f(\cos x)}}{{g(\sin x\cos x)}}dx} = \int\limits_a^b {\frac{{f\left( {\sin \left( {\frac{\pi }{2} - x} \right)} \right)}}{{g\left( {\cos \left( {\frac{\pi }{2} - x} \right)\sin \left( {\frac{\pi }{2} - x} \right)} \right)}}dx} = \int\limits_{\frac{\pi }{2} - b}^{\frac{\pi }{2} - a} {\frac{{f\left( {\sin u} \right)}}{{g\left( {\cos u\sin u} \right)}}du}$$Hope it helps ;) $$\frac{\cos(\pi/2-x)^2}{1+\cos(\pi/2-x)\sin(\pi/2-x)} = \frac{\sin(x)^2}{1+\sin(x)\cos(x)}$$ The integrands are both periodic with period $\pi$, so it suffices to verify the identity over any interval of length $\pi$, including $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Then, since both integrands are even functions, it suffices to check that the identity holds when integrating over $\left[0, \frac{\pi}{2}\right]$, that is, that $$\int_0^{\frac{\pi}{2}} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x =\int_0^{\frac{\pi}{2}} \frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x.$$ But one can show this simply by substituting $x = \frac{\pi}{2} - u$ on either side. • Your answer was the first I got, andthe first one I used. However for later readers who also stumble uppon this integral I feel the answer given by Anastasiya is clearer. – N3buchadnezzar Oct 7 '14 at 17:30 • The important thing is that you've learned something from posting here, which it seems you have. – Travis Oct 8 '14 at 3:23 $$I=\int_0^{\pi} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x$$ $$J=\int_0^{\pi/2} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x$$ Use $\int_a^b f(x)dx=\int_a^nf(a+b-x)dx$ and $\int_0^{2a}f(x)dx=\int_0^a f(x)dx+f(2a-x)dx$ $$I=\int_0^{\pi/2} \frac{(\cos (\pi-x))^2}{1 + \cos (\pi-x) \sin (\pi-x)} \,\mathrm{d}x+\frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x$$ $$I=\int_0^{\pi/2} \frac{(\cos x)^2}{1 - \cos x \sin x} \,\mathrm{d}x+\frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x$$ $$I=\int_0^{\pi/2} \frac{(\cos (\pi/2-x))^2}{1 - \cos (\pi/2-x) \sin (\pi/2-x)} \,\mathrm{d}x+\frac{(\cos (\pi/2-x))^2}{1 + \cos (\pi/2-x) \sin (\pi/2-x)} \,\mathrm{d}x$$ $$I=\int_0^{\pi/2} \frac{(\sin x)^2}{1 - \sin x \cos x} \,\mathrm{d}x+\frac{(\sin x)^2}{1 + \cos x \sin x} \,\mathrm{d}x$$ $$I=\int_0^{\pi/2} \frac{(\sin x)^2}{1 - \sin x \cos x} \,\mathrm{d}x+\frac{(\sin (\pi-x))^2}{1 - \cos (\pi-x) \sin(\pi-x)} \,\mathrm{d}x$$ $$I=\int_0^{\pi} \frac{(\sin x)^2}{1 - \sin x \cos x} \,\mathrm{d}x$$ Hope you can now show that: $$J=\int_0^{\pi/2} \frac{(\cos x)^2}{1 + \cos x \sin x} \,\mathrm{d}x=\int_0^{\pi/2} \frac{(\cos (\pi/2-x))^2}{1 + \cos (\pi/2-x) \sin (\pi/2-x)} \,\mathrm{d}x=\int_0^{\pi/2} \frac{(\sin x)^2}{1 + \sin x \cos x} \,\mathrm{d}x$$

http://stats.stackexchange.com/questions/32824/how-to-minimize-residual-sum-of-squares-of-an-exponential-fit

# How to minimize residual sum of squares of an exponential fit? I have the following data and would like to fit a negative exponential growth model to it: Days <- c( 1,5,12,16,22,27,36,43) Emissions <- c( 936.76, 1458.68, 1787.23, 1840.04, 1928.97, 1963.63, 1965.37, 1985.71) plot(Days, Emissions) fit <- nls(Emissions ~ a* (1-exp(-b*Days)), start = list(a = 2000, b = 0.55)) curve((y = 1882 * (1 - exp(-0.5108*x))), from = 0, to =45, add = T, col = "green", lwd = 4) The code is working and a fitting line is plotted. However, the fit is visually not ideal, and the residual sum of squares seems to be quite huge (147073). How can we improve our fit? Does the data allow a better fit at all? We could not find a solution to this challenge on the net. Any direct help or linkage to other websites/posts is greatly appreciated. - In this case, if you consider a regression model $\text{Emissions}_i=f(\text{Days}_i,a,b)+\epsilon_i$, where $\epsilon_i\sim N(0,\sigma)$, then you obtain similar estimators. By plotting the confidence regions, one can observe how these values are contained in the confindence regions. You cannot expect a perfect fit unless you interpolate the points or use a more flexible nonlinear model. – user10525 Jul 23 '12 at 13:12 I changed the title because "negative exponential model" means something different than described in the question. – whuber Jul 23 '12 at 13:15 Thanks for making the question clearer (@whuber) and thanks for your answer (@Procrastinator). How can I calculate and plot the confidence regions. And, what would be a more flexible nonlinear model? – Strohmi Jul 23 '12 at 13:18 You need an additional parameter. See what happens with fit <- nls(Emissions ~ a* (1- u*exp(-b*Days)), start = list(a = 2000, b = 0.1, u=.5)); beta <- coefficients(fit); curve((y = beta["a"] * (1 - beta["u"] * exp(-beta["b"]*x))), add = T). – whuber Jul 23 '12 at 13:20 @whuber - maybe you should post that as an answer? – jbowman Jul 23 '12 at 13:25 A (negative) exponential law takes the form $y=-\exp(-x)$. When you allow for changes of units in the $x$ and $y$ values, though, say to $y = \alpha y' + \beta$ and $x = \gamma x' + \delta$, then the law will be expressed as $$\alpha y' + \beta = y = -\exp(-x) = -\exp(-\gamma x' - \delta),$$ which algebraically is equivalent to $$y' = \frac{-1}{\alpha} \exp(-\gamma x' - \delta) - \beta = a\left(1 - u\exp(-b x')\right)$$ using three parameters $a = -\beta/\alpha$, $u = 1/(\beta\exp(\delta))$, and $b = \gamma$. We can recognize $a$ as a scale parameter for $y$, $b$ as a scale parameter for $x$, and $u$ as deriving from a location parameter for $x$. As a rule of thumb, these parameters can be identified at a glance from the plot: • The parameter $a$ is the value of the horizontal asymptote, a little less than $2000$. • The parameter $u$ is the relative amount the curve rises from the origin to its horizontal asymptote. Here, the the rise therefore is a little less than $2000 - 937$; relatively, that's about $0.55$ of the asymptote. • Because $\exp(-3) \approx 0.05$, when $x$ equals three times the value of $1/b$ the curve should have risen to about $1-0.05$ or $95\%$ of its total. $95\%$ of the rise from $937$ to almost $2000$ places us around $1950$; scanning across the plot indicates this took $20$ to $25$ days. Let's call it $24$ for simplicity, whence $b \approx 3/24 = 0.125$. (This $95\%$ method to eyeball an exponential scale is standard in some fields that use exponential plots a lot.) Let's see what this looks like: plot(Days, Emissions) curve((y = 2000 * (1 - 0.56 * exp(-0.125*x))), add = T) Not bad for a start! (Even despite typing 0.56 in place of 0.55, which was a crude approximation anyway.) We can polish it with nls: fit <- nls(Emissions ~ a * (1- u * exp(-b*Days)), start=list(a=2000, b=1/8, u=0.55)) beta <- coefficients(fit) plot(Days, Emissions) curve((y = beta["a"] * (1 - beta["u"] * exp(-beta["b"]*x))), add = T, col="Green", lwd=2) The output of nls contains extensive information about parameter uncertainty. E.g., a simple summary provides standard errors of estimates: > summary(fit) Parameters: Estimate Std. Error t value Pr(>|t|) a 1.969e+03 1.317e+01 149.51 2.54e-10 *** b 1.603e-01 1.022e-02 15.69 1.91e-05 *** u 6.091e-01 1.613e-02 37.75 2.46e-07 *** We can read and work with the entire covariance matrix of the estimates, which is useful for estimating simultaneous confidence intervals (at least for large datasets): > vcov(fit) a b u a 173.38613624 -8.720531e-02 -2.602935e-02 b -0.08720531 1.044004e-04 9.442374e-05 u -0.02602935 9.442374e-05 2.603217e-04 nls supports profile plots for the parameters, giving more detailed information about their uncertainty: > plot(profile(fit)) Here is one of the three output plots showing variation in $a$: E.g., a t-value of $2$ corresponds roughly to a 95% two-sided confidence interval; this plot places its endpoints around $1945$ and $1995$. - Oh, I almost forgot: res <- residuals(fit); res %*% res tells us that introducing the third parameter $u$ reduces the sum of squares to $2724$ (compared to $147073$ as stated in the question). – whuber Jul 23 '12 at 14:23 All well and good whuber. But maybe the OP had some reason to pick the exponential model (or maybe it is just because it is well known). I think first the residuals should be looked at for the exponential model. Plot them against potential covariates to see if there is structure there and not just large random noise. Before jumping into more sophisticated models try to see if a fancier model could possibly help. – Michael Chernick Jul 23 '12 at 14:25 Why don't you look at the original plot, Michael? It will make it abundantly obvious why at least one additional parameter is needed. Please note, too, that in a comment to the question the OP has asked, "what would be a more flexible nonlinear model?" One implication of the initial analysis offered in this answer is that it should be considered out of the ordinary to fit an exponential with fewer than three parameters: there must be some inherent constraint operating in such cases (such as an intrinsically determined unit of measurement or an intrinsic location for $x$). – whuber Jul 23 '12 at 14:32 I was not criticizing your answer! I did not see any residual plots. All I was suggesting is that plots of residuals vs potential covariates should be the first step in finding a better model. If I thought I had an answer to put up there I would have given an answer rather than raised my point as a constant. I thought you gave a great response and i was among the ones who gave you a +1. – Michael Chernick Jul 23 '12 at 14:58

https://scicomp.stackexchange.com/questions/37372/is-there-a-general-threshhold-for-which-a-large-condition-number-becomes-proble/37379

Is there a general threshhold for which a large condition number becomes “problematic?” I think the answer is probably no in the linear algebra community, but I'd say anything above $$10^8$$ and you're starting to lose too much precision, making the situation problematic. In some statistics settings, where they use OLS, I've seen it stated that even values over $$20$$ are problematic, but that seems pretty extreme. How do you guys assess whether a condition number is "problematic?" • I would consider anything above $1/\sqrt{\epsilon_{\text{mach}}}$ problematic, so it would be floating point system dependent. But it also depends on the size of the linear system. You can create small linear systems (3x3) with a condition number at O(1000) and the solution a computer finds could be out of acceptable tolerance. In general, I tell my students if they are using double precision the danger zone starts at 1000, and they need to be particularly careful if it is above $1/\sqrt{\epsilon_{\text{mach}}}$ – Abdullah Ali Sivas May 7 at 13:30 • @AbdullahAliSivas Is it also dependent on what method you're using to solve the linear system? If it's a direct method, I imagine a large condition number is (relatively) less problematic than an iterative method? Also, what is the logic behind $\frac{1}{\sqrt{\epsilon_{mach}}}$. I know what machine epsilon is, but I think I'm having trouble seeing the direct connection. – David May 7 at 13:35 • The condition number does not have much to do with stability of the method you are using. For example, say you are using the conjugate gradient method. The error coming from the matrix is at the mat-vec step. So the issue there would be caused by the condition number of the mat-vec operation not the condition number of the matrix. So both a direct method and CG would struggle similarly in that sense. But that doesn't have anything to do with the fact that an ill-conditioned problem is hard to solve independent of however stable the algorithm is. – Abdullah Ali Sivas May 7 at 14:08 • $1/\sqrt{\epsilon_{\text{mach}}}$ is some kind of heuristic for me; that is just where I notice things start to break. Like a small perturbation of $\epsilon_{\text{mach}}$ to the matrix (or the right-hand side vector) may cause upto an error of $\sqrt{\epsilon_{\text{mach}}}$ even if you assume that you have a perfect algorithm which does not introduce and propagate errors and is not affected by the floating point arithmetic. And my error tolerance is usually at $10^{-8}$ level which is close to $1/\sqrt{\epsilon_{\text{mach}}}$ in double precision :). – Abdullah Ali Sivas May 7 at 14:26 • @AbdullahAliSivas Why not make all of this into an answer? :-) – Wolfgang Bangerth May 7 at 16:10 This answer is my comments compiled and edited together. I apologize for the repetition. I would consider anything above $$1/\sqrt{\epsilon_{\text{mach}}}$$ problematic, so it would be floating point system dependent. But it also depends on the size of the linear system. You can create small linear systems (3x3) with a condition number at O(1000) and the solution a computer finds could be out of acceptable tolerance. In general, I tell my students if they are using double precision the danger zone starts at 1000, and they need to be particularly careful if it is above $$1/\sqrt{\epsilon_{\text{mach}}}$$. I want to note that the condition number does not have much to do with stability of the method you are using. Even when it is connected, for example, if you are using the conjugate gradient method, the error coming from the matrix is at the mat-vec step. So the issue there would be caused by the condition number of the mat-vec operation not the condition number of the matrix. As Federico Poloni pointed out in the comments, the condition number of the matrix-vector multiplication problem is bounded from above by the condition number of the matrix $$A$$, e.g. $$\|A\|\|A^{-1}\|$$. However, I think we should rather consider the slightly more general bound for the condition number of mat-vec $$Ax$$: $$\|A\|\|x\|/\|Ax\|$$. While we can immediately take the condition number of the problem equal to the condition number of the matrix, I usually see it written as $$\alpha\kappa(A)$$ with $$\alpha=(\|x\|/\|Ax\|)/(\|A^{-1}\|)$$. The idea, I believe, is that mat-vec and msolve are inverses of each other, and if one of them is well-conditioned, the other one will be ill-conditioned, similar to multiplication and division (some exceptions apply: $$1\times x$$ and $$x/1$$ are both well-conditioned). So, for matrices with high condition number msolves, it still may be the case that $$(\|x\|/\|Ax\|)\ll \|A^{-1}\|$$. Hence, the condition number of the mat-vec operation may be much lower than the condition number of the matrix. As an extreme example, if $$A$$ is singular $$Ax$$ is still meaningful and may be well-conditioned, but $$\kappa(A)=\infty$$ and $$A^{-1}x$$ is an ill-posed problem. So both a direct method and CG would struggle similarly if the elementary steps of the algorithm have issues related to input errors. But this is not the same thing as the conventional wisdom that an ill-conditioned problem is hard to solve independent of how stable the algorithm you employ is. Note that my arbitrary choice of $$1/\sqrt{\epsilon_{\text{mach}}}$$ is basically a heuristic for me and, in my experience, it is just where things start to break. As an example, a small perturbation of the relative magnitude $$\epsilon_{\text{mach}}$$ to the matrix (or the right-hand side vector) may cause up to an error of $$\sqrt{\epsilon_{\text{mach}}}$$, even if you assume that you have a perfect algorithm which does not introduce and propagate errors and is not affected by the floating point arithmetic. And my error tolerance is usually at $$10^{−8}$$ level which is close to $$\sqrt{\epsilon_{\text{mach}}}$$ in double precision :) • Thanks for the expanded answer! I still disagree on some points though: (1) if we are speaking about the usual relative condition number, then scalar matrix multiplication and division have both condition number 1, for each $x$. (2) I view matrix multiplication and solving linear systems as perfectly symmetrical operations, when seen as abstract maps: multiplying by $A$ isn't any different than multiplying by $A^{-1}$. The same vector-dependent modification to the condition number can be done for solving linear systems, by multiplying by $\alpha = ||b|| / ||A^{-1}b|| ||A||$. – Federico Poloni May 8 at 7:04 • I was thinking more in terms of this answer: math.stackexchange.com/a/3031746 . They are definitely symmetrical when thought as maps, and probably there are matrices for which msolve is well-conditioned but not mat-vec, I just haven't heard of them. – Abdullah Ali Sivas May 8 at 7:43

https://themortgagestudent.com/tag/550a36-what-is-the-cube-root-of-125

# what is the cube root of 125 The cube root of -8 is written as $$\sqrt[3]{-8} = -2$$. ... ∛125: 5 ∛216: 6 ∛1,000: 10 ∛1,000,000: 100 ∛1,000,000,000: 1000: The calculations were performed using this cube root calculator. When did organ music become associated with baseball? 5³ = 5 * 5 * 5 = 25 * 5 = 125 So the cube root of 125 is 5. So, in this case the cube root of 125 is 5. Here is the answer to questions like: What is the cube root of 125 or what is the cube root of 125? Someone help me with this ASAP ANSWER QUICK! The nearest previous perfect cube is … Thus, each edge of the cube is 5 cm long. That number is 5. Calculating n th roots can be done using a similar method, with modifications to deal with n. While computing square roots entirely by hand is tedious. Thus, each edge of the cube is 5 cm long. The cube root of 125 is what number cubes is 125. 80% of questions are answered in under 10 minutes Answers come with explanations, so that … See next answers. So, in this case the cube root of 125 is 5. Step 1) Set up 125 in pairs of two digits from right to left and attach one set of 00 because we want one decimal: … Learn more with Brainly! What is plot of the story Sinigang by Marby Villaceran? The material on this site can not be reproduced, distributed, transmitted, cached or otherwise used, except with prior written permission of Multiply. cube root of 125/512 = 5/8. Estimating higher n th roots, even if using a calculator for intermediary steps, is … The length of a side (edge) of a cube is equal to the cube root of the volume. Does the calculator support fractions? What is the birthday of carmelita divinagracia? While every effort is made to ensure the accuracy of the information provided on this website, neither this website nor its authors are responsible for any errors or omissions, or for the results obtained from the use of this information. The length of a side (edge) of a cube is equal to the cube root of the volume. Learn more with Brainly! 125 can be written as 125(e^0), 125(e^((2pi)i)), 125(e^((4pi)i)) where e is euler’s number, i is the imaginary unit, i^2=-1, and e^((theta)i)=cos(theta)+(i)(sin(theta)) where theta is an angle measured in radians. First we will find all factors under the cube root: 125 has the cube factor of 125. All Rights Reserved. Get free help! Find the interest rate … that Sally earned from the bank. Having trouble with your homework? Guess: 5.125 27 ÷ 5.125 = 5.268 (5.125 + 5.268)/2 = 5.197 27 ÷ 5.197 = 5.195 (5.195 + 5.197)/2 = 5.196 27 ÷ 5.196 = 5.196 Estimating an n th Root. Is evaporated milk the same thing as condensed milk? The cube root of a number answers the question "what number can I multiply by itself twice to get this number?". 5³ = 5 * 5 * 5 = 25 * 5 = 125 So the cube root of 125 is 5 Just right click on the above image, then choose copy link address, then past it in your HTML. The length of a side (edge) of a cube is equal to the cube root of the volume. For example, 5 is the cube root of 125 because 53 = 5•5•5 = 125, -5 is cube root of -125 because (-5)3 = (-5)•(-5)•(-5) = -125. Why don't libraries smell like bookstores? Yes, simply enter the fraction as a decimal floating point number and you will get the corresponding cube root. Who is the longest reigning WWE Champion of all time? Not sure about the answer? See also our cube root table from 1 to 1000. How to find the square root of 125 by long division method Here we will show you how to calculate the square root of 125 using the long division method with one decimal place accuracy. New questions in Mathematics. Therefore the cube roots of 125 are 5(e^0),5(e^((2pi)i/3)),5(e^((4pi)i/3)). 'CUBE ROOT OF 125' is a 13 letter phrase starting with C and ending with 5 Crossword clues for 'CUBE ROOT OF 125' Synonyms, crossword answers and other related words for CUBE ROOT OF 125 [five] We hope that the following list of synonyms for the word five will help you to finish your crossword today. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. How long does it take to cook a 23 pound turkey in an oven? 5(e^0)=5(1)=0 How long will the footprints on the moon last? For … As you can see the radicals are not in their simplest form. Let's check this with ∛125*1=∛125. Now extract and take out the cube root ∛125 * ∛1. Inter state form of sales tax income tax? The cube root of -64 is written as $$\sqrt[3]{-64} = -4$$. The cube root of 125 is 5 so therefore each edge is 5 cm which is about 2 inches A cube root of a number a is a number x such that x 3 = a, in other words, a number x whose cube is a. Please link to this page! Thus, each edge of the cube is 5 cm long. All information in this site is provided “as is”, with no guarantee of completeness, accuracy, timeliness or of the results obtained from the use of this information. 125 is said to be a perfect cube because 5 x 5 x 5 is equal to 125. Cube of ∛125=5 which results into 5∛1; All radicals are now simplified. For example, 5 is the cube root of 125 because 5 3 = 5•5•5 = 125, -5 is cube root of -125 because (-5) 3 = (-5)• (-5)• (-5) = -125. Therefore, the real cube root of 125 is 5. Copyright © 2020 Multiply Media, LLC. How will understanding of attitudes and predisposition enhance teaching? What is the conflict of the story of sinigang? So, in this case the cube root of 125 is 5. This is the lost art of how they calculated the square root of 125 by hand before modern technology was invented. That number is 5. The cube root of 125 is what number cubes is 125. Cube Root of 125. Volume to (Weight) Mass Converter for Recipes, Weight (Mass) to Volume to Converter for Recipes. Since 125 is a whole number, it is a perfect cube. Who of the proclaimers was married to a little person? ( cube root of 125 is 5 and cube root of 512 is 8) hope it helps you... i want full solution The Brain; Helper; Not sure about the answer? A cube root of a number a is a number x such that x3 = a, in other words, a number x whose cube is a. After 42 months, Sally earned $238 in simple interest. Sally deposited$850 into her bank account for 42 months. WHO WANNA JOIN MY … How long will it take to cook a 12 pound turkey? What is the contribution of candido bartolome to gymnastics? Cube roots (for integer results 1 through 10) Cube root of 1 is 1; Cube root of 8 is 2; Cube root of 27 is 3; Cube root of 64 is 4; Cube root of 125 is 5; Cube root of 216 is 6; Cube root of 343 is 7; Cube root of 512 is 8; Cube root of 729 is 9 What details make Lochinvar an attractive and romantic figure?

https://math.stackexchange.com/questions/2853732/prove-log-2x-log-xy-log-y8-geq-sqrt381

# Prove: $\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$ Prove that for every $x$,$y$ greater than $1$: $$\log_2(x)+\log_x(y)+\log_y(8)\geq \sqrt[3]{81}$$ What I've tried has got me to: $$\frac{\log_y(x)}{\log_y(2)}+\log_x(y)+3\log_y(2)\geq \sqrt[3]{81}$$ I didn't really get far.. I can't see where I can go from here, especially not what to do with $\sqrt[3]{81}$. This is taken out of the maths entry tests for TAU, so this shouldn't be too hard. • $\log_xy=\frac1{\log_yx}$ – MalayTheDynamo Jul 16 '18 at 19:04 • A neat thing about $\log_a b = \frac 1{\log_b a}$ and $\frac {\log_b c}{\log_b a} = \log_a c$.... It means $\log_a b* \log_b c = \log_a c$ and so $\log_a b*\log_b c* \log_c d*....... *\log_y z = \log_a z$...... – fleablood Jul 16 '18 at 19:40 We have $$\log_2x+\log_xy+\log_y8=\frac{\log x}{\log2}+\frac{\log y}{\log x}+\frac{3\log2}{\log y}$$ and since $2,x,y>1$, we can deduce that their logarithms are non-negative, and so will their quotients. Now use AM-GM: $$\sqrt[3]{\frac{\log x}{\log2}\cdot\frac{\log y}{\log x}\cdot\frac{3\log2}{\log y}}\le\frac{\frac{\log x}{\log2}+\frac{\log y}{\log x}+\frac{3\log2}{\log y}}3$$ giving $$\frac{\log x}{\log2}+\frac{\log y}{\log x}+\frac{3\log2}{\log y}\ge3\sqrt[3]3=\sqrt[3]{81}$$ as required. • While I did all these logarithmic exercises, I noticed that the AM-GM is actually a very common technique for the solution. Is there any reason for that or is it just a coincudence? – L0wRider Jul 17 '18 at 12:41 • @Maxim AM-GM is usually the second inequality that people learn (after $x^2 \ge 0$). So it's a good choice to use when writing a problem for an entrance exam, because many people are familiar with it. – Ovi Jul 17 '18 at 17:37 • @Ovi That's embarrassing. I learned about AM/GM/HM just last week :) – TheSimpliFire Jul 17 '18 at 19:22 • @TheSimpliFire Don't feel embarrased, I'm not saying that most people know about and how to use AM-GM. I'm saying that of the people who know inequalities, most know AM-GM; so if they put an inequality question, AM-GM is a good problem. However, the question of weather they should put inequalities questions is separate. – Ovi Jul 17 '18 at 21:39 Let's convert everything to $$\log$$ base $$2$$ so we have a common something to work with: $$\log_2 x + \dfrac {\log_2 y}{\log_2 x} + \dfrac {\log_2 8}{\log_2 y} \ge \sqrt [3]{81} = 3 \sqrt[3]{3}$$ Now let $$a = \log_2 x$$ and $$b = \log_2 y$$. The condition $$x, y > 1$$ implies that $$a, b > 0$$. So now we have to prove $$a + \dfrac ba + \dfrac {3}{b} \ge3 \sqrt[3]{3}$$ The cube root in there especially may remind us of AM-GM with $$3$$ terms. And indeed, the inequality $$\text{AM}\left(a, \dfrac ba, \dfrac 3b\right) \ge \text{GM}\left(a, \dfrac ba, \dfrac 3b\right)$$ gives exactly what is desired. Hint: Show that $a+\frac{b}{a}+\frac{3}{b}\geq \sqrt[3]{81}$ for all $a,b>0$, using AM-GM. Two things: $\log_a b = \frac 1{\log_b a}$ and $\frac {\log_b c}{\log_b a} = \log_a c$ so $\log_a b\log_b c = \frac {\log_b c}{\log_b a} = \log_a c$. And AM-GM says $\frac {a + b+ c}3 \ge \sqrt[3]{abc}$. So..... $\frac {\log_2 x + \log_x y + \log_y 8}3 \ge \sqrt[3]{\log_2 x \log_x y \log_y 8} = \sqrt[3]{\log_2 8}$

https://math.stackexchange.com/questions/560816/find-the-sum-of-the-series-sum-frac1nn1n2

# Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$ I got this question in my maths paper Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$ and find the sum if it exists. I managed to show that the series converges but I was unable to find the sum. Any help/hint will go a long way. Thank you. Using Partial Fraction Decomposition, $$\frac1{n(n+1)(n+2)}=\frac An+\frac B{n+1}+\frac C{n+2}$$ $$\implies 1=A(n+1)(n+2)+Bn(n+2)+Cn(n+1)$$ $$\implies 1=n^2(A+B+C)+n(3A+2B+C)+2A$$ Comparing the coefficients of the different powers (namely, $0,1,2$) of $n,$ we get $A=\frac12,B=-1,C=\frac12$ $$\implies\frac1{n(n+1)(n+2)}=\frac12\cdot\frac1n-\frac1{n+1}+\frac12\cdot\frac1{n+2}$$ $$=-\frac12\left(\underbrace{\frac1{n+1}-\frac1n}\right)+\frac12\left(\underbrace{\frac1{n+2}-\frac1{n+1}}\right)$$ Can you recognize the two Telescoping series? Hint $$\frac{2}{n(n+2)}=\frac{1}{n}-\frac{1}{n+2}$$ Now multiply both sides by $\frac{1}{n+1}$. • This is a good hint. – Mark Bennet Oct 28 '17 at 17:51 Hint. You may write $$\frac{1}{n(n+1)(n+2)} =\frac{1}{2}\left(\frac{1}{n}-\frac{1}{(n+1)}\right)+\frac{1}{2}\left(\frac{1}{(n+2)}-\frac{1}{(n+1)}\right)$$ giving two telescoping sums $$\sum_{n=1}^N\frac{1}{n(n+1)(n+2)} =\frac{1}{2}-\frac{1}{2(N+1)}+\frac{1}{2(N+2)}-\frac{1}{4}.$$ • That is a great clean answer!! +1 Not like that show off of Jack who always has to show his "ability" in complicating easy things. – Von Neumann Sep 20 '16 at 20:44 • @FourierTransform: the benefit of complicating things is that my approach also computes $$\sum_{k\geq 1}\frac{1}{k(k+1)(k+2)(k+3)(k+4)(k+5)}$$ in two steps. It is, in fact, equivalent to telescoping. – Jack D'Aurizio Sep 20 '16 at 20:47 • @DarioGutierrez Thanks. I just started with your identity:$\frac{1}{n(n+1)(n+2)} =\frac{1}{2}(\frac{1}{n}+\frac{1}{(n+2)})-\frac{1}{(n+1)}$, but $\frac{1}{2}(\frac{1}{n}+\frac{1}{(n+2)})-\frac{1}{(n+1)}=\frac{1}{2}\left(\frac{1}{n}-\frac{1}{(n+1)}\right)+\frac{1}{2}\left(\frac{1}{(n+2)}-\frac{1}{(n+1)}\right)$. – Olivier Oloa Sep 20 '16 at 20:51 • @FourierTransform It's always nice to have different points of view. That was what $\texttt{Jack D'Aurizio}$ did it. Usually, I receive a lot of negative comments or/and down-votes in my own answers because I posted some weird overkill answer. What 'some' people don't understand is that I'm adding something different to the bag of answers. By the way, behind me I have a guy who behaves like a 'MSE-police officer' who down-vote, propose to close, review my $\LaTeX$, change my editions, etc... a kind of insane user. Also, I enjoy your answers too which are 'full creative'. Best Regards. – Felix Marin Sep 20 '16 at 21:43 • @FourierTransform We should have different answers and different ideas, else, what would be the point of answering in the first place? Also, different ideas lead to learning and building of intuition, which is one reason I dislike the school system to some level. I agree with Felix, we enjoy both of your answers, and we want both. Sometimes, answers aren't meant for the OP, but for the community. – Simply Beautiful Art Sep 20 '16 at 22:04 There is an alternate method and is as follows. Notice that $$\frac{1}{n(n+1)(n+2)} = \frac{(n-1)!}{(n+2)!} = \frac{1}{2!} \, B(n,3)$$ where $B(x,y)$ is the Beta function. Using an integral form of the Beta function the summation becomes \begin{align} S &= \sum_{n=1}^{\infty} \frac{1}{n \, (n+1) \, (n+2)} \\ &= \frac{1}{2} \, \int_{0}^{1} \left( \sum_{n=1}^{\infty} x^{n-1} \right) \, (1-x)^{2} \, dx \\ &= \frac{1}{2} \, \int_{0}^{1} \frac{(1-x)^{2}}{1-x} \, dx = \frac{1}{2} \, \int_{0}^{1} (1-x) \, dx \\ &= \frac{1}{4} \end{align} This leads to the known result \begin{align} \sum_{n=1}^{\infty} \frac{1}{n \, (n+1) \, (n+2)} = \frac{1}{4}. \end{align} Using Euler's beta function, $$S=\sum_{k\geq 1}\frac{(k-1)!}{(k+2)!}=\sum_{k\geq 1}\frac{\Gamma(k)}{\Gamma(k+3)}=\frac{1}{2}\sum_{k\geq 1}B(k,3)$$ hence: $$S = \frac{1}{2}\int_{0}^{1}(1-x)^2\sum_{k\geq 1}x^{k-1}\,dx=\frac{1}{2}\int_{0}^{1}(1-x)\,dx = \color{red}{\frac{1}{4}}.$$ A straightforward generalization of this approach gives the identity: $$\sum_{n\geq 1}\frac{1}{n(n+1)\cdots(n+N)}=\color{red}{\frac{1}{N\cdot N!}}.$$ Writing $$\frac{1}{n(n+1)(n+2)} = \frac{1}{n+1}\left(\frac{1}{2n} - \frac{1}{2(n+2)}\right) = \frac{1}{2n(n+1)} - \frac{1}{2(n+1)(n+2)},$$ we see that the sum telescopes to $$\frac{1}{2(1)(2)} = \frac{1}{4}.$$ • This is the best form to use - simple and straightforward. – Mark Bennet Oct 28 '17 at 17:51 Alternatively, take $$\frac{1}{1-x} = \sum_{n=1}^{\infty} x^{n-1},$$ and integrate three times with lower limit $0$, giving \begin{align*} -\log{(1-x)} &= \sum_{n=1}^{\infty} \frac{x^n}{n} \\ x + (1-x)\log{(1-x)} &= \sum_{n=1}^{\infty} \frac{x^n}{n(n+1)} \\ \frac{3}{4}x^2 - \frac{1}{2}x - \frac{1}{2} (1-x)^2 \log{(1-x)} &= \sum_{n=1}^{\infty} \frac{x^n}{n(n+1)(n+2)}, \end{align*} and (as @Clement C reminds me) we then apply Abel's theorem to take the limit as $x \to 1$, which gives $1/4$ as the answer. • Isn't there a bit more to the argument, then? The original power series has radius of convergence $1$, so the equality and all integration theorems only apply on $(-1,1)$. To apply the conclusion to $x=1$ at the end (outside the open disc of convergence of the original series, and where the final LHS is only defined by continuity), don't you need Abel's theorem? – Clement C. Mar 26 '15 at 14:12 • I suppose, since you also have to take the limit to get $0\log{0} "=" 0$. I'll put a note. – Chappers Mar 26 '15 at 14:17 You are on the right track. Fix $N \geq 1$. Then \begin{align} \sum_{n=1}^{N} \frac{1}{n \cdot (n+1) \cdot (n+2)} &= \sum_{n=1}^N \left(\frac{1}{2n}-\frac{1}{n+1}+\frac{1}{2n+4}\right) = \frac{1}{2}\sum_{n=1}^N \frac{1}{n}-\sum_{n=1}^{N}\frac{1}{n+1}+\frac{1}{2}\sum_{n=1}^{N} \frac{1}{n+2}\\ &= \frac{1}{2}\sum_{n=1}^N \frac{1}{n}-\sum_{n=2}^{N+1}\frac{1}{n}+\frac{1}{2}\sum_{n=3}^{N+2} \frac{1}{n} \\ &= \frac{1}{2}\left(1+\frac{1}{2}\right) + \frac{1}{2}\sum_{n=3}^N \frac{1}{n}-\left(\frac{1}{2}+\frac{1}{N+1}\right)-\sum_{n=3}^{N}\frac{1}{n}+\\ &\quad \left(\frac{1}{N+1}+\frac{1}{N+2}\right)+\frac{1}{2}\sum_{n=3}^{N} \frac{1}{n} \\ \end{align} Can you continue from there? (the partial sums cancel out, and you only have a few remaining terms. Taking the limit $N\to\infty$ will give you the limit.) • Interesting approach... Different from everything I learned.... Thanks! – Frank Mar 26 '15 at 14:09 Here's another way to do it:$$\sum_{n\ge 1}\frac{1}{n(n+1)(n+2)}=\frac{1}{2}\sum_{n\ge 1}\bigg(\frac{1}{n(n+1)}-\frac{1}{(n+1)(n+2)}\bigg)=\frac{1}{2}\cdot\frac{1}{1\cdot 2}=\frac{1}{4}.$$One advantage is an easy generalisation to a problem for which the partial fractions decomposition would get thorny:$$\sum_{n\ge 1}\frac{1}{\sum_{j=0}^k(n+j)}=\frac{1}{k!\cdot k}.$$

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# Analytic Trigonometry Pdf Trigonometry is a special subject of its own, so you might like to visit: Introduction to Trigonometry; Trigonometry Index. Most of the topics that appear here have already been discussed in the Algebra book and often the text here is a verbatim copy of the text in the other book. Find the length of side AB in the figure below. There are two values of x that satisfy this condition: x=ÅÅpÅÅ 6 and x=p. Following a unified approach, the authors obtain estimates for these sums similar to the classical I. Algebra And Trigonometry With Analytic Geometry. Euler’s Formula and Trigonometry Peter Woit Department of Mathematics, Columbia University September 10, 2019 These are some notes rst prepared for my Fall 2015 Calculus II class, to give a quick explanation of how to think about trigonometry using Euler’s for-mula. Since angles are so important in the analysis of the muscu-loskeletal system, trigonometry is a very useful biomechanics tool. Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this book popular among students year after year. The trigonometric functions sine and cosine have four important limit properties: You can use these properties to evaluate many limit problems involving the six basic trigonometric functions. Collaborative Project — Analytic Trigonometry 1. Resources are IEB and CAPS aligned. new and clearer proofs, expansion of some problem sets, a review of Analytic Trigonometry, a few remarkable airbrush figures, and bonuses on pages previously only partially used. 2, Using fundamental identities and verifying trigonometric identities Video 1: Solving trig identities A; AHSMESA, 8:45. The book presents the theory of multiple trigonometric sums constructed by the authors. An "identity" is a tautology, an equation or statement that is always true, no matter what. Check the book if it available for your country and user who already subscribe will have. 4 Analytic Trigonometry 69 Chapter 4 Analytic Trigonometry 4. We apply the novel inequality to the generalized trigonometric functions and establish several Redheffer-type inequalities for these functions. Substituting 0 for x, you find that cos x approaches 1 and sin x − 3 approaches −3; hence, Example 2: Evaluate. If we know the length of two sides of the triangle, we are able to work out the. Trigonometric Identities Sum and Di erence Formulas sin(x+ y) = sinxcosy+ cosxsiny sin(x y) = sinxcosy cosxsiny cos(x+ y) = cosxcosy sinxsiny cos(x y) = cosxcosy+ sinxsiny tan(x+ y) = tanx+tany 1 tanxtany tan(x y) = tanx tany 1+tanxtany Half-Angle Formulas sin 2 = q 1 cos 2 cos 2 = q 1+cos 2 tan 2 = q 1+cos tan 2 = 1 cosx sinx tan 2 = sin 1+cos. CHAPTER 5: ANALYTIC TRIGONOMETRY 5. So this book is not just about mathematical content but is also about the process of learning and doing mathematics. This study guide provides practice questions for all 34 CLEP exams. 2 Verifying Trigonometric Identities 5. Unit 8 Analytic Trigonometry. Improve your math knowledge with free questions in "Trigonometric identities I" and thousands of other math skills. Evaluate the function at each specified value of the independent variable and simply: f(x) = x 2 + 1 (a) f(t 2) (b) f(t + 1)remember: just put x3 for x 3 (place commas between multiple answers) in order. Why you should learn it Fundamenta l trigonometric. 5 Trigonometric Form of a Complex Number Selected Applications Triangles and vectors have many real-life applications. We dare you to prove us wrong. We will use all the ideas we've been building up as we've been studying vectors to be able to solve these questions. Right Triangle Definition. Powered by Create your own unique website with customizable templates. With the knowledge of trigonometry anything from calculus to vector algebra can be solved. Yes, Analytical Trigonometry isn't particularly exciting. net : Allows you online search for PDF Books - ebooks for Free downloads In one place. Similar analyses are studied in the car industry, where numerical simulation is used in virtually every aspect of design and car production. sec x Matches (d). Chapter 5 Analytic Trigonometry Section 5. 294-296, Exercises # 1-105 (every other odd) (4) 4. 2, Using fundamental identities and verifying trigonometric identities Video 1: Solving trig identities A; AHSMESA, 8:45. Round to the nearest tenth. Some examples are. c_n = {1 \over T}\int\limits_T {x(t)e^{ - jn\omega _0 t} dt} \quad \quad Analysis \cr} The derivation is similar to that for the Fourier cosine series given above. is in Quadrant II. 3) Equation (16. Free PDF download of RD Sharma Class 11 Solutions Chapter 11 Trigonometric Equations solved by Expert Maths Teachers on NCERTBooks. Barnett, Analytic Trigonometry is a text that students can actually read, understand, and apply. the vector by the trigonometric functions. 5 j 4AXlBlf krNimgthst9s O Gr seesse nr 7v1e ed L. This teacher's edition accompanies the sold-separately Precalculus with Trigonometry and Analytical Geometry and contains a copy of the student text and curriculum/lesson plans for the entire book. Galois Theory Emil Artin. Sine and cosine in a right angle triangle, finding angle in a triangle using sine and cosine. sec x Matches (d). The following picture shows the. (A triangle cannot have more than one right angle) The standard trigonometric ratios can only be used on. 5 Graphs of Sine and Cosine Functions. Ready for PreCalculus? Find out with our FREE PreCalculus Assessment Test. Use the hints provided. jmap by topic To access Regents, Practice and Journal Worksheets, Lesson Plans, Videos and other resources, click on the State Standard in the last column below. Using Trigonometry To Find Lengths Date_____ Period____ Find the missing side. Polyhedra and Non-Polyhedra. About ACT math practice problems worksheet pdf. 536 Chapter 7 Analytic Trigonometry What you should learn •Recognize and write the fundamental trigonometric identities. the vector by the trigonometric functions. Part IV Answer Key. Algebra and Trigonometry with Analytic Geometry [with CD-ROM] by Earl W. 1) 16 28? 2) 30 24? 3) 45 13? 4) 31 37? 5) 15 37? 6) 18 34? 7) 56 33 65?. Books to Borrow. App to discover the relationship between the trigonometric functions of sine, cosine and tangent and the side length ratios of a right triangle. Then we find A and B. Read & download eBooks for Free: anytime!. A quick evaluation of the alterations may be made by comparing old and new pages 3-4, 54-56, 76-77, 118, 179-186,. 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The idea of this book is to give an extensive description of the classical complex analysis, here ”classical” means roughly … [Download] Complex Analysis PDF | Genial eBooks Download the eBook Complex Analysis in PDF or EPUB format and read it directly on your mobile phone, computer or any device. Euler's formula states that for any real number x: = ⁡ + ⁡,. All things Sine, cosine and Tangent: links, you tube vids, formulas etc. 115 Franklin Turnpike, #203, Mahwah, NJ 07430. Student’s Mistakes and Misconceptions on Teaching of Trigonometry Nevin ORHUN(1) Abstract:Trigonometry is an unseparable part of mathematics in high school. 3 sec secxx x 10. These are PDF files suitable for an ebook reader. There are six functions that are the core of trigonometry. Algebra and Trigonometry offers a wealth of. Powered by Create your own unique website with customizable templates. If we know the length of two sides of the triangle, we are able to work out the. 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The word “Trigonometry” comes from the Greek words: ‘ Trigonon ’ meaning ‘triangle’ and ‘ metron ’ meaning a ‘measure’. 2 sin2x — sin x —3 0, 6. Following a unified approach, the authors obtain estimates for these sums similar to the classical I. Average Value of a Function. oop&system analysis design analysis ebook pdf links; trigonometry by loney pdf. Trigonometry, 8th Edition Algebra and Trigonometry, 3rd Edition Just-in-Time Algebra and Trigonometry for Calculus (4th Edition) Algebra and Trigonometry: Structure and Method, Book 2 Algebra and Trigonometry with Analytic Geometry (with CengageNOW Printed Access Card). Trigonometry Functions and Unit Circle TEST STUDY GUIDE Test covers: Given a right triangle, find 6 trig functions. Trigonometry Trigonometry is derived from Greek words trigonon (three angles) and metron ( measure). 25sin 92 x 6. 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Type your expression into the box to the right. This course is structured to not leave you behind in the dust. Description: Intuitive Guide to Fourier Analysis – Chapter 1. Essential Math 10 - Trigonometry Problems. As early as in the work of Balthazard et al. 3 sec secxx x 10. EbookNetworking. co s 4 x = (cos2 x)2 = a 1 + cos 2x 2 b 2 Use cos 2 u = 1 + cos 2u 2. UNIT 1: ANALYTICAL METHODS FOR ENGINEERS Unit code: A/601/1401 QCF Level: 4 Credit value: 15 OUTCOME 1 TUTORIAL 2 EXPONENTIAL, TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS Unit content 1 Be able to analyse and model engineering situations and solve problems using algebraic methods. (Precalculus by Larsen, Hostetler) We have numbered the videos for quick reference so it's reasonably obvious that each subsequent video presumes knowledge of the previous videos. Suppose sinx=ÅÅ1ÅÅ 2. Trigonometry plays a major role in industry, where it allows manufacturers to create everything from automobiles to zigzag scissors. 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Even though the subject is is easy, it is sometimes complicated for students to get their heads around basics concepts like angles, what pi is, angles in a circle and their use, right triangle using sine and cosine. Chapter 3: Graphing Linear Functions. 4 Sum and Difference Formulas 5. Suppose sinx=ÅÅ1ÅÅ 2. 7 Trigonometric Equations and Inequalities. Cole offers clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications. The Trigonometry problems with solutions that we provide are correct ones that you can rely on. View any or all of the videos to the right. 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Download ANALYTIC TRIGONOMETRY WITH APPLICATIONS 11TH EDITION book pdf free download link or read online here in PDF. tan x csc x sin x cos x 1 sin x 1 cos x 15. Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Read online ANALYTIC TRIGONOMETRY WITH APPLICATIONS 11TH EDITION book pdf free download link book now. On this page you can read or download 8 4 trigonometry worksheet glencoe geometry answers in PDF format. The text is suitable for a typical introductory Algebra & Trigonometry course, and was developed to be used flexibly. tan 162 x 5. It presented a concise account of the main results then known, but was on a scale which limited the amount of detailed discussion possible. Substituting 0 for x, you find that cos x approaches 1 and sin x − 3 approaches −3; hence, Example 2: Evaluate. 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New Book Geometry and Trigonometry for Calculus (Wiley Self-Teaching Guides). 1) 16 28? 2) 30 24? 3) 45 13? 4) 31 37? 5) 15 37? 6) 18 34? 7) 56 33 65?. Swokowski and Jeffery A. Download ANALYTIC TRIGONOMETRY WITH APPLICATIONS 11TH EDITION book pdf free download link or read online here in PDF. The word “Trigonometry” comes from the Greek words: ‘ Trigonon ’ meaning ‘triangle’ and ‘ metron ’ meaning a ‘measure’. 3cos 4cos 42 xx 4. Powered by Create your own unique website with customizable templates. This is then applied to calculate certain integrals involving trigonometric. If we know the length of two sides of the triangle, we are able to work out the. 3 Vectors in the Plane 6. Average Value of a Function. Mathematics: Overall Analysis Grade-wise Weightage: 11th (44%), 12th (56%) Chapters with more than one questions: Application of Derivative (3) Matrices and Determinants (2) Vectors and 3D (2) Sequence and Series (2). 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An Amusing Equation: From Euler’s formula with angle …, it follows that the equation: ei… +1 = 0 (2) which involves five interesting math values in one short equation. Right Triangle Trigonometry Project Due Date: Background: Trigonometry is the only way to mathematically figure out the length of a side in a right triangle given another side and an angle or to find out the measure of an angle given two sides. 2 Law of Cosines 6. Analytic Trigonometry details the fundamental concepts and underlying principle of analytic geometry. It plays an important role in surveying, navigation, engineering, astronomy and many other branches of physical science. Thanks to Andrew Solomon for this correction. The material covered includes many. Unit 5: Analytic Trig Review Solve each equation. The examination covers the subjects of Algebra, Trigonometry, and Analytic. Department of Mathematics, Purdue University 150 N. This is referred to as ‘spectral analysis’ or analysis in the ‘frequency. 1219 83° 3) tan Y = 0. Trigonometric Ratios of certain Angles. Circular function definitions, where is any angle. Find all of the values of x in the interval @0,2 pD for which 4 sin2 x=1. All books are in clear copy here, and all files are secure so don't worry about it. Basic Trigonometric Equations An equation that contains trigonometric functions is called a trigonometric equation. Below are a few standard hints. There is no "sum of squares" formula, i. Analytic geometry is a great invention of Descartes and Fermat. 4 new maths courses and all free material. Given the value of one trig ratio, find the other 5 trig ratios. With the knowledge of trigonometry anything from calculus to vector algebra can be solved. Pines Use this website to download homework and review power points for upcoming tests. 2 Law of Cosines 6. tan x csc x sin x cos x 1 sin x 1 cos x 15. Remembering the definitions 4 6. HINT COLLECT LIKE TERMS HINT EXTRACT SQUARE ROOTS 1. This is a 4 part worksheet: Part I Model Problems. This course is structured to not leave you behind in the dust. Use the hints provided. For the most complete and up-to-date course information, contact the instructor Instructor. identities that it knows about to simplify your expression. China Phone: +86 10 8457 8802. Kouba And brought to you by : eCalculus. 12 KB (Last Modified Yesterday at 11:57 AM). Vedamurthy and Dr. 7) 55 51? 68° 8) 19 27? 45° 9) 34 55. Each worksheet contains 50 questions with answers. solu-tions of the planar Laplace equation. Answers Pre-requisite- Chapter 2. 3/9 Solving Trig Equations 3/10 Solving Trig Equations 3/13 Sum and Difference Formulas 3/14 Pi Day 3/15 Optional. CHAPTER 5: ANALYTIC TRIGONOMETRY 5. Problem 1: Students are asked to text in one thing they already know about reciprocal trig functions. Solve right triangles. Comments 1. 39,41,47,49. 2 Factoring Formulas A. 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Current search Algebra Trigonometry And Analytic Geometry. no formula for. 1 - Trigonometry review Recall that the three most useful trigonometric ratios are relationships between the lengths of the sides in a right triangle as defined by the following table: Full ratio name Standard abbreviation Ratio sine(A) sin(A) A s nuse O H cosine(A) cos(A) side adjacent to A. (A triangle cannot have more than one right angle) The standard trigonometric ratios can only be used on. Older, obsolete versions are posted below: Math 221 – First Semester Calculus. no formula for. [pdf] Exploring C By Yashavant Kanetkar free Pdf download www. Trigonometric Functions V. There are three primary ones that you need to understand completely: Sine (sin) Cosine (cos) Tangent (tan) The other three are not used as often and can be derived from the three primary functions. 6820 47° 6) sin C = 0. As we know that book Trigonometry (11th Edition) has many kinds or. 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The word “Trigonometry” comes from the Greek words: ‘ Trigonon ’ meaning ‘triangle’ and ‘ metron ’ meaning a ‘measure’. Sinusoidal function, harmonic motion, periodic functions, applications. The best thing to try is using trigonometric identities (see transc. Once you feel you mastered one type of problem you get stumped on the next. functions, identities and formulas, graphs: domain, range and transformations. 4cos2x + 4cosx —3 = 0, O < 9. 4KB 2019-10-25. sec x Matches (d). All books are in clear copy here, and all files are secure so don't worry about it. Solve Trig Problems With Double- or Half-Angles. This PDf download consists of the following… Trigonometry. Basic Trigonometry involves the ratios of the sides of right triangles. Pre-req for Math Analysis/Trigonometry Honors (For students entering Math Analysis/Trig Honors in 2019) Students entering Math Analysis/Trig Honors should make sure they are proficient with these topics before school begins on August 19, 2019. 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Algebra and Trigonometry with Analytic Geometry 13th Edition by Earl W. 010 Summer II 2019 Trigonometry and Analytic Geometry MCS 212 @10:00 -11:45 AM M - F J Montemayor Disclaimer This syllabus is current and accurate as of its posting date, but will not be updated. simple & compound interest, loan payments, store credit. Formulas Perfect Square Factoring: Difference of Squares: Difference and Sum of Cubes: B. Cole is the author of 'Algebra and Trigonometry with Analytic Geometry (College Algebra and Trigonometry)', published 2011 under ISBN 9780840068521 and ISBN 0840068522. UNIT 1: ANALYTICAL METHODS FOR ENGINEERS Unit code: A/601/1401 QCF Level: 4 Credit value: 15 OUTCOME 1 TUTORIAL 2 EXPONENTIAL, TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS Unit content 1 Be able to analyse and model engineering situations and solve problems using algebraic methods. All books are in clear copy here, and all files are secure so don't worry about it. Page 482 #'s 5-17 odd. 3 sec secxx x 10. 2 Law of Cosines. Simply download the PDF below and start putting in the answers into MyMathLab. Euler's formula states that for any real number x: = ⁡ + ⁡,. Chapter 7: Polynomial Equations and Factoring. 1) sin 75° 2) tan 105° Use the angle difference identity to find the exact value of each. This study guide provides practice questions for all 34 CLEP exams. 417721 0321304349 0. All cheat sheets, round-ups, quick reference cards, quick reference guides and quick reference sheets in one page. 2 Factoring Formulas A. Remembering the definitions 4 6. Trig Definition Math Help. There is no "sum of squares" formula, i. Students learn to determine angles and side lengths in 30-60-90 and 45-45-90 right triangles using the law of sines and the law of cosines, as well as how to identify similar triangles and determine proportions using proportionality. The / value at 95% was 1. Pedoe or D. Princeton Asia (Beijing) Consulting Co. 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After completing this course, students are expected to be able to solve problems showing all appropriate details in the following topics: 1. Recall that this means that Kis a commutative unitary ring equipped with a structure of vector space over k so that the multiplication law in Kis a bilinear map K K!K. All the exercise of Chapter 4 Inverse Trigonometric Functions RD Sharma Class 12 questions with Solutions to help you to revise complete Syllabus and Score More marks in JEE Mains, JEE Advanced, and Engineering entrance exams. students to think about where the triangle is applicable to AC circuit analysis, and not just to use it blindly. Trigonometry. Analytic geometry is a great invention of Descartes and Fermat. Use the hints provided. Use calculator to determine the value of sine, cosine or tangent. A NALYTIC TRIGONOMETRY is an extension of right triangle trigonometry. 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The / value at 95% was 1. pdf: File Size: 135 kb: File Type: pdf. 2tan2x— 10. How high is the kite (from the ground)? Basic Steps: 1) Draw a picture 2) Label the parts 3) Isolate the triangle 4) Solve 5) Answer the question. 1 The inverse sine, cosine, and tangent functions 1. Save up to 80% by choosing the eTextbook option for ISBN: 9781133037644, 113303764X. Download ANALYTIC TRIGONOMETRY WITH APPLICATIONS 11TH EDITION book pdf free download link or read online here in PDF. In ch 1 we first defined angles – our way of measuring them was based on a circle We then narrowed our focus to angles of a triangle and explored similarity of triangles Slideshow 2169034 by dannon. 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https://math.stackexchange.com/questions/2186634/how-to-draw-a-sublattice-to-exhibit-diagonalization/2186851

# How to draw a sublattice to exhibit diagonalization? Given the matrix: $$A=\begin{pmatrix} 3 & 1 \\ -1 & 2 \\ \end{pmatrix}.$$ Let $V =\mathbb Z^2$ and $L = AV$. We want to find basis for $V$ and $L$ and draw the sublattice that exhibit the diagonalization. I found the diagonal form of the matrix to be: $$\begin{pmatrix} 1 & 0 \\ 0 & -7 \\ \end{pmatrix}$$ I think we need that because the solution says the basis for $V$ is $\{(1,2) , (0,1)\}$ while the basis for $L$ is $\{(1,2),(0,7)\}$. But it says we can conclude that from the sublattice. So, can someone briefly point out how I can sketch the sublattice? EDIT: Also my attempt at drawing it: • Do you mean that $A$ is your name for the first matrix you display and $V$ is the module $\mathbb Z^2$? And how did you find that purported diagonalized matrix? It doesn't have the same determinant as the original one ($-7$ versus $7$), doesn't have the same trace ($-6$ versus $5$), and neither $1$ nor $-7$ are eigenvalues as far as I can see. – Henning Makholm Mar 14 '17 at 18:29 • @HenningMakholm I think the OP has correctly calculated the Smith normal form of the given matrix. Why should quantities such as the determinant, trace, or eigenvalues be preserved under this transformation? – André 3000 Mar 14 '17 at 20:19 • @Quasicoherent: Hmm, possibly. I was thrown off by the matrix being described as the "diagonalization" of the matrix. – Henning Makholm Mar 14 '17 at 20:22 • In my textbook they call it the "diagonal form" sorry about that! – Lana Mar 14 '17 at 20:23 • @HenningMakholm Ah okay, I see why that is confusing. Computing the Smith normal form corresponds to changing bases in both the domain and codomain of the linear map, so it's not like usual diagonalization where we choose the same basis in both the domain and codomain. – André 3000 Mar 14 '17 at 20:25 The row and column operations you use in computing the Smith normal correspond to invertible matrices $P,Q \in \operatorname{GL}_2(\mathbb{Z})$ such that $$D = PAQ$$ where $D$ is the diagonal matrix you found. (Since $-1$ is invertible in $\mathbb{Z}$, we can actually take $D$ to be $$D = \begin{pmatrix} 1 & 0\\ 0 & 7\end{pmatrix}$$ which is what I will use for the rest of the answer.) In this case, I get $$D = \left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right) \qquad P = \left(\begin{array}{rr} 0 & 1 \\ 1 & -4 \end{array}\right) \qquad Q =\left(\begin{array}{rr} 1 & 2 \\ 1 & 1 \end{array}\right) \, .$$ We can interpret these as change of basis matrices for $V$. (For more on this, see this post or this post.) The change of basis matrix $P$ contains the information we seek: since $P^{-1} = \begin{pmatrix} 4 & 1\\ 1 & 0\end{pmatrix}$, then the set $$\{v_1, v_2\} = \left\{\begin{pmatrix} 4\\ 1\end{pmatrix}, \begin{pmatrix} 1\\ 0\end{pmatrix}\right\}$$ is a basis for $V$ such that $L = v_1 \mathbb{Z} \oplus 7v_2 \mathbb{Z}$. As for drawing the sublattice, take a look at pp. 4-5 of this set of notes by Keith Conrad. He draws a lattice and sublattice with respect to so-called unaligned and aligned bases, which I've copied below. I think this is the sort of picture you have in mind. All right, below is my attempt at drawing the lattice $V$ and its sublattice $L$. The blue parallelograms are the fundamental parallelograms of $V$ and $L$ using my basis, and the red is the same for the book's. This shows quite clearly that both answers are correct. • It's {(1,2),(0,1)} that's a basis for v, no? – Lana Mar 14 '17 at 23:03 • That's not what I got. I'll edit to add a few more details. Could this be a result of you choosing $-7$ rather than $+7$ as the bottom-right entry of $D$? – André 3000 Mar 14 '17 at 23:08 • I just added the sample answer – Lana Mar 14 '17 at 23:10 • ok I tried to draw according to your reference lol.. not the prettiest drawing – Lana Mar 14 '17 at 23:17 • I have a feeling that both my answer and the book's are correct, meaning that the answer is not unique. You can see that $(4,1)^T$ and $(7,0)^T$ definitely generate $L$ since $(3,-1)^T = (7,0)^T - (4,1)^T$ and $(1,2)^T = 2 \cdot (4,1)^T - (7,0)^T$, so my answer works. – André 3000 Mar 14 '17 at 23:26

https://stonedynamicsco.com/what-does-oiypld/topological-sort-problems-b4c8cf

A topological sort takes a directed acyclic graph and produces a linear ordering of all its vertices such that if the graph $$G$$ contains an edge $$(v,w)$$ then the vertex $$v$$ comes before the vertex $$w$$ in the ordering. Also try practice problems to test & improve your skill level. if the graph is DAG. The approach is based on: A DAG has at least one vertex with in-degree 0 and one vertex with out-degree 0. There are 2 vertices with the least in-degree. Topological sorting of vertices of a Directed Acyclic Graph is an ordering of the vertices v 1, v 2,... v n in such a way, that if there is an edge directed towards vertex v j from vertex v i, then v i comes before v j. In the previous post, we discussed Topological Sorting and in this post, we are going to discuss two problems based on it. Topological sorting forms the basis of linear-time algorithms for finding the critical path of the project, a sequence of milestones and tasks that controls the length of the overall project schedule. For the given graph, following 2 different topological orderings are possible-, For the given graph, following 4 different topological orderings are possible-. Remove vertex-C since it has the least in-degree. 2/24. SPOJ TOPOSORT - Topological Sorting [difficulty: easy] UVA 10305 - Ordering Tasks [difficulty: easy] UVA 124 - Following Orders [difficulty: easy] UVA 200 - Rare Order [difficulty: easy] Every day he makes a list of things which need to be done and enumerates them from 1 to n. However, some things need to be done before others. We will first create the directed Graph and perform Topological Sort to it and at last we will return the vector which stores the result of Topological Sort.NOTE if Topological Sorting is not possible, return empty vector as it is mentioned in the problem statement.Let’s see the code. P and S must appear before R and Q in topological orderings as per the definition of topological sort. Topological Sort Examples. LeetCode 210 - Course Schedule II Problem : Alien Dictionary Topological Sort will be : b d a c and it will be our result.Let’s see the code. Topological Sort. Excerpt from The Algorithm Design Manual: Topological sorting arises as a natural subproblem in most algorithms on directed acyclic graphs. Your email address will not be published. Topological Sorting for a graph is not possible if the graph is not a DAG. Scheduling problems — problems that seek to order a sequence of tasks while preserving an order of precedence — can be solved by performing a topological sort … then ‘u’ comes before ‘v’ in the ordering. If you're thinking Makefile or just Program dependencies, you'd be absolutely correct. Topological sort Medium Accuracy: 40.0% Submissions: 42783 Points: 4 . Which of the following statements is true? • Question: when is this problem tractable? So, here graph is Directed because [0,1] is not equal to [1,0]. Directed acyclic graphs are used in many applications to indicate the precedence of events. In the previous post, we have seen how to print topological order of a graph using Depth First Search (DFS) algorithm. Remove vertex-3 and its associated edges. Note that for every directed edge u -> v, u comes before v in the ordering. It is quite obvious that character appearing first in the words has higher preference. It is easy to notice that this is exactly the problem of finding topological … The graph does not have any topological ordering. ... ordering of V such that for any edge (u, v), u comes before v in. It is a simple Topological Sort question. The Topological Sort Problem Let G = (V;E)be a directed acyclic graph (DAG). So, let’s start. Remove vertex-2 and its associated edges. Problem link— SPOJ TOPOSORT Topological Sorting /* Harun-or-Rashid CSEDU-23rd Batch */ Select Code #include using namespace std; #define white 0 … The graph is directed because … (The solution is explained in detail in the linked video lecture.). The goal of topological sortis to produce a topological order of G. COMP3506/7505, Uni of Queensland Topological Sort on a DAG. We wish to organize the tasks into a linear order that allows us to complete them one at a time without violating any prerequisites. Save my name, email, and website in this browser for the next time I comment. Each test case contains two lines. There may exist multiple different topological orderings for a given directed acyclic graph. Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge uv, vertex u comes before v in the ordering. Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge u v, vertex u comes before v in the ordering. Given a sorted dictionary of an alien language having N words and k starting alphabets of standard dictionary. Find any Topological Sorting of that Graph. There's actually a type of topological sorting which is used daily (or hourly) by most developers, albeit implicitly. Scheduling or grouping problems which have dependencies between items are good examples to the problems that can be solved with using this technique. Problem: Find a linear ordering of the vertices of $$V$$ such that for each edge $$(i,j) \in E$$, vertex $$i$$ is to the left of vertex $$j$$. For example, a … A typical Makefile looks like this: With this line we define which files depend on other files, or rather, we are defining in which topological orderthe files should be inspected to see if a rebuild … The main part of the above problem is to analyse it how approach the above problem step by step, how to reach to the point where we can understand that Topological Sort will give us the desired result.Now let’s see another problem to make concept more clear. There are a total of n courses you have to take, labeled from 0 to n-1.Some courses may have prerequisites, for example, to take course 0 you have to first take course 1, which is expressed as a pair [0,1].Given the total number of courses and a list of prerequisite pairs, return the ordering of courses you should take to finish all courses.There may be multiple correct orders, you just need to return one of them. no tags Sandro is a well organised person. The main function of the solution is topological_sort, which initializes DFS variables, launches DFS and receives the answer in the vector ans. Any of the two vertices may be taken first. Then in the next line are sorted space separated values of the alien dictionary.Output:For each test case in a new line output will be 1 if the order of string returned by the function is correct else 0 denoting incorrect string returned.Your Task:You don’t need to read input or print anything. Also go through detailed tutorials to improve your understanding to the topic. Remove vertex-2 since it has the least in-degree. I just finish the Course Schedule II problem and get a neat and simple answer. We learn how to find different possible topological orderings of a given graph. Topological Sort is a linear ordering of the vertices in such a way that, Topological Sorting is possible if and only if the graph is a. Topological Sort or Topological Sorting is a linear ordering of the vertices of a directed acyclic graph. Explanation: There are a total of 4 courses to take. It is a simple Topological Sort question. Review: Topological Sort Problems; Leetcode: Sort Items by Groups Respecting Dependencies Topological Sorting¶ To demonstrate that computer scientists can turn just about anything into a graph problem, let’s consider the difficult problem of stirring up a batch of pancakes. What does DFS Do? While the exact order of the items is unknown (i.e. Find any Topological Sorting of that Graph. From every words, we will find the mismatch and finally we will conclude the graph.Let’s take the another example to make things clear. Problems; tutorial; Topological Sorting; Status; Ranking; TOPOSORT - Topological Sorting. A topological sort takes a directed acyclic graph and produces a linear ordering of all its vertices such that if the graph $$G$$ contains an edge $$(v,w)$$ then the vertex $$v$$ comes before the vertex $$w$$ in the ordering. A topological ordering is possible if and only if the graph has no directed cycles, i.e. Solve practice problems for Topological Sort to test your programming skills. if there is an edge in the DAG going from vertex ‘u’ to vertex ‘v’. As a particular case of Alexandro spaces, it can be considered the nite topological spaces. Remove vertex-D since it has the least in-degree. PRACTICE PROBLEMS BASED ON TOPOLOGICAL SORT- Problem-01: Find the number of different topological orderings possible for the given graph- Solution- The topological orderings of the above graph are found in the following steps- Step-01: Write in-degree of each vertex- Step-02: His hobbies are Topological Sort pattern is very useful for finding a linear ordering of elements that have dependencies on each other. Your email address will not be published. Explanation: There are total of 2 courses to take. Input: The first line of input takes the number of test cases then T test cases follow . | page 1 Graph : b->d->a->cWe will start Topological Sort from 1st vertex (w). Then, update the in-degree of other vertices. We have discussed two problems on Graph and hope both problem and approach to them is clear to you. Input: The first line of input takes the number of test cases then T test cases follow . An acyclic graph always has a topological sort. Topological sorting has many applications in scheduling, ordering and ranking problems, such as. If you have doubts comment below.It’s Ok if you are not able to solve the problem, just learn the approach to tackle it. Topological Sort Topological sorting problem: given digraph G = (V, E) , find a linear ordering of vertices such that: for any edge (v, w) in E, v precedes w in the ordering A B C F D E A B E C D F Not a valid topological sort! See all topologicalsort problems: #topologicalsort. I think above code doesn’t require any explanation. no tags Sandro is a well organised person. Abhishek is currently pursuing CSE from Heritage Institute of Technology, Kolkata. Consider the following directed acyclic graph-, For this graph, following 4 different topological orderings are possible-, Few important applications of topological sort are-, Find the number of different topological orderings possible for the given graph-, The topological orderings of the above graph are found in the following steps-, There are two vertices with the least in-degree. Note : The input prerequisites is a graph represented by a list of edges, not adjacency matrices.You may assume that there are no duplicate edges in the input prerequisites. Remove vertex-3 since it has the least in-degree. Topological Sort¶ Assume that we need to schedule a series of tasks, such as classes or construction jobs, where we cannot start one task until after its prerequisites are completed. You have solved 0 / 6 problems. Topological Sorting Topological sorting or Topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge ( u v ) from … The first line of … For example, another topological … Remove vertex-D and its associated edges. Let’s draw the directed graph for the above example. For above example, order should be 0 1 2 3. Given a digraph , DFS traverses all ver-tices of and constructs a forest, together with a set of source vertices; and outputs two time unit arrays, . The first line of each test case contains two integers E and V representing no of edges and the number of vertices. Remove vertex-C and its associated edges. In the above example, From first two words, it is clear that t comes before f. (t->f)From 2nd and 3rd words, it is clear that w comes before e. (w->e)From 3rd and 4th words, it is also clear that r comes before t. (r->t)Lastly, from 4th and 5th words, it is clear that e comes before r. (e->r)From above observation, order will be (w->e->r->t->f).Hope you got the Logic, it is highly recommended to think of the approach to solve this problem before moving ahead. In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. For example, the pictorial representation of the topological order [7, 5, 3, 1, 4, 2, 0, 6] is:. And if the graph contains cycle then it does not form a topological sort, because no node of the cycle can appear before the other nodes of the cycle in the ordering. If it is impossible to finish all courses, return an empty array. • Output: is there a topological sort that falls in L? Using the DFS for cycle detection. Now, update the in-degree of other vertices. He has a great interest in Data Structures and Algorithms, C++, Language, Competitive Coding, Android Development. Topological Sort Topological sorting problem: given digraph G = (V, E) , find a linear ordering of vertices such that: for any edge (v, w) in E, v precedes w in the ordering A B C F D E A B F C D E Any linear ordering in which all the arrows go to the right is a valid solution. Learning new skills, Content Writing, Competitive Coding, Teaching contents to Beginners. Since, we had constructed the graph, now our job is to find the ordering and for that Topological Sort will help us.We already have the Graph, we will simply apply Topological Sort on it. So the correct course order is [0,1] . a b a b b a a b a b b a... not in L! 14.4.1. I make a simple version for this Course Schedule problem. For example, a topological sorting of the following graph is “5 4 … Subscribe to see which companies asked this question. Topological Sorting for a graph is not possible if the graph is not a DAG. Topological Sort | Topological Sort Examples. Watch video lectures by visiting our YouTube channel LearnVidFun. There are $n$ variables with unknown values. (d->a)From 3rd and 4th words, a comes before c. (a->c)Lastly, from 4th and 5th words, b comes before d. (b->d)From above observation order will be (b->d->a->c).Let’s see the code for constructing the Graph. Lecture 8: DFS and Topological Sort CLRS 22.3, 22.4 Outline of this Lecture Recalling Depth First Search. Here’s simple Program to implement Topological Sort Algorithm Example in C Programming Language. Implementation of Source Removal Algorithm. Objective : Find the order of characters in the alien language.Note: Many orders may be possible for a particular test case, thus you may return any valid order.Input:The first line of input contains an integer T denoting the no of test cases. Implementations. In a real-world scenario, topological sorting can be utilized to write proper assembly instructions for Lego toys, cars, and buildings. Practice Problems. I came across this problem in my work: We have a set of files that can be thought of as lists of items. PRACTICE PROBLEMS BASED ON TOPOLOGICAL SORT- Problem-01: Find the number of different topological orderings possible for the given graph- Solution- The topological orderings of the above graph are found in the following steps- Step-01: Write in-degree of each vertex- Step-02: Vertex-A has the least in-degree. For example, consider below graph: For some variables we know that one of them is less than the other. PRACTICE PROBLEMS BASED ON TOPOLOGICAL SORT- Problem-01: Find the number of different topological orderings possible for the given graph- Solution- The topological orderings of the above graph are found in the following steps- Step-01: Write in-degree of each vertex- … | page 1 Hope code is clear, we constructed the graph first and then did Topological Sort of that Graph, since graph is linear, Topological Sort will be unique. | page 1 To practice previous years GATE problems on Topological Sort. Comment: Personally, I prefer DFS solution as it's more straightforward for this problem # Topological Sort | Time: O(m * n) | Space: O(m * n) # m: height of matrix # n: width of matrix # Slower than the DFS approach due to graph and indegree construction. Also since, graph is linear order will be unique.Let’s see a example. graph can contain many topological sorts. Query [2, 0] means course 0 should be completed before course 2, so (0 -> 2) represents course 0 first, then course 2.Now, our objective is to return the ordering of courses one should take to finish all courses. So, following 2 cases are possible-. 2.Initialize a queue with indegree zero vertices. We will be discussing other applications of Graph in next post.That’s it folks..!!! The given graph is a directed acyclic graph. | page 1 Input. So, remove vertex-B and its associated edges. ?Since, problem statement has some ‘n’ keys along with some sets which are dependent on each other, clearly it is a Graph Question.In the above problem, n represent the number of vertices and prerequisites define the edges between them.Also, [0,1] means course 1 should be completed first, then course 0. Topological Sorting for a graph is not possible if the graph is not a DAG. Problems. R. Rao, CSE 326 6 Step 1: Identify vertices that have no incoming edge •The “ in-degree” of these vertices is zero A B C F D E Also go through detailed tutorials to improve your understanding to the topic. To take course 1 you should have finished course 0. the ... – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow.com - id: 275642-ZDc1Z Let’s take an example. Running time is about 264ms. To take course 3 you should have finished both courses 1 and 2. Now, the above two cases are continued separately in the similar manner. Topological Sort Algorithms. Another correct ordering is [0,2,1,3]. A common problem in which topological sorting occurs is the following. # - Note that we could have not built graph and explore the neighbors according to matrix in topological sort proccess. So, remove vertex-1 and its associated edges. So, remove vertex-A and its associated edges. The recipe is really quite simple: 1 egg, 1 cup of pancake mix, 1 tablespoon oil, and $$3 \over 4$$ cup of milk. Directed acyclic graphs are used in many applications to indicate the precedence of events. right with no problems) . (b->a)From 2nd and 3rd words, d comes before a. Also go through detailed tutorials to improve your understanding to the topic. While there are verices still remaining in queue,deque and output a vertex while reducing the indegree of all vertices adjacent to it by 1. Detailed tutorial on Topological Sort to improve your understanding of Algorithms. The Problem Statement is very much clear and I don’t think any further explanation is required.NOTE that prerequisite [0,1] means course 1 should be completed first, then course 0.How to approach the problem..? You have to check whether these constraints are contradictory, and if not, output the variables in ascending order (if several answers are possible, output any of them). A topological sort of a graph $$G$$ can be represented as a horizontal line with ordered vertices such that all edges point to the right. Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge uv, vertex u comes before v in the ordering. The time-stamp structure. Given a Directed Graph. An Example. We are all familiar with English Dictionary and we all know that dictionary follows alphabetical order (A-Z).Alien Dictionary also follows a particular alphabetical order, we just need to find the order with the help of given words.There are some N words given which uses k letters, we have to find the order of characters in the alien language.Hope, problem statement is clear to you, please refer to above examples for more clarity. In this task you have to find out whether Sandro can solve all his duties and if so, print the correct order. 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Circumference C= Diameter Area Radius π Learn the formula for circumference: πd Learn the formula for the area of a circle: A= πr 2 Learn the value of pi correct to d. Calculate the circumference of a circle. Calculate the area of the square, and divide that by the surface area of the facing-out side of a brick. How to Use the Material Needed Calculator (Circle) This calculator is easy to use. Paper 4 (Calculator) Circle Theorems Past Paper Questions Arranged by Topic Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. If the perimeter of the semi-circle is needed, add together the lengths of one-half of the circumference + diameter. 14159) and diameter is the distance across the circle (along the edge of the half circle). In the result you will get all unknown variables presented. When you substitute the values into the formula, you obtain: A=πr 2 /2 A = (3. Set up each circle's circumference to its diameter. Thus the surface area is: SA = 2(pR 2)+ 2pRH. perimeter (circumference) of a semicircle: pi times radius. A connected section of the circumference of a circle. Suppose we want to start from 45 degrees. Online Circumference Memory Game. A Trapezium is a four-sided polygon with two non-adjacent parallel sides or one set of parallel sides. Important Questions for Class 10 Maths Chapter 12 Areas Related to Circles Areas Related to Circles Class 10 Important Questions Very Short Answer (1 Mark) Question 1. Find A, C, r and d of a circle. Calculate the total square inches in column F by using =E2*D2. out Finds circumference of circle ----- Enter radius: 8 Circumference of Circle: 50. Paper 4 (Calculator) Circle Theorems Past Paper Questions Arranged by Topic Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Students learn about finding the circumference of a circle as a multiple of π and as a decimal correct to 3 significant figures. A circle has a circumference of ìcm. (a) Calculate the area of the whole target. Then tap or click the Calculate button. A = \frac { 1 }{ 2 } × π × { r }^{ 2 } where π = \frac { 22 }{ 7 } (or) we can also take π as 3. Circumference of a circle is defined as the distance around it. Then, do the following: Measure the diameter of each circle by folding in half. Once you have the radius, you can calculate the circumference with 2 x Pi x R. Yes, the area of this rectangle would be π R 2. Instructions to Candidates. The result is A=5/4*Pi*a². Online Circumference Memory Game. Find (a) the circumference of a circle with radius 6 centimeters and (b) the radius of a circle with circumference 31 meters. Write the formula that you will use (e. ©P j260 r1w2 d 8K fukt 5a8 rS Moof qtcwxaJr1eI fLELOCD. 70 So, the circumference is about 37. It is calculated just by multiplying the diameter of the circle with π value. When a circle is divided into two equal halves, it forms two semicircles. !Shown below is a compound shape made from a rectangle and semi-circle. An Arc is a part of the circumference of a circle. The laundry basket that Mrs. I can calculate the volume of cylinders. o Calculate the circumference of a circle. • The measure of a semicircle, or an arc that is equal to half of a circle, is 180º. A Quonset roof is typically a half-round shape of any length that can support or accommodate a cathedral or domed interior ceiling structure. 14 and “r” is the radius of the given semi circle. Draw a picture to represent the window. Find the circumference of a circle which has a radius of 4 cm. What is the radius of the cup? Round your answer to a suitable degree of accuracy. Where the perpendicular bisectors of the chords intersect is the expected center of the concentric circles forming the original path. G) Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Find the magnitude of the electric field at the point P -- the center of the semicircle. Solution : Here r = 4. The radius is: r=D/2 r=30. We'll use 3. I can calculate the volume of cylinders. Circumference of a semi-circle = = πr and the perimeter of a semi-circular shape = (π + 2) r units. A sector of a circle) is made by drawing two lines from the centre of the circle to the circumference. Calculate the exact area of the shaded sector. Because the semicircle is only half of this divide your answer by 2. lcm 2Tt x 4cm 9. Here we are writing a simple C program that calculates the area and circumference of circle based on the radius value provided by user. How to find the area of a circle: The area of a circle can be found by multiplying pi ( π = 3. on your calculator as an approximation for 𝜋𝜋. TOPIC: Circumference, measurement, advanced calculations INTERACTIVITY: Participants will measure the radius of the circumference of the rainbow (semicircle). Answer: The shaded area is 314. Let me ask you a question: How do you calculate the circumference of a circle? As you know, the formula for the circumference of a circle is: C = π · d or C = 2 · π · r The circumference is equal to π times the length of the diameter, or it is equ. Follow these simple instructions: 1. Then tap or click the Calculate button. If you know the length of the radius, you can calculate its perimeter using the following formula:. An easy to use, free perimeter calculator you can use to calculate the perimeter of shapes like square, rectangle, triangle, circle, parallelogram, trapezoid, ellipse, and sector of a circle. Circumference C= Diameter Area Radius π Learn the formula for circumference: πd Learn the formula for the area of a circle: A= πr 2 Learn the value of pi correct to d. The radius of a circle is 1. We have step-by-step solutions for your textbooks written by Bartleby experts!. [5] A rubber gasket forms a seal between the glass and the window frame. Equation of a Circle Through Three Points Calculator. Students can play by themselves or with a partner. Online Circle Tool. And in this case, r is equal to 6. Step-by-step explanation: Here we note that the shape consists of two small circles and one larger circle. A segment is the shape formed between the chord and the arc. You can use numerous different inputs and choose to submit the radius, the diameter or the circumference of the circle. The radius of a circle is any of the line segments from its center to its perimeter. In this article, we will consider a geometric figure that does not involve line segments, but is instead curved: the circle. The total height of the window is 90cm and the total width is 60cm. Then work out 37. Give your answers to 2 d. In this equation, "C" represents the circumference of the circle, and "d" represents its diameter. The only basic figure is the semi-circle, that appears four times. Calculate the diameter of the. This circumference to diameter calculator is used to find the diameter of a circle given its circumference. Thus the surface area is: SA = 2(pR 2)+ 2pRH. TOPIC: Circumference, measurement, advanced calculations INTERACTIVITY: Participants will measure the radius of the circumference of the rainbow (semicircle). For example, if the circumference in this example equals 11 inches, multiply 11 by 0. Perimeter Of A Circle With Examples Example 1: If the perimeter of a semi-circular protractor is 66 cm, find the diameter of the protractor (Take π = 22/7). If the handle is 28 cm long, what is the diameter of the pot? Give your answer correct to 3 significant figures. 14 or 22 — 7 for 𝛑. Program to calculate Area and Circumference based on user input. Smith can determine where to place it. • Write the formula: Area = πr2 • Substitute the words with the measurements you have been given. A circle has a radius of 𝑟 cm and a circumference of cm. The ratio circumference. Textbook solution for Single Variable Calculus: Early Transcendentals 8th Edition James Stewart Chapter 3. The perimeter of a semicircle is half of the circumference plus the. 0 2cm 0 4cm. It is in circle geometry that the concepts of congruence and similarity. Explain in words how to determine the perimeter of a semicircle. What is 282. Perimeter of a Semicircle Formulas & Calculator. The notation for semicircle and major arc is similar to that of minor arc. A major arc is an arc that is larger than a semicircle. Therefore, all we need to do is add the measurement of each side of the shape and we will have the perimeter. The perimeter (our constraint) is the lengths of the three sides on the rectangular portion plus half the circumference of a circle of radius $$r$$. Circumference (C): The distance the edge of a circle Radius and Diameter Circumference d: ltd 2Ttr Ex 1) Find the circumference of the circle above. They will also have to move the leprechaun around the rainbow throughout the activity. 14 r is the radius of the semicircle 2. We can find the perimeter of a semicircle with the help of this below formula: where, R = Radius of the semicircle Use our below online perimeter of a semicircle calculator to find the perimeter. ©P j260 r1w2 d 8K fukt 5a8 rS Moof qtcwxaJr1eI fLELOCD. Next we calculate the circumference of the full circle. Circumference is equal to 2 pi r. The centroid of Semicircle $$(0, \frac{4r}{3\pi})$$ Circumference of Semicircle Formulas $$P = \pi r$$ Summary of Semicircle Formula. Its unit length is a portion of the circumference. (3) diagram not to scale (b) Find AOB , giving your answer in radians. Add them to the table above, measure the circumference and diameter for each, and then compute the ratio Circumference Diameter for each object. Find three other circular objects. Our circumference calculator provides circumference of a circle by entering its radius. It depends on what characteristics of the semicircle you are given. Give your answers to 2 d. The surface of the pond is a semicircle of radius 1. Write a formula that expresses the value of in terms of 𝑟 and 𝜋. The width of each small ring is 3 centimetres. Our area of a sector calculator is flexible and reliable. Give your answer to a sensible degree of accuracy. The region bounded by the circle is also often referred to as a circle (as in when we speak of the area of a circle), and the curve is referred to as the perimeter or the circumference of the circle. An arc is a part of the circumference. So it's equal to 2 pi times 6, which is going to be equal to 12pi. This is a great start. These slightly more advanced circle worksheets require students to calculate area or circumference from different measurements of a circle. The missing length is 13. Andlearning. The earlier worksheets in this section require calculating the area and the circumference given either a radius or a diameter. • Volume: The volume of a three-dimensional object is a measure of the total space it occupies, measured in cubic units. How to find the area of a circle: The area of a circle can be found by multiplying pi ( π = 3. c) The answer of 16. You are not making allowances for the space needed for mortar, if any, and you are including bricks for the spaces at the corners of the box. We have step-by-step solutions for your textbooks written by Bartleby experts!. Find the circumference of the fl ying disc. Thus the surface area is: SA = 2(pR 2)+ 2pRH. Click here to learn about how to calculate the circumference of a circle. Calculate Areas & Properties of Annulus Rings Ring shapes are essentially formed by one circle within another. MEMORY METER. Examples of Circle and Semi-circle functions We look at a number of examples of circle and semi-circle functions, sketch their graphs, work out their domains and ranges, determine the centre and radius of a circle given its function, etc. The radius for this circle is 45. Another fact that you need to keep in mind about inscribed angles is the fact that any inscribed angle that intercepts a semicircle has to be a right angle. If d=2r where r is the radius, then C=2πr. 3 Problem 56E. C 5pd or C 52pr 26. This figure consists of a rectangle and semicircle. The length of the circumference is given by the formula: C = πd, where d is the diameter of the circle. The distance around a rectangle or a square is as you might remember called the perimeter. Assume that the radius of a circle is 21 cm. Clearly stating the units of your answers, calculate do Co(reck the circumference of his £2 coin, giving your answer to an appropriate degree of accuracy. Tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle. And in this case, r is equal to 6. A chord is a straight line joining any two parts of the circumference. If the handle is 28 cm long, what is the diameter of the pot? Give your answer correct to 3 significant figures. To find the perimeter, P, of a semicircle, you need half of the circle's circumference, plus the semicircle's diameter: P = 1 2 ( 2 π r ) + d The 1 2 and 2 cancel each other out, so you can simplify to get this perimeter of a semicircle formula. Calculate the area of one surface of the table mat. o Pi (π) Objectives. Area of a Semicircle In the case of a circle, the formula for area, A, is A = pi * r^2, where r is the circle’s radius. If someone could please help with the following question? What is the formula to calculate the distance perpendicular from the section line to the nearest edge of the circumference at any point X along the section line? Thanks in advance. Step 1) Write the formula Step 2) Substitute what you know Step 3) Calculate. A sector of a circle) is made by drawing two lines from the centre of the circle to the circumference. 1 foot ≈ 0. Its unit length is a portion of the circumference. The area of a circle is. This is the perimeter of the semi-circle!. perimeter (circumference) of a semicircle: pi times radius. A segment is the shape formed between the chord and the arc. A major arc is an arc that is larger than a semicircle. 14(15) Replace d with 15. WIIat is the perimeter of the figure shown below (semicircle attached to rectangle)? Give the exact answer. Find the circumference of the quad-rant with radius 4. Shapes and Figures. Finally, you can find the diameter - it is simply double the radius: D = 2 * R = 2 * 14 = 28 cm. Please find teh circumference of the basket so Mrs. !Shown below is a compound shape made from a rectangle and semi-circle. When the area is fixed and the perimeter is a minimum, or when the perimeter is fixed and the area is a maximum, use Lagrange multipliers to verify that the length of the rectangle is twice its height. The arc that remains as a part of the semicircle is half as long as the circumference of the whole circle. Write a formula that expresses the value of 𝐶𝐶 in terms of 𝑟𝑟 and 𝜋𝜋. Title: Print Layout - Mathster Created Date: 20140105104657Z. A connected section of the circumference of a circle. This method yields a ceiling estimate of the number of bricks you will need. First, calculate the circumference of the whole circle. 14 (2 cm); C = 6. Find its arc length A. (Answer in units of N/C. That’s because a semi circle has an arc length that is exactly 1/2 of the circumference of a complete circle. A semicircle is an arc that is “half” of a circle. If we know the diameter then we can calculate the area of a circle using formula: A=π/4*D² (D is the diameter). Draw a diagram of a circle and label all given information 2. G7-M3-Lesson 18: More Problems on Area and Circumference 1. Floor Area Area of the floor. You can also select units of measure for both input data and results. Use this calculation to calculate the area of a circle. 8 11 14 4 7 6 Now that we have all the lengths of the sides, we can simply calculate the perimeter by adding the lengths together to get 4+14+11+8+7+6=50. The perimeter of the semicircle also includes the straight line that connects the two ends of that arc. C = πd Circumference of a circle C = 3. • Circumference: The circumference of a circle is the distance around the outside of a circle and is denoted. 7 m and 9557 m2) [2+3](Curved S. out Finds circumference of circle ----- Enter radius: 8 Circumference of Circle: 50. WIIat is the perimeter of the figure shown below (semicircle attached to rectangle)? Give the exact answer. To calculate area and circumference we must know the radius of circle. 24 Circumference of a Semi-circle formula Circumference of a Semi-circle = π * radius C Program to find the circumference of a semi-circle. DCO is a straight line. The radius of a circle is 1. What is 282. This is also known as the longest chord of the circle. ) cm² (to. Radius: r Diameter: d Circumference: C Area: K : d = 2r C = 2 Pi r = Pi d K = Pi r 2 = Pi d 2 /4 C = 2 sqrt(Pi K) K = C 2 /4 Pi = Cr/2 To read about circles, visit The Geometry Center. We'll use 3. The calculations are done "live": How to Calculate the Area. Program to calculate Area and Circumference based on user input. Find the diameter. Leave your answer in terms of π 1 Area = ∏ x 11 x 11 = 121∏ cm2 Diameter of circle = 22 cm Circumference = 22∏ cm Circumference = 6π m Area = 32 π m2 = 9 π m2 Circle = π x 52 = 25π Semi-circle = (π x 22) / 2 = 4. If the area of a circle is equal to sum of the areas of two circles of diameters 10 cm and 24 cm, calculate the diameter of the […]. A sector is the part of a circle between two radii. Contrast radius and diameter. Show Answer. The circle has a diameter of 30 cm. If you have the radius instead of the diameter, multiply it by 2 to get the diameter. Magnetic field around a circular wire is calculated by the formula; B=2πk. [5] A rubber gasket forms a seal between the glass and the window frame. Yes, the area of this rectangle would be π R 2. For interactive applets, worksheets, and more videos, please visit http://www. Do NOT confuse a 'sector' with a 'segment'. Input: Store the constant pI in M1: M1 = 3. asked by tom on February 7, 2014; Math. As a semicircle is half of a circle, the length of the arc of the semicircle will be 𝜋𝑑 divided by two. That is, we want -90 + a / total_semi_circles = -45, so a / total_semi_circles = 45. 7cm 5cm Ttx8 = 25. lcm 2Tt x 4cm 9. You can get your required area of the section by just putting radius value and angle. Using “ ” to represent the diameter of the circle, write an algebraic expression that will result in the perimeter of a. Find the diameter or radius of a circle using the formulas: C = πd; C = 2πr. Give your answer to a sensible degree of accuracy. Find three other circular objects. For a complete analysis of a circular arc please check out The Complete Circular Arc Calculator. Circumference of a circle 1 Work out the circumference of each circle. Use this calculation to calculate the area of a circle. A line that is drawn straight through the midpoint of a circle and that has its end points on the circle border is called the diameter (d). Ex2) 14 in. The figure below is in the shape of a semicircle. Ellipse In geometry, an ellipse is a regular oval shape, traced by a point moving in a plane so that the sum of its distances from two other points (the foci) is constant, or resulting when a cone is cut by an oblique plane that does not. ! C=2"r) 3. How to Use the Material Needed Calculator (Circle) This calculator is easy to use. Need to subtract a semicircle? BC. Write a formula that expresses the value of in terms of 𝑟 and 𝜋. Area of a Circle Calculator. If you need an anticipatory set you might want to use this online circumference memory game. In order to find the area of semi circle use the below given formula. Enter the radius: 1 The area of circle is: 3. (AQA June 2007 Intermediate Paper 1 NO Calculator) A semi-circle is cut from a circle. Define tangent. Its length is always less than half of the circumference. Note that measuring circumference requires the use of 𝜋, an irrational number customarily approximated as 3. 2) Chemistry periodic calculator. Unit 2: Two-Dimensional Geometry. The semi-circle has a diameter of 20 cm. : We know the diameter of the semi circles = the width of the rectangle, y Therefore the circumference of the semi circles = pi*y The total perimeter divide both sides by 2: total area = 2 semicircle area + rectangular potion area A = Replace x A = is the area as a function of y:. Calculate Perimeter Of a Square Calculate Perimeter Of a Rectangle Calculate Perimeter Of a Triangle. The ratio of the circumference C to diameter d of both circles simplify to the same value, 3. Calculate E at center of semicircle: A uniformly charged insulating rod of length L = 12 cm is bent into the shape of a semicircle as shown. Arcs are measured in two ways: as the measure of the central angle , or as the length of the arc itself. I can calculate the radius and diameter of a circle when I know the Circumference. Area and Circumference Worksheets. The perimeter formula is one of the easier formulas to remember in math! The perimeter of a shape is the distance around the outside of the shape. 1416… Browse more Topics Under Areas Related To Circles. In the result you will get all unknown variables presented. Step 1) Write the formula Step 2) Substitute what you know Step 3) Calculate. asked by tom on February 7, 2014; Math. 19 8 cm 5 cm 18 A wheel of radius 20 cm rolls without slipping on level ground. I suspect you may be visiting this page to calculate the area of an arc, or piece of a circle, because semi circle can sometimes be considered ambiguous. An arc is a segment of a 6 cm Chapter 50 cm A semicircle is of a whole circle. Circumference calculator is a free tool used to calculate the circumference of a circle when the radius is given. Circumference is the linear distance around the circle edge. Solve for a missing angle in a tangent problem. The perimeter (our constraint) is the lengths of the three sides on the rectangular portion plus half the circumference of a circle of radius $$r$$. If r is the radius of a circle, then (i) Circumference = 2πr or πd, where d = 2r is the diameter of the circle. The other two sides should meet at a vertex somewhere on the circumference. Perform the following steps: 1. Solutions for the assessment Circles, Perimeters and Sectors 1) Area = 56. The radius of this circle is 8 cm. For example, if the circumference in this example equals 11 inches, multiply 11 by 0. The diameter of a circle is known as the straight line segment which passes through the center of the circle. • Volume: The volume of a three-dimensional object is a measure of the total space it occupies, measured in cubic units. Area of a Circle Calculator. Revision exercise 3. The distance around a circle is called the circumference. A semicircle is an arc that is “half” of a circle. You can also select units of measure for both input data and results. your calculator as an approximation for 𝜋. Find the perimeter of the shape. The area of a semicircle of radius r is given by A = int_0^rint_(-sqrt(r^2-x^2))^(sqrt(r^2-x^2))dxdy (1) = 2int_0^rsqrt(r^2-x^2)dx (2) = 1/2pir^2. She wants to move the basket to another place in her classroom. The circumference of the traffi c circle is approximately 16. If you know the length of the radius, you can calculate its perimeter using the following formula:. The diameter of a circle is known as the straight line segment which passes through the center of the circle. A line that is drawn straight through the midpoint of a circle and that has its end points on the circle border is called the diameter (d). A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass. When a circle is divided into two equal halves, it forms two semicircles. You are given a semicircle of radius 1 ( see the picture on the left ). Circumference Word Problems For each problem, you are expected to complete all of the following: 1. C 5pd or C 52pr 26. The circumference is the outside edge of the circle. 3 divide circumference in inches by the width of the wallblocks you want to use. The handle of a paint pot is half the circumference of the pot (a semi–circle). Semicircle Radius using Circumference Calculation. Draw the diagram (as above on right) to understand the problem and arrive at the required answer. If you want to calculate the outer boundary of a semicircle - the circumference or perimeter of a semicircle - you need to be careful to not fall into any traps. A semicircle is exactly half a circle. your calculator as an approximation for 𝜋. The POWER function will take any number and raise it to the power of any other number. Semicircle is half of a circle. 14159, which is equal to the ratio of the circumference of any circle to its diameter. Calculate E at center of semicircle: A uniformly charged insulating rod of length L = 12 cm is bent into the shape of a semicircle as shown. Perimeter Of Sector calculator provides for the same. Find the circumference of the circles below. • The measure of a semicircle, or an arc that is equal to half of a circle, is 180º. Then r r r is a constant, and its derivative is 0 0. Work out the circumference of the circle. the areas of two circles are in a ratio 4:9 find the ratio between their circumference AB is a line segment whose midpoint is M. Need to subtract a semicircle? BC. Leave your answer in terms of π 1 Area = ∏ x 11 x 11 = 121∏ cm2 Diameter of circle = 22 cm Circumference = 22∏ cm Circumference = 6π m Area = 32 π m2 = 9 π m2 Circle = π x 52 = 25π Semi-circle = (π x 22) / 2 = 4. The figure shows a circle within a square. Circumference of a circle (C) = 2πr, where π is pi and “r” is radius. 56 This is the length of just the curvy part of the semi-circle. Just enter the measurement you know. 1% Convert the percentages from a pie chart to central angle measures?. The result is A=5/4*Pi*a². Perimeter and Area of Circle and Semicircle; Arcs of a Circle. In the result you will get all unknown variables presented. The area of a circle will be shown in the selected units. Another circle is inscribed in the semi–circle so that it touches the diameter at point A and also touches the circumference of the semi– circle. The area of a semi-circle is 1/2 πr 2. The rectangle has a width of 15 meters and a height of 12 meters. Click here to learn about how to calculate the circumference of a circle. 3 divide circumference in inches by the width of the wallblocks you want to use. 14 r is the radius of the semicircle 2. Circumference = 2 × π × radius = 2 × 3 × 21 [Using 3 as an estimated value for π]. The perimeter of the semicircle also includes the straight line that connects the two ends of that arc. Area and Circumference of a Circle (Grades 6-7) Given a Slope and Points, calculate the missing coordinate Determine if the Points are Collinear. For example, if the circumference in this example equals 11 inches, multiply 11 by 0. Its unit length is a portion of the circumference. Answer by noguf (162) (pi)x(diameter)/2 + diameter. A semicircle is an arc that is “half” of a circle. Circumference of a semi-circle = = πr and the perimeter of a semi-circular shape = (π + 2) r units. More About Arc. Circumference to Diameter Calculator. It can also be thought of as a sector with an angle of 180 degrees. Step 1: say the radius of the field is r, then the total area of the field = (r 2)/2. In the article below, we provide the semicircle. Degree measure of a minor arc: Defined as the same as the measure of its corresponding central angle. Perimeter of a triangle calculation using all different rules: SSS, ASA, SAS, SSA, etc. The circumference of a circle is found using the circumference formula: Circumference = 2 * π * radius So, all you have to do is plug in the radius into the formula, and approximate π to be 3. Contrast radius and diameter. Perimeter Formulas and Circumference of a Circle Formula. 28 cm 2 The circumference of a bicycle wheel is 50. The ratio circumference. Determine the length of this semicircle, as measured along the curve. A compass is an instrument used to draw circles or the parts of circles called arcs. You can click on the link about the calculator to learn more about the area of semi circles and it’s relationship to arc length. The circumference of a circle is an arc measuring 360 o. From geometry, you know that "A(r) = πr 2" indicates the area of a circle given in terms of the value of the radius r, while "C(r) = 2πr" indicates the circumference given in terms of the radius r. 2 cm 2 Work out the circumference of each circle. This math worksheet was created on 2010-03-06 and has been viewed 81 times this week and 53 times this month. The notation for semicircle and major arc is similar to that of minor arc. The only basic figure is the semi-circle, that appears four times. Welcome to The Circumference and Area of Circles (A) Math Worksheet from the Measurement Worksheets Page at Math-Drills. Work out the circumference of the circle. the way around a circle. Calculate the total square inches in column F by using =E2*D2. Area and Circumference of a Circle (Grades 6-7) Given a Slope and Points, calculate the missing coordinate Determine if the Points are Collinear. Online calculators and formulas for a hemisphere and other geometry problems. 7%) A: 4 (14. The radius is: r=D/2 r=30. We can substitute 7 for r in the formula C = 2πr to end up with C = 2π(7) = 14π ≈ 44 inches. It progresses nicely and finishes with some challenging problems at the end borrowed from MEP. Calculate the area of a semicircle of radius 1. The perimeter of a circle is known as its circumference. Click on "Calculate" button to get results instantly. The circumference of a figure is the sum of all the side lengths. A compass is an instrument used to draw circles or the parts of circles called arcs. 14159) and diameter is the distance across the circle (along the edge of the half circle). A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. Multiply the portion of the circumference the minor arc comprises by the circumference of the circle to calculate the length of the minor arc. It is calculated just by multiplying the diameter of the circle with π value. The circumference of a circle of a radius 𝑟𝑟 is 2𝜋𝜋. Calculate the circumference of the circle. 14 * radius. You can find out more about Pi here. The formula is:. Point A lies on the diameter of a semi–circle with the radius 8 cm and the center O. the semi circle touches the rectangle at a b and c. To calculate the area of a semicircle we use the formula: Area of a semicircle = π × (radius)2 2 A = πr2 2 Calculate the area of this semicircle. I can calculate the radius and diameter of a circle when I know the Circumference. To calculate the perimeter of Shape 2 add the length of the three sides of the rectangle and the circumference of the semi-circle. 7cm 5cm Ttx8 = 25. base area × height Step 2: Total area = Area of circle + Area of rectangle Mensuration. Arc Measure. Contrast radius and diameter. ) Homework Equations E = K((Q)/(r^2)). Plugging our radius of 3 into the formula, we get C = 6π meters or approximately 18. For example, if the circumference in this example equals 11 inches, multiply 11 by 0. A B C AB = 1. These slightly more advanced circle worksheets require students to calculate area or circumference from different measurements of a circle. 06 = Calculate the length of the circumference of each circle, rounding your answer to one decimal place: 62. Construct the Gnomon to the calculated dimensions and stand vertically along the centre line. A circle is drawn with radius 'r' touching all the three semi-circles. where r is the radius of the circle and π is the ratio of a circle’s circumference to its diameter. This online calculator will find and plot the equation of the circle that passes through three given points. We have step-by-step solutions for your textbooks written by Bartleby experts!. Formulas, explanations, and graphs for each calculation. They have to find the circumference of 6 different circles and then play a memory matching game. Find the perimeter of the shape. Find the circumference of a circle which has a radius of 4 cm. This gives us the lengths of all the sides as shown in the figure below. We can substitute 7 for r in the formula C = 2πr to end up with C = 2π(7) = 14π ≈ 44 inches. Formulas, explanations, and graphs for each calculation. thx [3] 2020/04/05 02:53 Male / 40 years old level / An engineer / Useful / Purpose of use. The distance around a rectangle or a square is as you might remember called the perimeter. Round to the nearest hundredth for each circumference. These slightly more advanced circle worksheets require students to calculate area or circumference from different measurements of a circle. Calculate ‘d’ and the total area inside the track. What is the arc-length of a semi-circle with radius of 6 cm? (to 2 decimal places) A car wheel has a circumference of 1. It progresses nicely and finishes with some challenging problems at the end borrowed from MEP. Step 1) Write the formula Step 2) Substitute what you know Step 3) Calculate. You can calculate the circumference from total circle angle 360. For example, if the circumference in this example equals 11 inches, multiply 11 by 0. In order to find the area of semi circle use the below given formula. To calculate area and circumference we must know the radius of circle. 2 Calculate the length of the circumference of each circle, rounding your answers to one decimal place: 10 = 31. Do NOT confuse a 'sector' with a 'segment'. Calculate the circumference and the area of the circle. Consider the diagram of a semicircle shown. 69 km 0 266. If you have the radius instead of the diameter, multiply it by 2 to get the diameter. The diagram shows a trapezium with two identical semi-circles. To establish a major arc as a semicircle some additional notation would be required. Step 2 Since you only need half of the. I can use the information I know to calculate the length of the radius of the. What is the radius of the cup? Round your answer to a suitable degree of accuracy. Formula Used: Radius of a Semicircle, r = Diameter (d) / 2 or r = Circumference / Pi Where, Pi is a constant = 3. If a circle has a circumference of 8. Get the result. As learning progresses students are challenged to find the perimeter of semi-circles and sectors. Calculate the exact area of the shaded sector. Step 1: say the radius of the field is r, then the total area of the field = (r 2)/2. What are the circumference and area of the circle? 2. Well, that and the formula. The length of the rectangle is 15 cm. A circle has a circumference of ìcm. Step-by-step explanation: Here we note that the shape consists of two small circles and one larger circle. - calculate the percentage in each category - calculate the central angle for each category for a circle graph - create and label a circle graph using your data Student Grades c: o: 2 c: 10 135. In a similar manner, we can calculate the length of the other missing side using 14−8=6. Remember that the perimeter is the distance round the outside. Do NOT confuse a 'sector' with a 'segment'. Input: Store the constant pI in M1: M1 = 3. In our semicircle, the diameter is 35 centimetres. In the result you will get all unknown variables presented. Common Core Standard: 7. Welcome to The Circumference and Area of Circles (A) Math Worksheet from the Measurement Worksheets Page at Math-Drills. For each swimming pool below, calculate: the perimeter of the swimming pool. asked by Bunny head on December 2, 2018; Math. Draw a picture to represent the window. Write the formula that you will use (e. This means that in any circle, there are 2 PI radians. (3 marks) _____ A cylinder has a radius of 5 cm. Formula: Area = 3. The sum of the central angles in any circle is 360°. The diagram shows a 400-m running track. The top part of the door is a semicircle. By that logic, the arc length of a quarter circle is exactly 1/4 of the circumference of a circle. If a circle has a circumference of 8. Arcs are measured in two ways: as the measure of the central angle , or as the length of the arc itself. So that's straightforward, area 36pi, we leverage pi r squared to figure out that the radius was 6, and then from that we were able to figure out that the circumference was 12pi. 7 cm Length of Y 3. Find the circumference of the object. You can also select units of measure for both input data and results. What is the arc-length of a semi-circle with radius of 6 cm? (to 2 decimal places) A car wheel has a circumference of 1. show your working. Angle in a semicircle We want to prove that the angle subtended at the circumference by a semicircle is a right angle. Program to calculate Area and Circumference based on user input. Perimeter and the area of a semicircle Perimeter of a semicircle. Set up each circle's circumference to its diameter. 14) by the square of the radius; If a circle has a radius of 4, its area is 3. You know circles are round. The widest part of a tea cup has a circumference of 24 cm. Second circle theorem - angle in a semicircle. o Circumference. The area of a circle is given by Pi*Radius^2 where Pi is a constant approximately equal to 3. Round this ratio to the nearest thousandth. side a: side b: distance h:. Find the diameter of the circle to the nearest tenth. This is the length of just the curvy part of the semi-circle. The circumference of a figure is the sum of all the side lengths. The perimeter of a semicircle can be calculated using the following formula: if you know the length of the radius. SOLUTION a. \$ cc circumference-of-circle. This math worksheet was created on 2010-03-06 and has been viewed 81 times this week and 53 times this month. And in this case, r is equal to 6. 2 Calculate the length of the circumference of each circle, rounding your answers to one decimal place: 10 = 31. The diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie. The floor is defined as a circle equal to the diameter of the base of the dome. circle theorems rules pdf DA is a tangent to the circle. 2831852 m e t e r s. MEMORY METER. Another circle is inscribed in the semi–circle so that it touches the diameter at point A and also touches the circumference of the semi– circle. The ratio circumference. Perimeter of composite shapes. Lesson Plan Title : Finding circumference, diameter, and radius. In the result you will get all unknown variables presented. ) cm² (to. n r tM Nazdhe t 9wKi7t6h l wI2nOf vi1nIi pt se F BGFe jo Lmteztjr 8yE. radius: distance from center of circle to any point on it. The circumference is the outside edge of the circle. They find the area and circumference of circles given a radius or diameter measurement. This formula can also be given as: C = 2πr, where r is the radius. Pi ( π ): It is a number equal to 3. Perimeter of a semicircle = (1 2 Suppose we want to calculate the circumference of a circle with radius r r r. Examples: how to work out the area of a trapezium?. 2, but that is the best I can do with the crude calculator on the computer and a scratch piece of paper. Then tap or click the Calculate button. 14159, which is equal to the ratio of the circumference of any circle to its diameter. Ex2) 14 in. (a) Calculate the area of the whole target. Area of semi-circle formula is derived from the formula of a circle. So, to find the diameter, divide the circumference by. The POWER function will take any number and raise it to the power of any other number. That's the. You can also use it to find the area of a circle: A = π * R² = π * 14² = 615. The area is generally 1/2*Pi*r². Students, teachers, parents, and everyone can find solutions to their math problems instantly. The circumference of C i r c l e 2 is 6. Calculator online for a capsule. o Pi (π) Objectives. C = πd Circumference of a circle C = 3. The calculations are done "live": How to Calculate the Area. Angle in a semicircle We want to prove that the angle subtended at the circumference by a semicircle is a right angle. o Calculate the circumference of a circle. We have step-by-step solutions for your textbooks written by Bartleby experts!. Formulas, explanations, and graphs for each calculation. C = 2π r Write formula for. Program to calculate Area and Circumference based on user input. 77 feet/2 r=15. 7 ft O C r d. For example, if the circumference in this example equals 11 inches, multiply 11 by 0. 2, but that is the best I can do with the crude calculator on the computer and a scratch piece of paper. where r is the radius of the circle and π is the ratio of a circle’s circumference to its diameter. I am stuck on put a circle around the number that is not a square number 9 16 36 48 81 100. Calculate the area and circumference of this circle, leaving your answer in terms of π 4. The diameter of the larger semicircle is subtended by the two smaller semicircles the small semicircle closer to the left of the internal circumference of the larger semicircle is shaded one while the one on the right is without color This calculator calculates for the radius length width or chord height or sagitta apothem angle and area of an. (c) Find the area of the white. Perimeter Of Sector calculator provides for the same. The area of a circle is given by Pi*Radius^2 where Pi is a constant approximately equal to 3. Use our circumference calculator to determine the area of a circle. 8cm This time, we have a semicircle. Step 2: Divide the circumference in half. The radius of a circle measuring 20 cm. Perimeter of a triangle calculation using all different rules: SSS, ASA, SAS, SSA, etc. !Work out the area. 56 + 8 = 20. Given OA = 4, find the radius of the inscribed circle. I use this product as part of a grade 7 unit of study on the area and circumference of circles. This video shows how to find the area of a rectangle with a semi-circle. You know circles are round. (c) Find the area of the white. the circumference. n r tM Nazdhe t 9wKi7t6h l wI2nOf vi1nIi pt se F BGFe jo Lmteztjr 8yE. Circumference Measuring the _____ of the circle. But, the mathematical description of circles can get quite confusing, since there is a set equation for a circle, including symbols for the radius, and center of the circle. 14 or 22 — 7 for 𝛑. your calculator as an approximation for 𝜋. Calculate the area of one surface of the table mat. Calculate quickly surface area or volume of sphere. (a) Calculate the area of glass needed to glaze the window. First, you must apply the formula for calculate the area of a semicircle, which is: A=πr 2 /2 A is the area of the semicircle. that the perimeter of the semi-circle is. Calculate the shaded area. Step 2: Divide the circumference in half. I can calculate thecircumference of a circle when I know the diameter. In column H, calculate the dollars per square inch of pizza, using =G2/F2. Define tangent. It is possible to inscribe a rectangle by placing its two vertices on the semicircle and two vertices on the x-axis. Add them to the table above, measure the circumference and diameter for each, and then compute the ratio Circumference Diameter for each object. DCO is a straight line. Tracing paper may be used. (c) Given that APB is 1. Use your calculator to find the maximum area. 14 or the π button on your calculator. Draw the diagram (as above on right) to understand the problem and arrive at the required answer. 4 Find the radius of a circle. Follow 93 views (last 30 days) Timothy States on 7 Feb 2016.

http://math.stackexchange.com/questions/95562/how-to-calculate-z4-frac1z4-if-z2-z-1-0/95596

# How to calculate $z^4 + \frac1{z^4}$ if $z^2 + z + 1 = 0$? Given that $z^2 + z + 1 = 0$ where $z$ is a complex number, how do I proceed in calculating $z^4 + \dfrac1{z^4}$? Calculating the complex roots and then the result could be an answer I suppose, but it's not quite elegant. What alternatives are there? - From $z^2 + z + 1 = 0$, we have $$z +\frac{1}{z}=-1.$$ Taking square, we get $$z^2 +\frac{1}{z^2}+2=1,$$ which implies that $$z^2 +\frac{1}{z^2}=-1.$$ I think you will know what to do next. - I see... Thank you for the quick answer! –  Mihai Bişog Jan 1 '12 at 11:54 $\displaystyle{z^{2^k}+\frac{1}{z^{2^k}}=-1}$ for all $k$. –  Jonas Meyer Jan 1 '12 at 11:54 @Jonas: nice observation! –  Paul Jan 1 '12 at 12:06 Dickson polynomials FTW! –  Jyrki Lahtonen Jan 1 '12 at 14:20 Another way to justify that $z+\dfrac1{z}=-1$ is that due to the coefficient symmetry, if $z$ is a root, then $\dfrac1{z}$ is a root as well. By Vieta, then, the sum of those two roots ought to be the negative of the coefficient of the linear term... –  J. M. is back. Jan 1 '12 at 15:15 Essentially the same calculation also follows from the observation that $x^2+x+1=\phi_3(x)$ is the third cyclotomic polynomial. So $z^3-1=(z-1)(z^2+z+1)=0$, and hence $z^3=1$ for any solution $z$. Therefore $$z^4+\frac{1}{z^4}=z\cdot z^3+\frac{(z^3)^2}{z^4}=z+z^2=-1.$$ - Very clever! :D –  J. M. is back. Jan 1 '12 at 12:08 Very nice! If you had not posted this already, I would have used the sum of geometric series (instead of cyclotomic polynomials) to write $$0=z^2+z+1=\frac{z^3-1}{z-1}\Rightarrow z^3=1$$ and $$z^4+\frac{1}{z^4}=(z^3)z+\frac{1}{(z^3)z}=z+\frac{1}{z}=\frac{z^2+1}{z}=\frac{‌​-z}{z}=-1.$$ –  Dilip Sarwate Jan 1 '12 at 15:48 \begin{align*} z^4+\frac1{z^4}&=(-z-1)^2+\frac1{(-z-1)^2}\\ &=z^2+2z+1+\frac1{z^2+2z+1}\\ &=(-z-1)+2z+1+\frac1{(-z-1)+2z+1}\\ &=z+\frac1{z}=\frac{z^2+1}{z}=\frac{-z-1+1}{z}=-1 \end{align*} - What would be wrong with systematic reduction using the minimal polynomial? Nothing! –  Jyrki Lahtonen Jan 1 '12 at 12:25 Here is an alternative approach: let's consider $z^{8}+1$ , and then divide by $z^{4}$. By using geometric series, notice that $$z^{8}+z^{7}+z^{6}+z^{5}+z^{4}+z^{3}+z^{2}+z+1=\left(z^{2}+z+1\right)\left(z^{6}+z^{3}+1\right)=0.$$ Now, as $z^{2}+z+1=0$, we know that both $z^{7}+z^{6}+z^{5}=0$ and $z^{3}+z^{2}+z=0$, and hence $$z^{8}+z^{4}+1=0$$ so that $z^{8}+1=-z^{4}$. Thus, we conclude that $$z^{4}+\frac{1}{z^{4}}=-1.$$ - Different people see different things in the relation $z^2+z+1=0$. Just as Jyrki did, I see the third cyclotomic polynomial. Its roots are $-1/2 \pm \sqrt{-3}/2$, the two primitive cube roots of unity. Call one of these $\omega$ and see that $\omega^3=1$, so that $\omega^4=\omega$ and $\omega^{-4}=\omega^2$. Their sum is $-1$, from the defining relation. - Hint $\$ Exploit innate symmetry: for $\rm\ y = z^{-1}$ you know $\rm\ yz\ (=\: 1)\:$ and $\rm\ y+z\ (=\: z^{-1}\!+z\: =\: -1)\$ Thus you know $\rm\ \ \ y^2 + z^2\ =\ (y\ +\ z)^2-\ 2\:(y\:z)$ hence you know $\rm\ y^4 + z^4\ =\ (y^2\! + z^2)^2 - 2\:(y\:z)^2$ For more on symmetric polynomials see the Wikipedia article on Newton's identities. - To downvoter: if something is not clear then please feel welcome to ask questions and I will be happy to elaborate. –  Bill Dubuque Jan 1 '12 at 22:52 $$z^2+z+1=0$$ Clearly $z \not= 0$ and therefore $$z+\frac{1}{z}=-1.$$ Note that $$z^{n+1}+\frac{1}{z^{n+1}}=(z+\frac{1}{z})(z^n+\frac{1}{z^n})-(z^{n-1}+\frac{1}{z^{n-1}}).$$ Therefore if we define the function $U:N \to N$ as $$U_n=z^n+\frac{1}{z^n}$$ then we have $U(0)=2$ and $U(1)=-1.$ Also $$U(n+1)=-U(n)-U(n-1)$$ for all $n \in N.$ Using this recurrence you can calculate $z^n+\frac{1}{z^n}$ for any $n \in N.$ -

http://mathematica.stackexchange.com/questions/22349/mathematica-for-teaching-orthographic-projection

# Mathematica for teaching orthographic projection Edit: All the four answers to this question are great, and if you're interested, you should take a look at all the answers. Nevertheless, belisarius' code was accepted since it was closest to what I had in mind. I'm trying to use Mathematica to help students better understand 3-dimensional views of objects, specifically the "plan view", the "side elevation view" and the "front elevation view", as elaborated upon here. Given the following example "building", I was able to combine Graphics3D primitives to simulate the object using the following code. dee = Graphics3D[{Green, Opacity[0.6], Cuboid[{0, 0, 0}, {2/3, 1, 1}], Red, Polygon[{{{2/3, 1/2, 0}, {2/3, 1/2, 1}, {3/2, 1/2, 0}}, {{2/3, 1, 0}, {2/3, 1, 1}, {3/2, 1, 0}}, {{2/3, 1/2, 0}, {2/3, 1, 0}, {3/2, 1, 0}, {3/2, 1/2, 0}}, {{2/3, 1/2, 1}, {2/3, 1, 1}, {3/2, 1, 0}, {3/2, 1/2, 0}}}]}] However, as you can see, the code is rather convoluted, with a great deal of time spent making the red prism. In this case, it was still feasible to manually generate the prism, but what I'm interested in is, Is there a way to generate a 3D object which is made up of a combination of prisms simply by identifying the vertices at the corner of the object? If you're interested, my code to allow students to see the three views is as follows, with an example result below. planviewer = TableForm@{{"3d", "Side", "Front", "Top"}, {Show[#, ViewPoint -> {1, -1, 1}], Show[#, ViewPoint -> {∞, 0, 0}], Show[#, ViewPoint -> {0, -∞, 0}], Show[#, ViewPoint -> {0, 0, ∞}]}, {"", Show[#, ViewPoint -> Right], Show[#, ViewPoint -> Front], Show[#, ViewPoint -> Above]}} &; The following code allows them to manipulate the viewpoint dynamically. Manipulate[ Show[dee, ViewPoint -> {10^a, -(10^b), 10^c}], {a, 0, 3}, {b, 0, 3}, {c, 0, 3}] - If you want to teach about the different View* functions and what they do, you might find this answer useful. –  The Toad Mar 30 '13 at 16:59 There are some tricks ... Specifying the prism's vertices is enough (you don't need to take care of the faces) if you use some undocumented methods for finding the convex hull: v = {{2/3, 1/2, 0}, {2/3, 1/2, 1}, {2/3, 1, 0}, {2/3, 1, 1}, {3/2, 1/2, 0}, {3/2, 1, 0}}; prism@v_:=Cases[ComputationalGeometryMethodsConvexHull3D[v,GraphicsMeshFlatFaces->False], _GraphicsComplex, Infinity]; Graphics3D[{Green, Opacity[0.6], Cuboid[{0, 0, 0}, {2/3, 1, 1}], Opacity[.6], Red, prism@v}] - This is easily turned into a function, polyhedron[v_] := Cases[.. Very nice. (+1) –  Michael E2 Mar 30 '13 at 18:21 @MichaelE2 Thanks! Rewritten as a function now. –  belisarius Mar 30 '13 at 18:43 @belisarius how do you find out about all these undocumented questions? I was trying to search for it, and saw some similar questions on this site, but I think that's the first time I've seen it. Edit: ninja'd by bill. –  Vincent Tjeng Mar 31 '13 at 3:30 It's not on this list, by the way :) : mathematica.stackexchange.com/questions/809/… –  Vincent Tjeng Mar 31 '13 at 3:38 @VincentTjeng I don't remember how I found this one, probably following some InternalTrace. You can find similar things trying Names["ComputationalGeometryMethods*"] –  belisarius Mar 31 '13 at 5:55 GraphicsComplex is probably what you are looking for. For example, define the vertices for the 3D polygon: v = {{2/3, 1/2, 0},{2/3, 1/2, 1},{3/2, 1/2, 0},{2/3, 1, 0},{2/3, 1, 1},{3/2, 1, 0}}; and make a list of which vertices should connect to each other: i = {{1, 2, 3}, {4, 5, 6}, {1, 2, 5, 4}, {1, 3, 6, 4}, {2, 3, 6, 5}}; The first two elements of i represent the two triangular faces of the prism; the final three elements are the three rectangular faces. This is plotted using Graphics3D[{Opacity[.8], Red, GraphicsComplex[v, Polygon[i]]}] or you can plot the green cube plus the red prism together: Graphics3D[{Green, Opacity[0.6], Cuboid[{0, 0, 0}, {2/3, 1, 1}], Opacity[.6], Red, GraphicsComplex[v, Polygon[i]]}] which gives the figure below. The code is modified from the documentation in GraphicsComplex where you can find all sorts of neat 3D tricks. If they have access, I think the students would benefit from being able to manipulate the 3D illustration themselves -- this seems like a good application for deployed .cdf's. - Thanks for the answer! Would it be possible for the one to write some code that immediately identifies which of vertices should be connected to each other to form the faces, keeping in mind that the faces of the prism may not always be in the same plane? Also, I notice that the part of the red prism joining the green cuboid is actually a triangle, and not a rectangle ... is that a problem or simply a visual illusion? –  Vincent Tjeng Mar 30 '13 at 10:21 The little triangle on the adjoining face is an illusion that has to do with the shading -- when you rotate the graphic it changes. But to your first question, the answer is no. After all, how can it know what edges you want to connect if you don't tell it? You will get something no matter what edges you connect -- only one of these is the figure you presumably want. –  bill s Mar 30 '13 at 10:33 @VincentTjeng: yes, you can connect the vertices automagically because the faces must form the convex hull. See my answer –  belisarius Mar 30 '13 at 18:54 Very nice... how does one find "undocumented features"? –  bill s Mar 31 '13 at 3:17 bill I don't remember how I found this one, probably following some InternalTrace. You can find similar things trying Names["ComputationalGeometryMethods*"] –  belisarius Mar 31 '13 at 5:59 Take a look at the output of PolyhedronData[{"Prism", 3}, "Faces"]}, which may help. Note this isn't exactly a direct answer, but I think the optimal solution here is a tool that would let you... well, like this: whatThe[n_] := Module[{list = {}, grid = False}, Grid@List@{ EventHandler[#, {"KeyDown", "k"} :> (grid = Not[grid])] &[ Graphics3D[{ EventHandler[MouseAppearance[Sphere[#, .06], Style["\[RightUpDownVector]\[LeftUpDownVector]", Darker@Blue, Large]], {"MouseClicked" :> AppendTo[list, ##]}] & /@ Tuples[Range[n], 3], (*Opacity[.3],*)Dynamic[Polygon[list]], Thick, Dynamic[Line[list]], Thin, Lighter[Gray, .8], Dynamic[If[grid, Line[ Select[Subsets[Tuples[Range[n], 3], {2}], EuclideanDistance @@ ## == 1 &]], {}]]}, Boxed -> True, ImageSize -> Large, Axes -> True]], Graphics3D[Dynamic[Polygon[list]]], EventHandler[Dynamic[Column[list]], {"MouseClicked" :> (CopyToClipboard[list]; Beep[])}]} ]; whatThe[4] You click on points to add them to a list. MAGIC!!! It takes some getting used to... here's one of my attempts at creating a prism: I couldn't tell what order you have to construct the list in for the resulting polygon to be reasonable, but at the very least it might help you pick some points out. It's also another example of just how ridiculously powerful Mathematica is. The basic working version of this took me like 4 minutes to make. O.O (Note: A simple improvement might be to use a hotkey to cut the list up as you build the polygon). Update. Using belisarius's convex hull code: whatThe[n_] := Module[{list = {}, faces, grid = False}, faces[v_] := Cases[ComputationalGeometryMethodsConvexHull3D[v, GraphicsMeshFlatFaces -> False], _GraphicsComplex, Infinity]; Grid@List@{ EventHandler[#, {"KeyDown", "k"} :> (grid = Not[grid])] &[ Graphics3D[{EventHandler[ MouseAppearance[Sphere[#, .06], Style["\[RightUpDownVector]\[LeftUpDownVector]", Darker@Blue, Large]], {"MouseClicked" :> AppendTo[list, ##]}] & /@ Tuples[Range[n], 3], (*Opacity[.3],*)Dynamic[faces[list]], Thin, Lighter[Gray, .8], Dynamic[If[grid, Line[ Select[Subsets[Tuples[Range[n], 3], {2}], EuclideanDistance @@ ## == 1 &]], {}]]}, Boxed -> True, ImageSize -> Large, Axes -> True]], Graphics3D[Dynamic[faces[list]]], EventHandler[Dynamic[Column[list]], {"MouseClicked" :> (CopyToClipboard[list]; Beep[])}]}]; - +1 amazing. and it helps with the first part of translating the problems in the students' worksheets to the prism I want. now I just need to modify the code such that it accepts multiple convex hulls representing the multiple prisms :) –  Vincent Tjeng Mar 31 '13 at 2:58 You can get away with specifying even less than all the points. A (right) prism is a planar figure extruded in an orthogonal direction. It would suffice, then, to give the coordinates of that figure (thereby describing one of the two "ends" of the prism) and its intended height. The figure in the question can therefore be created as a collection of two prisms (no Cuboid if you don't care to use it) like this: Graphics3D[{Opacity[0.6], Green, prism[{{0, 0, 0}, {2/3, 0, 0}, {2/3, 1, 0}, {0, 1, 0}}, 1], Red, prism[{{2/3, 1, 0}, {3/2, 1, 0}, {2/3, 1, 1}}, 1/2]}] The calculations come down to 1. Find a basis for the polygon's plane by orthogonalizing its vertices relative to their barycenter; 2. Perform the extrusion by taking the cross product of the basis elements (normalized to the desired height) and adding that to each of the polygon's vertices; and 3. Describe how the vertices are connected to form the two polygonal ends and each of the sides. Here is such a solution. Its arguments are pt, a list of the (3D) vertices around the (nondegenerate) polygon, and height, the height of the prism. It produces a 3D graphics object. prism[pt_List, height_] := Block[{n = Length[pt], normal}, normal = Cross @@ Orthogonalize[# - Mean[pt] & /@ pt][[1 ;; 2]]; GraphicsComplex[pt~Join~(height normal + # & /@ pt), {Polygon[Range[n]], Polygon[Range[2 n, n + 1, -1]], Polygon[{#, n + #, n + Mod[# + 1, n, 1], Mod[# + 1, n, 1]}] & /@ Range[n]}]] To see it in action, let's generate a polygon with a random position and orientation: basis = RandomReal[NormalDistribution[0, 1], {2, 3}]; pt = ({Cos[#], Sin[#]} & /@ Range[0, 2 \[Pi], 2 \[Pi]/7]).basis Here is the original polygon as a solid black "floor" drawn with two prisms, one with a positive height and another with a negative height: Graphics3D[{Opacity[0.75], prism[pt, 1], prism[pt, -1/2], Opacity[1], Black, Polygon[pt]}] This approach is easily extended to non-right prisms by replacing height by the extrusion vector (describing one of the "vertical" edges of the prism) and using that instead of height normal in the code. - nice application of math :) just a quick question, when you code as prism[pt_List, height_], pt_List is written as such since it is a list, right? so if i were to code a function f[a_String, b_String], then the variables would be a, b and they must be strings? –  Vincent Tjeng Mar 31 '13 at 3:35 –  whuber Mar 31 '13 at 13:19 Got it, thank you! –  Vincent Tjeng Apr 1 '13 at 3:01

https://math.stackexchange.com/questions/2998130/matrix-representation-of-mixed-derivatives

# Matrix representation of Mixed derivatives Imagine we have the problem $$\begin{cases} -\frac{d^2u}{dx^2} = f(x), x \in [0,L] \\ u(0) = 0 \\ u(L) = 0 \end{cases}$$ We know that we can approximate the second derivative using this formula:$$\frac{d^2u(x)}{dx^2} \approx \frac{u(x+h)-2u(x)+u(x-h)}{h^2}$$ if we define $$u_{k} := u(x_{k})$$; $$\ \ x_{k} = kh$$ and $$\ \ k = 0,1,2,...,N$$. $$h$$ is known as the mesh size or step size. We get: $$\frac{d^2u_{k}}{dx^2} \approx \frac{u_{k+1}-2u_{k}+u_{k-1}}{h^2} = \frac{u_{k-1}-2u_{k}+u_{k+1}}{h^2}$$ for $$k=1,...,N-1$$ Since $$u(0) = u_{0} = 0$$ and $$u(L) = u(x_{N}) = u_{N} = 0$$ we get the following matrix representation of the second derivative operator $$$$\frac{d^2}{dx^2} \approx L_{2} = \frac{1}{h^2}\left(\begin{matrix} -2 & 1 & & 0\\ 1 & \ddots & \ddots & \\ & \ddots & \ddots & 1 \\ 0 & & 1 & -2 \end{matrix} \right)$$$$ Then to approximate the solution of the differental equation we solve the system: $$-L_{2}\hat{u} = \hat{f}$$ where $$\hat{f} = [ f(x_{1}) \ f(x_{2}) \ ... \ f(x_{N-1}) ]^T$$ and $$\hat{u} = [ u_{1} \ ... \ u_{N-1} ]^T$$ The matrix representation of the laplacian operator using the Kronecker product is: $$\Delta =\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} = L_{2}\otimes I_{n_{x}} + I_{n_{y}} \otimes L_{2}$$ $$\textbf{Note that I have used L_{2} and kronecker product to get a matrix representation}$$ of the laplacian operator. With this in mind. I want to approximate the first derivative using central difference: $$\frac{du_{k}}{dx} \approx \frac{ u_{k+1}-u_{k-1} }{ 2h } = \frac{ -u_{k-1} +u_{k+1} }{ 2h }$$ Since $$u(0) = u_{0} = 0$$ and $$u(L) = u(x_{N}) = u_{N} = 0$$ we get the following matrix representation of the first derivative $$$$\frac{d}{dx} \approx L_{1} = \frac{1}{2h}\left(\begin{matrix} 0 & 1 & & 0\\ -1 & \ddots & \ddots & \\ & \ddots & \ddots & 1 \\ 0 & & -1 & 0 \end{matrix} \right)$$$$ I want to use $$L_{1}$$ and kronecker product to get the matrix representation of $$\frac{\partial^2 }{\partial y \partial x }$$ and $$\frac{\partial^2 }{\partial x \partial y }$$ $$\textbf{My questions are:}$$ 1. Is this possible? 2. if question 1. is affirmative, what is the matrix representation of the mixed derivatives using $$L_{1}$$ and kronecker product? 3. if question 1. is negative, how can we get the matrix representation of the mixed derivatives using a simple matrix and kronecker product? 4. Do you know a book or document( article or other ) that explain in detail this or something similar? $$\textbf{I want to use kronecker product because it is fast and easy to implement in}$$ matlab or octave. By the way I tried to use this formula $$\frac{\partial^2u_{k,j} }{\partial x \partial y } = \frac{ u_{k+1,j+1}+u_{k-1,j-1}-u_{k+1,j-1}-u_{k-1,j+1} }{4h^2}$$ But it was hard to see a pattern. Thank you! Once you have the matrix $$\bf L_1$$ to represent the divided difference of $$f(x)$$ given as a vertical vector, then if you transpose the whole you'll get a horizontal vector for the difference. So if $$f(x,y)$$ is represented as a matrix $$\bf F_{\, x,\, y}$$ , the matrix representing $${\partial \over {\partial x\partial y}}f(x,y)$$ will be $$\bf L_1 \bf F_{\, x, \,y} \bf L_1^T$$ • Honestly I can not imagine the matrix... – tnt235711 Nov 14 '18 at 11:45 • @tnt235711: added – G Cab Nov 14 '18 at 12:19 • @tnt235711: is it ok now for you ? – G Cab Nov 15 '18 at 16:55 Now using $$\mathbf{L_{1}}$$ and $$\textbf{kronecker product}$$: $$\frac{\partial^2 }{\partial x \partial y} = \Big((L_{1})_{n_{x}} \otimes I_{n_{y}} \Big)\Big( I_{n_{x}} \otimes (L_{1})_{n_{y}} \Big) = { (L_{1})_{n_{x}} \otimes (L_{1})_{n_{y}} }$$ I have used the mixed-product property for kronecker product. $$(\mathbf{A} \otimes \mathbf{B})(\mathbf{C} \otimes \mathbf{D}) = (\mathbf{AC}) \otimes (\mathbf{BD})$$. In a similar way we get: $$\frac{\partial^2 }{\partial y \partial x} = (L_{1})_{ n_{y} } \otimes (L_{1})_{n_{x}}$$

https://math.stackexchange.com/questions/1610277/how-to-solve-an-initial-value-problem-if-my-function-is-implicit

# How to solve an initial value problem if my function is implicit? I have the differential equation: $\dfrac{dy}{dt} = \dfrac{1}{2y+3}$ with the initial value of $y(0) = 1$ So I solve the diffeq via separation of variables: $y^2 + 3y = t + c$ But from here, how do I solve for the function? Am I "allowed" to use the initial value information with an implicit function such as this, and give an implicit function as my answer, as below? $y^2 + 3y = t + 4$ As far as I can tell, it should be perfectly fine to leave my answer as an implicit function. And I don't see a way to solve for $y$ here. TO be honest, I used a diffeq solver to check my work, and that tool did come up with an explicit solution, though when I tried to see where it came from I had no idea. • This is a quadratic equaion for $y$. – kmitov Jan 13 '16 at 3:25 • @kmitov But i do not want to solve for a particular value of $y$, I want to solve for a particular solution $y(t)$. – user278703 Jan 13 '16 at 3:30 Yes, you should use your known initial condition to solve for $c$. Proceeding to solve for $y(t)$, we write $y^2+3y+\frac{9}{4}-\frac{9}{4}=\left(y+\frac{3}{2}\right)^2-\frac{9}{4}$ by completing the square. Now, we write $\left(y+\frac{3}{2}\right)^2-\frac{9}{4}=t+4.$ and solve for $y$ in terms of $t$. Adding $\frac{9}{4}$, taking the square root, and subtracting $\frac{9}{2}$, we get: $$y(t)=\sqrt{t+\frac{25}{4}}-\frac{3}{2}$$ EDIT: I had misplaced a $3$. Edit: To address your comment: Note that we can write $y(t)=\sqrt{t+\frac{25}{4}}-\frac{3}{2}=\sqrt{\frac{1}{4}(4t+25)}-\frac{3}{2}=\frac{1}{2}\sqrt{(4t+25)}-\frac{3}{2}=\frac{1}{2}(\sqrt{(4t+25)}-3).$ The reason that $c$ is different is because $c$ is some arbitrary constant. In this case, we have already solved for it. • But $\frac{9}{2} + \frac{9}{2}$ doesn't equal 3?? And $\frac{9}{2} * \frac{9}{2}$ doesn't equal $\frac{9}{4}$?? How can you factor $y^2 + 3y + \frac{9}{4}$ and get $(y+ \frac{9}{2})^2$? Shouldn't the correct factoring have $\frac{3}{2}$ instead of $\frac{9}{2}$? – user278703 Jan 13 '16 at 3:46 • I'm not factoring. I'm completing the square. For $x^2+bx$, we can add and subtract $(\frac{b}{2})^2$ to get $x^2+bx+(\frac{b}{2})^2-(\frac{b}{2})^2$, which is equal to $(x+\frac{b}{2})^2-(\frac{b}{2})^2$. Now, let $b=3$. – zz20s Jan 13 '16 at 3:51 • How? If you factor $y^2 + 3y + \frac{9}{4}$ it equals $(y+\frac{3}{2})^2$. If we remove the $-\frac{9}{4}$ from each side of the equation you wrote, it is $y^2 + 3y + \frac{9}{4} = (y+\frac{9}{2})^2$, which is wrong. Edit: I have seen your edit, thanks – user278703 Jan 13 '16 at 3:56 • No problem. Do you understand the solution now? – zz20s Jan 13 '16 at 4:02 • I understand, When I solve the diffeq on an online solver it gives $y(t) = \frac{1}{2}(-\sqrt{4t+c} - 3)$. I'm not sure where they got the $4$, or the $-3$, but I guess I'll play with it and see. – user278703 Jan 13 '16 at 4:04 From where you arrived, $y^2+3y=t+c$ Substituting $y(0)=1$, we get $c=4$ Hence, we get $y^2+3y-4=t$ Proceed by completing the square for $y^2+3y-4=(y+\frac{3}{2})^2-\frac{7}{4}$ $(y+\frac{3}{2})^2-\frac{7}{4}=t$ Then we get $y=\frac{1}{2}(3±\sqrt{4t+7})$

https://math.stackexchange.com/questions/2944252/finding-the-probability-that-someone-has-the-disease-given-they-test-positive-o

# Finding the probability that someone has the disease, given they test positive on two tests This question is from the textbook "Introduction to Probability - Blitzstein & Hwang." I was studying for a class when I came across an example problem that I solved, but got a slightly different result than the textbook. Here's the problem in question, paraphrased: "Fred tests for a disease which afflicts 1% of the population. The test's accuracy is deemed 95%. He tests positive for the first test, but decides to get tested for a second time. Unfortunately, Fred also tests positive for the second test as well. Find the probability that Fred has the disease, given the evidence." $$\$$ My approach is as follows: Let $$D$$ be the event that Fred has the disease, $$T_1$$ be the event that the first test result is positive, and $$T_2$$ be the event that the second test is also positive. We want to find $$P(D\ |\ T_1,\ T_2)$$. We are also able to condition on $$T_1$$ (i.e. the event that the first test result is positive). This would give us: $$P(D\ |\ T_1,\ T_2) = \frac{P(T_2\ |\ D,\ T_1)P(D\ |\ T_1)}{P(T_2\ |\ T_1)}$$ From my calculations: $$P(T_2\ |\ D,\ T_1)\ =\ P(T_2\ |\ D)\ =\ 0.95$$ $$P(D\ |\ T_1)\ \approx\ 0.16$$ $$P(T_2\ |\ T_1)\ =\ \frac{P(T_1 ,\ T_2)}{P(T_1)}\ =\ \frac{P(T_1,\ T_2,\ D)\ +\ P(T_1,\ T_2,\ D^c)}{P(T_1,\ D)\ +\ P(T_1,\ D^c)}\ =\ \frac{0.0115}{0.059}\ \approx\ 0.19$$ $$\$$ $$P(D\ |\ T_1,\ T_2)\ =\ \frac{0.95\ \times\ 0.16}{0.19}\ =\ 0.8$$ Therefore, I concluded that there is an 80% chance that Fred has the disease, given that both the first and second test results are positive. $$\$$ The problem is that the textbook has taken a different approach of using the odds form of Bayes' rule, which resulted in a conclusion slightly different from mine (0.78), and I'm having trouble understanding how that conclusion came to be. $$\$$ Textbook approach is as follows: $$\frac{P(D\ |\ T_1,\ T_2)}{P(D^c\ |\ T_1,\ T_2)}\ =\ \frac{P(D)}{P(D^c)}\ \times\ \frac{P(T_1,\ T_2\ |\ D)}{P(T_1,\ T_2\ |\ D^c)}$$ $$=\ \frac{1}{99}\ \times\ \frac{0.95^2}{0.05^2}\ =\ \frac{361}{99}\ \approx\ 3.646$$ which "corresponds to a probability of 0.78." $$\$$ Here are the specific questions I have: 1. Is my approach wrong? A 0.02 difference is a pretty big difference. 2. How did the author derive the equation: $$P(D\ |\ T_1,\ T_2)\ =\ P(D)P(T_1,\ T_2\ |\ D)$$ 1. What does the author mean when he/she says "3.646 corresponds to a probability of 0.78?" $$\$$ Any feedback is appreciated. Thank you! • A very similar question was asked on this site about 6 years ago. Here is the link to it: math.stackexchange.com/questions/198677/… – David Oct 7 '18 at 0:25 • Thanks for the feedback! I actually read that question while looking for other answers. It didn't address the questions that I had, though, which led me to ask my own. – Seankala Oct 7 '18 at 2:26 I have some quibbles with the way the textbook sets up its question, beginning with the assumption that both positive and negative tests each have the same likelihood to be correct, and even more so the assumption that the outcomes of two tests on the same person are independent in probability. In real life, I would want to explore both of those points further before advising Fred. But let's ignore those objections for the sake of being able to compute something based on the given information, and assume each administration of the test has the same chance to give a correct result, even when two tests are administered one after the other on the same person. 1. Is my approach wrong? A 0.02 difference is a pretty big difference. The two methods are equivalent. The apparent discrepancy is due to roundoff. The textbook finds an odds ratio of $$361:99,$$ which is exact (insofar as the $$1\%$$ and $$95\%$$ are exact). Since this is $$P(D) : P(D^C),$$ the probability is given by $$P(D) = \frac{P(D)}{P(D) + P(D^C)} = \frac{361}{361 + 99} \approx 0.78478,$$ which the text rounds to $$0.78.$$ (Since you asked about this as a separate part of the question, I'll explain in more detail below.) In your approach, $$P(T_1,\ T_2,\ D)\ +\ P(T_1,\ T_2,\ D^c) = 0.0115$$ is an exact result, and so is $$P(T_1,\ D)\ +\ P(T_1,\ D^c) = 0.059,$$ but $$0.0115 / 0.059 \approx 0.19492.$$ Meanwhile, $$P(T_1 \mid D) \approx 0.16102.$$ If we carry all these digits into the computation rather than rounding off to two places immediately, we find that $$P(D\mid T_1,\ T_2) = \frac{0.95\times 0.16102}{0.19492} \approx 0.78478.$$ That is, keeping five digits we get the same answer as the textbook method (if it retained five digits), and if we round to two digits only at the end (as the textbook does) we naturally would round the same way, to $$0.78.$$ I think an argument could be made for keeping only one digit of precision in the answer (how precise is that "$$1\%$$" anyway?), in which case both answers round to $$0.8.$$ 1. How did the author derive the equation ... They didn't. Instead, the fact is that $$P(D\mid T_1,\ T_2) = \frac{P(T_1,\ T_2\mid D)P(D)}{P(T_1,\ T_2)}$$ and $$P(D^C\mid T_1,\ T_2) = \frac{P(T_1,\ T_2\mid D^C)P(D^C)}{P(T_1,\ T_2)}.$$ When you compute the ratios of the two probabilities $$\frac{P(D\mid T_1,\ T_2)}{P(D^C\mid T_1,\ T_2)},$$ you get factors of $$P(T_1,\ T_2)$$ in both the numerator and the denominator, and these factors cancel each other. 1. What does the author mean when he/she says "3.646 corresponds to a probability of 0.78?" As I hinted above, $$3.646$$ is an odds ratio; or as I would rather say, the odds ratio is $$3.646 : 1.$$ An odds ratio of $$1:1$$ corresponds to a $$50\%$$ chance, that is, each possibility is equally likely, whereas a $$2:3$$ odds ratio describes something that happens twice for every three times it does not happen. in general, if the probability of something is $$p,$$ its odds ratio is $$p : (1 - p),$$ that is, $$\frac{p}{1 - p} : 1.$$ If we say $$p = P(D \mid T_1,\ T_2),$$ then $$P(D^C \mid T_1,\ T_2) = 1 - P(D \mid T_1,\ T_2) = 1 - p,$$ and what the textbook has computed is that $$\frac{p}{1 - p} \approx 3.646,$$ that is, on average in situations like this, when both tests come up positive, there will be $$3.646$$ cases in which the tests were both correct for each case in which both tests were incorrect. That means there are $$3.646$$ accurate positives for every $$3.646 + 1$$ times the test comes out positive both times, which gives a probability of $$\frac{3.646}{3.646 + 1} \approx 0.78.$$ The way I worked the probability, however, was to take the fraction $$\frac{p}{1 - p} = \frac{361}{99}$$ and directly extract an odds ratio of $$361:99$$ from it. This means I can wait until the very end before doing any roundoff, but in other respects it's the same as the textbook's method. In both cases the odds are simply $$kp : k(1 - p),$$ where $$k$$ is whatever constant you have to multiply each side by in order to produce either $$361:99$$ or $$3.646:1$$ from the odds ratio $$p : (1 - p).$$ • Hello. Thanks for the detailed and easy-to-understand answer! You've cleared up everything I was wondering regarding this concept. You were also correct in that my hasty rounding-off led to a different answer. The $0.19$ that I got is actually $0.194915$, which gives me a final answer of $0.779827$, which is $78\%$. Thanks for your explanation regarding the odds ratio as well, I was having some difficulty grasping what that is. – Seankala Oct 7 '18 at 2:12 • It's just a textbook exercise problem. The percentages were given by the authors of the textbook. I wouldn't say it's "meaningless" as it provides something that students like myself can practice with. In this example it's assumed that there is only "one population" and Fred is a part of it. How is that a bad assumption? – Seankala Oct 7 '18 at 11:49 • @David So your objections are that Fred might not be a member of "the" population, that the $1\%$ figure might be inaccurate or outdated. And indeed in real life these are things to consider, in addition to the unstated assumptions made about the frequency of errors of the test. But mathematics is never about absolute answers to real-life questions; it is about seeing what conclusions can be drawn from a given set of assumptions. To refuse to answer that question mathematically because the assumptions might be wrong, seems like a cheap excuse to me. – David K Oct 7 '18 at 11:49 • @David The link you gave to the other question is a useful one, as the answers to that question include a real-life example that shows why we should consider it important that a disease has a very low incidence in the population. – David K Oct 7 '18 at 11:53 • In real life, people only usually get tested for a rare disease if there is some suspicion or risk factor involved, in which case those risk factors will change your prior significantly. If Fred was only sent in to test because he had a family history of this rare disease, then your prior is going to be more than 1%. But this is a math exercise from a textbook. The point is to solve it with only what information we know, ceteris paribus. – Zubin Mukerjee Oct 7 '18 at 13:56 The conditional probability (or Bayes's rule) must be: $$P(D\ |\ T_1\cap T_2) = \frac{P(D\cap T_1\cap T_2)}{P(T_1\ \color{red}{\cap}\ T_2)}=\frac{P(D)\cdot P(T_1|D)\cdot P(T_2|D\cap T_1)}{P(T_1\ \cap\ T_2)}=\\ \frac{0.01\cdot 0.95^2}{0.01\cdot 0.95^2+0.99\cdot 0.05^2}\approx 0.7848.$$ • That's nice, but that doesn't mean it applies to this question as stated. – David Oct 6 '18 at 16:41 • @David would you care to elaborate why you think Bayes formula does not apply to a problem so clearly amenable to conditioning... – Nap D. Lover Oct 7 '18 at 2:12 • Fred is a part of the population! I think that you're right, that the question should have stated it explicitly, but mentioning a population that is completely unrelated to Fred would be a hilarious and unlikely red herring by the textbook authors. Why would they just randomly mention a population in a question about Fred, if Fred wasn't part of the population? Once you do assume Fred is part of the population, you can no longer ignore the 1% affliction rate. – Zubin Mukerjee Oct 7 '18 at 14:03 • It was fun commenting but given the assumptions are true, I agree with these answers here. It was just an exercise on being accurate about assumptions so many next time someone posts a question like this, they will explicitly state the "missing" info. I deleted all my previous answers and comments to clean this up. – David Oct 7 '18 at 15:40

https://math.stackexchange.com/questions/1591534/for-numerical-integration-is-it-true-that-higher-degree-of-precision-gives-bett

# For numerical integration, is it true that higher degree of precision gives better accuracy always? In case of numerical integration, is it true that higher degree of precision always gives better accuracy? Justify your answer. I know the definition of degree of precision. For Trapezoidal and Simpson's 1/3 rule they are 1 and 3 respectively. Simpson's 1/3 gives better accuracy than Trapezoidal rule. Then whether the above statement is true always. If not, why? If yes, then why we learn Trapezoidal/ Simpson rule? Why we shall not establish/go for higher and higher DOP from generalized Newton-Cote's rule or other general quadrature formula . • Usually, but not always. Consider the function $f(x) = \left\{\matrix{1 & x < 1/2\\0 & x \geq 1/2}\right.$. The trapesoidal rule with $n=3$ points ($x=0,1/2,1$) gets the integral $\int_0^1f(x){\rm d}x$ exactly, but if we use an even number of points we will always be a bit off the true result no matter how large $n$ is. – Winther Jan 3 '16 at 8:00 • @Winther Please give me the answer when higher degree of precision gives better accuracy and when higher degree of precision does not give better accuracy with analytical justification. – user1942348 Jan 3 '16 at 8:10 • That is almost impossible I think (for general functions). The key to the proofs of the order of the different methods is that as long as functions have bounded derivatives (of some order) then the error will grow roughly as $\frac{1}{N^k}$ for some some integer $k$ (and consequently go to zero when $N\to\infty$). However we usually don't have any control if $N=38$ will give better results than $N=40$. – Winther Jan 3 '16 at 8:12 • @Winther What do you mean by " a bit off the true result"? – user1942348 Jan 3 '16 at 8:18 • @winther Somewhere I read that In particular, "when range of integration is large", the statement above is not correct. Whether it ("when range of integration is large, higher DOP gives less accuracy") is correct. If correct, would you explore why? – user1942348 Jan 3 '16 at 8:24 Increasing the precision, both in terms of the order of the method and in the number of gridpoints used, usually (most of the time) leads to a more accurate estimate for the integral we are trying to compute. However this is not always the case as the following (aritficial) example shows. $$\bf \text{Example where higher order does not imply better accuracy}$$ Let $$f(x) = \left\{\matrix{1 & x < \frac{1}{2}\\0 & x \geq \frac{1}{2}}\right.$$ and consider the integral $I=\int_0^1f(x){\rm d}x = \frac{1}{2}$. If we use the trapezoidal rule with $n$ gridpoints then $$I_{n} = \frac{1}{n}\sum_{i=1}^{n}\frac{f(\frac{i-1}{n})+f(\frac{i}{n})}{2} \implies I_n = \left\{\matrix{\frac{1}{2} & n~~\text{odd}\\\frac{1}{2} - \frac{1}{n} & n~~\text{even}}\right.$$ so for $n=3$ we have the exact answer which is better than any even $n$ no matter how large it is. This shows that increasing the number of gridpoints does not always improve the accuracy. With Simpson's rule we find $$I_n = \frac{1}{3n}\sum_{i=1}^{n/2}f\left(\frac{2i-2}{n}\right)+4f\left(\frac{2i-1}{n}\right)+f\left(\frac{2i}{n}\right) \implies I_n = \left\{\matrix{\frac{1}{2} - \frac{1}{3n}&n\equiv 0\mod 4\\\frac{1}{2}&n\equiv 1\mod 4\\\frac{1}{2} + \frac{2}{3n} & n\equiv 2\mod 4\\\frac{1}{2} - \frac{5}{6n} & n\equiv 3\mod 4}\right.$$ so even if Simpson's rule has higher order we see that it does not always do better than the trapezoidal rule. $$\bf \text{What does higher degree of precision really mean?}$$ If we have a smooth function then a standard Taylor series error analysis gives that the error in estimating the integral $\int_a^bf(x){\rm d}x$ using $n$ equally spaced points is bounded by (here for Simpsons and the trapezoidal rule) $$\epsilon_{\rm Simpsons} = \frac{(b-a)^5}{2880n^4}\max_{\zeta\in[a,b]}|f^{(4)}(\zeta)|$$ $$\epsilon_{\rm Trapezoidal} = \frac{(b-a)^3}{12n^2}\max_{\zeta\in[a,b]}|f^{(2)}(\zeta)|$$ Note that the result we get from such an error analysis is always an upper bound (or in some cases an order of magnitude) for the error apposed to the exact value for the error. What this error analysis tell us is that if $f$ is smooth on $[a,b]$, so that the derivatives are bounded, then the error with a higher order method will tend to decrease faster as we increase the number of gridpoints and consequently we typically need fewer gridpoints to get the same accuracy with a higher order method. The order of the method only tell us about the $\frac{1}{n^k}$ fall-off of the error and says nothing about the prefactor in front so a method that has an error of $\frac{100}{n^2}$ will tend to be worse than a method that has an error $\frac{1}{n}$ as long as $n\leq 100$. $$\bf \text{Why do we need all these methods?}$$ In principle we don't need any other methods than the simplest one. If we can compute to arbitrary precision and have enough computation power then we can evaluate any integral with the trapezoidal rule. However in practice there are always limitations that in some cases forces us to choose a different method. Using a low-order method requires many gridpoints to ensure good enough accuracy which can make the computation take too long time especially when the integrand is expensive to compute. Another problem that can happen even if we can afford to use as many gridpoints as we want is that truncation error (errors due to computers using a finite number of digits) can come into play so even if we use enough points the result might not be accurate. Other methods can elevate these potential problems. Personally, whenever I need to integrate something and has to implement the method myself I always start with a low-precision method like the trapezoidal rule. This is very easy to implement, it's hard to make errors when coding it up and it's usually good enough for most purposes. If this is not fast enough or if the integrand has properties (e.g. rapid osccilations) that makes it bad I try a different method. For example I have had to compute (multidimensional) integrals where a trapezoidal rule would need more than a year to compute it to good enough accuracy, but with Monte-Carlo integration the time needed was less than a minute! It's therefore good to know different numerical integration methods in case you encounter a problem where the simplest method fails. • Your answer is very impressive. Thanks a lot. You have given an example of $f(x)$. It is not differentiable at x=1/2. For such non-smooth function, the Trap is better than Simp. Is there any example where $f(x)$ is continuous and smooth but Trap gives better accuracy than Simp? – user1942348 Jan 4 '16 at 13:50 • @user1942348 Yes there is. For example take $f(x)$ and connect $x = 1/2 - \epsilon$ to $x = 1/2 + \epsilon$ with a line (smoothed at the corners). Take $\epsilon$ to be tiny and you'll get pretty much the same result as I got here. – Winther Jan 4 '16 at 14:21 • I just googled and see that for periodic function with limits of integration one with its period, Trap works better. self.gutenberg.org/articles/trapezoidal_rule Do you know is this correct. Can you give such example for me? – user1942348 Jan 4 '16 at 15:55 • @user1942348 I think the statement on that page is that Trap does better than what we naively would expect from looking at the error term for functions that oscillate, not that it neccesarily is much better than say Simpsons. I tried some random test cases like $\sin(2\pi x)$ and $[x(1-x)]^4$ and Trap does indeed do better than expected and the acctual error as function of $n$ is similar to Simpsons for these two functions ($\epsilon \propto 1/n^6$ for the latter function). – Winther Jan 4 '16 at 16:49 • @user1942348 btw let me know if you have Mathematica and I can add the code I used to test this so you can play with it and see for yourself. – Winther Jan 4 '16 at 16:53

https://mathhelpboards.com/threads/find-all-pairs-m-k-of-integer-solutions.6882/

Find all pairs (m,k) of integer solutions anemone MHB POTW Director Staff member Find all solutions to $(m^2+k)(m+k^2)=(m-k)^3$, where $m$ and $k$ are non-zero integers. Opalg MHB Oldtimer Staff member Find all solutions to $(m^2+k)(m+k^2)=(m-k)^3$, where $m$ and $k$ are non-zero integers. Multiply out the brackets: $$\displaystyle m^3 + m^2k^2 + mk + k^3 = m^3 -3m^2k + 3mk^2 - k^3$$, which (after dividing by the non-zero $k$) simplifies to $2k^2 + (m^2-3m)k + (m+3m^2) = 0.$ The discriminant of that quadratic in $k$ is $(m^2-3m)^2 - 8(m+3m^2)$, and that has to be a square, say $n^2$. So $$\displaystyle n^2 = (m^2-3m)^2 - 8(m+3m^2) = m^4 - 6m^3 - 15m^2 - 8m = m(m-8)(m+1)^2$$ and therefore $m(m-8) = (m-4)^2 - 16$ must also be a square. The only way that can happen is if $m-4 = \pm5$. Thus $m = 9$ or $-1$. If $m=9$ then the quadratic for $k$ has solutions $k= -21$ and $k=-6$. If $m=-1$ then the only solution for $k$ is $k=-1$. That gives three possible solutions, $(m,k) = (9,-21),\ (9,-6),\ (-1,-1)$. Klaas van Aarsen MHB Seeker Staff member therefore $m(m-8) = (m-4)^2 - 16$ must also be a square. The only way that can happen is if $m-4 = \pm5$. Why is that? Opalg MHB Oldtimer Staff member therefore $m(m-8) = (m-4)^2 - 16$ must also be a square. The only way that can happen is if $m-4 = \pm5$. Why is that? If $x^2-16 = y^2$ then $16 = x^2-y^2 = (x+y)(x-y)$. The only factorisation of $16$ giving positive integer values for $x$ and $y$ is $x+y=8$, $x-y=2$, so that $x=5$ and $y=3$. anemone MHB POTW Director Staff member Thank you so much for participating, Opalg! And welcome back to the forum as well! But Opalg, I am sorry to tell you that you missed another solution set, which is $(m, k)=(8, -10)$. My solution: By expanding and simplifying the given equation, we get $$\displaystyle m^3+m^2k^2+mk+k^3=m^3-3m^2k+3mk^2-k^3$$ $$\displaystyle 2k^3+(m^2-3m)k^2+mk+3m^2k=0$$ Since $k \ne 0$, divide through the equation above by $k$, we have $$\displaystyle 2k^2+(m^2-3m)k+m+3m^2=0$$ and solve for $k$ using the quadratic formula yields $$\displaystyle k=\frac{-(m^2-3m) \pm \sqrt{(m^2-3m)^2-4(2)(m+3m^2)}}{2(2)}$$ $$\displaystyle \;\;\;=\frac{3m-m^2 \pm \sqrt{(m+1)^2(m(m-8))}}{4}$$ $$\displaystyle \;\;\;=\frac{3m-m^2 \pm (m+1)\sqrt{m(m-8)}}{4}$$ Recall that the expression inside the radical must be greater than or equal to zero, thus, $$\displaystyle m(m-8)\ge 0$$ and this gives $m\le 0$ and $m \ge 8$. Also, bear in mind that we're told $m, k$ are both integer values, so, $m(m-8)$ has to be a squre. But if we apply the AM-GM inequality to the terms $m$ and $m-8$, we find $$\displaystyle \frac{m+m-8}{2} \ge \sqrt{m(m-8)}$$ $$\displaystyle m-4 \ge \sqrt{m(m-8)}$$ $m\ge 8$ implies $m-4 \ge 4$ and this further implies $\sqrt{m(m-8)} \le 4$ and now, we will solve for $m$ by considering $\sqrt{m(m-8)}=0$ or $1$ or $2$ or $3$or $4$ and we will take whichever solution that gives the integer values of $m$. $$\displaystyle \sqrt{m(m-8)}= 4$$ gives irrational $m$ values. $$\displaystyle \sqrt{m(m-8)}= 3$$ gives $m=-1, 9$. $$\displaystyle \sqrt{m(m-8)}= 2$$ gives irrational $m$ values. $$\displaystyle \sqrt{m(m-8)}= 1$$ gives irrational $m$ values. $$\displaystyle \sqrt{m(m-8)}= 0$$ gives irrational $m=0, 8$ values but $m>0$, hence $m=8$. Thus, by substituting each of the value of $m$ to the equation (*) above to find for its corresponding $k$ value, we end up with the following 4 solution sets, namely $(m, k)=(-1, -1), (8, -10), (9, -6), (9, -21)$ Opalg MHB Oldtimer Staff member therefore $m(m-8) = (m-4)^2 - 16$ must also be a square. The only way that can happen is if $m-4 = \pm5$. Why is that? Oops, I should have taken I like Serena's query more seriously. I was thinking that the only way you could have $x^2-16 = y^2$ was in the situation $5^2 - 4^2 = 3^2$. But in the solution to this problem, the situation $4^2 - 4^2 = 0^2$ is also possible. That leads to the solution $m=8$ and $k=-10$.

https://mathhelpboards.com/threads/review-my-solution-trigonometry-proof.5778/

TrigonometryReview my solution: Trigonometry proof sweatingbear Member For a triangle with sides $$\displaystyle a$$, $$\displaystyle b$$, $$\displaystyle c$$ and angle $$\displaystyle C$$, where the angle $$\displaystyle C$$ subtends the side $$\displaystyle c$$, show that $$\displaystyle c \geqslant (a+b) \sin \left( \frac C2 \right)$$ _______ Let us stipulate $$\displaystyle 0^\circ < C < 180^\circ$$ and of course $$\displaystyle a,b,c > 0$$. Consequently, $$\displaystyle 0^\circ < \frac C2 \leqslant 90^\circ \implies 0 < \sin^2 \left( \frac C2 \right) \leqslant 1$$ (we include $$\displaystyle 90^\circ$$ for the angle $$\displaystyle \frac C2$$ in order to account for right triangles). Law of cosines yield $$\displaystyle c^2 = a^2 + b^2 - 2ab \cos ( C )$$ Using $$\displaystyle \cos (C) \equiv 1 - 2\sin^2 \left( \frac C2 \right)$$, we can write $$\displaystyle c^2 = a^2 + b^2 - 2ab + 4ab\sin^2 \left( \frac C2 \right)$$ Now since $$\displaystyle 0 < \sin^2 \left( \frac C2 \right) \leqslant 1$$, $$\displaystyle a^2 + b^2 - 2ab + 4ab\sin^2 \left( \frac C2 \right)$$ will either have to equal $$\displaystyle a^2 + b^2 - 2ab + 4ab$$ or be greater than it. Thus $$\displaystyle c^2 \geqslant a^2 + b^2 - 2ab + 4ab = a^2 + 2ab + b^2 = (a+b)^2$$ A similar argument can be made for $$\displaystyle (a+b)^2$$ versus $$\displaystyle (a+b)^2 \sin^2 \left( \frac C2 \right)$$: $$\displaystyle (a+b)^2$$ will either have to equal or be greater than the latter expression, due to the values $$\displaystyle \sin^2 \left( \frac C2 \right)$$ can assume. Therefore $$\displaystyle c^2 \geqslant (a+b)^2 \geqslant (a+b)^2 \sin^2 \left( \frac C2 \right)$$ We can finally conclude $$\displaystyle c \geqslant (a+b) \sin \left( \frac C2 \right)$$ Thoughts? MarkFL Staff member I see nothing wrong with your proof. This is how I would prove it. Please refer to the following diagram: I would postulate concerning the angle bisector $m$ of $\angle C$ and the altitude $h$, we must have: (1) $$\displaystyle m\ge h$$ That is, the shortest distance between a point and a line is the perpendicular distance. Now, the area $T$ of the triangle may be written in these ways: $$\displaystyle T=\frac{1}{2}ch=\frac{1}{2}m(a+b)\sin\left(\frac{C}{2} \right)$$ Note: I have made use of the formulas (for a general triangle): $$\displaystyle T=\frac{1}{2}bh$$ $$\displaystyle T=\frac{1}{2}ab\sin(C)$$ Thus, we have: $$\displaystyle ch=m(a+b)\sin\left(\frac{C}{2} \right)$$ And from (1), we therefore conclude: $$\displaystyle c\ge (a+b)\sin\left(\frac{C}{2} \right)$$ sweatingbear Member I see nothing wrong with your proof. This is how I would prove it. Please refer to the following diagram: View attachment 1085 I would postulate concerning the angle bisector $m$ of $\angle C$ and the altitude $h$, we must have: (1) $$\displaystyle m\ge h$$ That is, the shortest distance between a point and a line is the perpendicular distance. Now, the area $T$ of the triangle may be written in these ways: $$\displaystyle T=\frac{1}{2}ch=\frac{1}{2}m(a+b)\sin\left(\frac{C}{2} \right)$$ Note: I have made use of the formulas (for a general triangle): $$\displaystyle T=\frac{1}{2}bh$$ $$\displaystyle T=\frac{1}{2}ab\sin(C)$$ Thus, we have: $$\displaystyle ch=m(a+b)\sin\left(\frac{C}{2} \right)$$ And from (1), we therefore conclude: $$\displaystyle c\ge (a+b)\sin\left(\frac{C}{2} \right)$$ Thank for the feedback and the excellent alternative solution! Deveno Well-known member MHB Math Scholar A good rule of thumb in mathematics I live by is this: If something is true, you should be able to provide two proofs, and one of them should have a picture. This is a perfect example of what I mean.

http://mathhelpforum.com/discrete-math/93273-divide-3-a.html

# Math Help - divide by 3? 1. ## divide by 3? hi folks, I hope this is the right place to post this! I have no idea how to proceeed with the following question: Show that for all positive integral values of n [HTML]7 ⁿ + 2 ²ⁿ⁺¹[/HTML] is divisible by 3. I tried a few terms as follows: n = 1. 7 ^ 1 + 2 ^ 3 = 7 + 8 = 15 n = 2. 7 ^ 2 + 2 ^ 5 = 49 + 32 = 81 n = 3. 7 ^ 3 + 2 ^ 7 = 343 + 128 = 471 and these are all divisible by 3 but how do I handle the general case? I thought of expanding the expression i.e. 2.2 ^ n. 2 ^ n - (1 - 8) ^ n and using a binomial on the second term but it doesn't get me anywhere. I guess I am trying to calculate the sum of a series and show that it has a factor of 3 but I can't see how to do it. Any ideas? regards and thanks Simon p.s. sorry about the formating. the HTML stuff doesn't seem to come out so I resorted to the ^ symbol which is pretty unpretty! 2. Hi, Originally Posted by s_ingram I thought of expanding the expression i.e. 2.2 ^ n. 2 ^ n - (1 - 8) ^ n and using a binomial on the second term but it doesn't get me anywhere. I guess I am trying to calculate the sum of a series and show that it has a factor of 3 but I can't see how to do it. Any ideas? I suggest writing $7^n+2^{2n+1}=(2^2+3)^n+2^{2n+1}$ and then using the binomial theorem to expand $(2^2+3)^n$. Originally Posted by s_ingram p.s. sorry about the formating. the HTML stuff doesn't seem to come out so I resorted to the ^ symbol which is pretty unpretty! To enter math. equations you can use LaTeX. (see http://www.mathhelpforum.com/math-help/latex-help/) 3. $(2^{2}+3)^{n} = 2^{2n} + \binom{n}{1}2^{2(n-1)}3^{1} + ... + \binom{n}{k}2^{2(n-k)}3^{k} + ... + 3^{n} =$ $= 2^{2n} + 3[\binom{n}{1}2^{2(n-1)} + ... + \binom{n}{k}2^{2(n-k)}3^{k-1} + ... + 3^{n-1}]$ So we can write: $3[\binom{n}{1}2^{2(n-1)} + ... + \binom{n}{k}2^{2(n-k)}3^{k-1} + ... + 3^{n-1}] + 2^{2n}(1+2) =$ $3[\binom{n}{1}2^{2(n-1)} + ... + \binom{n}{k}2^{2(n-k)}3^{k-1} + ... + 3^{n-1}] + 2^{2n}(3)$ Now factor out a 3 from both terms. $3[(\binom{n}{1}2^{2(n-1)} + ... + \binom{n}{k}2^{2(n-k)}3^{k-1} + ... + 3^{n-1}) + 2^{2n} ]$ And because we have a number times 3, the result is divisible by 3. Note: If there aint a careless mistake somewhere in this mess, I am surprised. 4. thanks to Twig and flyingsquirrel. You guys make it look so easy! 5. If you do it by induction here is the last step. $\begin{array}{rcl} {7^{n + 1} + 2^{2n + 3} } & = & {7^{n + 1} + 7 \cdot 2^{2n + 1} - 7 \cdot 2^{2n + 1} + 2^{2n + 3} } \\ {} & {} & {7\left( {7^n + 2^{2n + 1} } \right) - 2^{2n + 1} \left( {7 - 2^2 } \right)} \\ {} & {} & {7\left( {7^n + 2^{2n + 1} } \right) - 2^{2n + 1} \left( 3 \right)} \\ \end{array}$ 6. Hello, s_ingram! How about an inductive proof? Show that for any positive integer $n \!:\;\;7^n + 2^{2n+1}$ is divisible by 3. Verify $S(1)\!:\;\;7^1 + 2^3 \:=\:7+8 \:=\:15$ ... divisible by 3. Assume $S(k)\!:\;\;7^k + 2^{2k+1} \:=\:3a\;\text{ for some integer }a.$ Add $6\!\cdot\!7^k + 3\!\cdot\!2^{2k+1}$ to both sides: . . $7^k + {\color{blue}6\!\cdot\!7^k} + 2^{2k+1} + {\color{blue}3\!\cdot\!2^{2k+1}} \;=\;3a + {\color{blue}6\!\cdot\!7^k + 3\!\cdot\!2^{2k+1}}$ . . $(1 + 6)\!\cdot\!7^k + (1 + 3)\!\cdot\!2^{2k+1} \;=\;3\left(2a + 2\!\cdot\!7^k + 2^{2k+1}\right)$ . a multiple of 3 . . . . . . . . $7\!\cdot\!7^k + 2^2\!\cdot2^{2k+1} \;=\;3b\;\;\text{ for some integer }b$ Therefore: . . $7^{k+1} + 2^{2k+3} \;=\;3b$ We have proved $S(k+1).$ . . The inductive proof is complete. 7. Hi Soroban, now that is a smart proof. Too smart! How did you think adding those two terms? Once you see them it all fits but finding them is the real trick! I have never really been impressed with proofs by induction, but I am now! Usually I just try to add the next term in the series and ensure that it has a corresponding impact on the expression fo the sum but with 7 sup k+1 + 2 sup 2k+1 I couldn't even imagine what the next term would be! 8. Or you can just note that $7^n\equiv 1^n \equiv 1 \mod 3$, $2^{2n+1} \equiv (-1)^{2n+1} \equiv -1 \mod 3$, so that $7^n+2^{2n+1} \equiv 0 \mod 3$. 9. Originally Posted by s_ingram Hi Soroban, now that is a smart proof. Too smart! How did you think adding those two terms? Once you see them it all fits but finding them is the real trick! I have never really been impressed with proofs by induction, but I am now! Usually I just try to add the next term in the series and ensure that it has a corresponding impact on the expression fo the sum but with 7 sup k+1 + 2 sup 2k+1 I couldn't even imagine what the next term would be! Hi $7^{n+1} + 2^{2n+3} = 7(7^{n} + 2^{2n+1})-7 \cdot 2^{2n+1} + 2^{2n+3} = 7 \cdot 3a + 2^{2n+1}(-7+2^2) = 7 \cdot 3a -3 \cdot 2^{2n+1}$ Therefore $7^{n+1} + 2^{2n+3} = 3(7a-2^{2n+1})$

https://mathhelpboards.com/threads/another-solid-of-revolution-problem.1116/

# [SOLVED]Another Solid of Revolution problem ##### Member I'm finding myself stuck again. This time it is more in the set up then the solving. Find the volume of the following solid of revolution. The region bounded by $$\displaystyle y=\frac{1}{x^2}$$, y=0, x=2, x=8, and revolved about the y-axis. I am trying to use the shell method to solve this, as it seems the best scenario in this situation. This is my set-up, $$\displaystyle 2\pi\int_2^8 (radius)(height)dx$$ $$\displaystyle 2\pi\int_2^8 (x+2)(\frac{1}{x^2})dx$$ $$\displaystyle 2\pi\int_2^8 (\frac{1}{x}+\frac{2}{x^2})dx$$ Now here is where I am getting into a snag. Following this through I am coming up with a completely different solution to the manual, and the solution manual is showing the first integration step as $$\displaystyle 2\pi\int_2^8 (\frac{1}{x})dx$$, but I can't figure out how they are coming up with that integral. It is showing a final solution of $$\displaystyle \pi\ln(16)$$ Any help, or a kick in the right direction, would be greatly appreciated. Mac #### Chris L T521 ##### Well-known member Staff member [snip] Now here is where I am getting into a snag. Following this through I am coming up with a completely different solution to the manual, and the solution manual is showing the first integration step as $$\displaystyle 2\pi\int_2^8 (\frac{1}{x})dx$$, but I can't figure out how they are coming up with that integral. It is showing a final solution of $$\displaystyle \pi\ln(16)$$ Any help, or a kick in the right direction, would be greatly appreciated. Mac [/snip] That's because the radius of your solid of revolution is just $x$, not $x+2$. $x$ is always measured from the $y$-axis (i.e. $x=0$), so there's no need to add 2 to this value; in this case, $x$ just happens to be between 2 and 8. On the other hand, the radius would be $x+2$ if you were revolving the region about the line $x=-2$. I hope this clarifies things! #### CaptainBlack ##### Well-known member I'm finding myself stuck again. This time it is more in the set up then the solving. Find the volume of the following solid of revolution. The region bounded by $$\displaystyle y=\frac{1}{x^2}$$, y=0, x=2, x=8, and revolved about the y-axis. I am trying to use the shell method to solve this, as it seems the best scenario in this situation. This is my set-up, $$\displaystyle 2\pi\int_2^8 (radius)(height)dx$$ $$\displaystyle 2\pi\int_2^8 (x+2)(\frac{1}{x^2})dx$$ $$\displaystyle 2\pi\int_2^8 (\frac{1}{x}+\frac{2}{x^2})dx$$ Now here is where I am getting into a snag. Following this through I am coming up with a completely different solution to the manual, and the solution manual is showing the first integration step as $$\displaystyle 2\pi\int_2^8 (\frac{1}{x})dx$$, but I can't figure out how they are coming up with that integral. It is showing a final solution of $$\displaystyle \pi\ln(16)$$ Any help, or a kick in the right direction, would be greatly appreciated. Mac Where did a radius of $$x+2$$ come from? Why not $$x$$? CB

http://math.stackexchange.com/questions/56688/please-help-me-to-prove-the-convergence-of-gamma-n-1-frac12-frac1

Please help me to prove the convergence of $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}-\ln n$ Please help me to prove that $\gamma_{n} = 1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{n}-\ln n$ converge and it's limit $\gamma \in [0,1]$. Then, using $\gamma_{n}$ find the sum: $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{1}{n}$. - Welcome to MSE! I'd suggest though you'd ask question in a less imperative way and in a more friendly way. Plus, have you managed to find something? Any approaches? What I mean is, you're not a book, you're human, so act like one, it makes us more willing to answer. =) –  Patrick Da Silva Aug 10 '11 at 11:49 The following question addresses the convergence –  Sasha Aug 10 '11 at 11:56 possible duplicate of What is $\lim_{n \to \infty}\sum_{k=1}^n 1/k$ –  user17762 Aug 10 '11 at 12:14 Sorry because my question was imperative. My english is not very good and I translated word by word from my language. I will be more carefully next time. –  NumLock Aug 10 '11 at 12:28 In the sense that if you read the answer given, all the good stuff's in there. Although they don't compute the alternated sum. I don't agree it's a duplicate. –  Patrick Da Silva Aug 10 '11 at 12:56 The following question addresses the convergence, so assume $\gamma_n$ converges and denote the limit as $\gamma$. Now consider $\sum_{n=1}^{2m} (-1)^{n+1} \frac{1}{n}$ and split it into even and odd terms $\sum_{n=1}^m \frac{1}{2n-1} - \sum_{n=1}^m \frac{1}{2n}$. Then complete the sum over odd integers to sum over consequtive integers: $\sum_{n=1}^{2m} \frac{1}{n} - 2 \sum_{n=1}^m \frac{1}{2n}$. Then subtract logarithms to form $\gamma_{2m} - 2 \gamma_m + \log(2)$, like so $$( \sum_{n=1}^{2m} \frac{1}{n} - \log(2m)) - ( \sum_{n=1}^m \frac{1}{n} - \log(m)) + \log(2).$$ In the limit $m \to \infty$ it becomes $\gamma - \gamma + \log(2) = \log(2)$. - Thank you for your help too! –  NumLock Aug 10 '11 at 12:34 You can use the mean value theorem to prove first that $$\frac{1}{n+1}<\ln(n+1)-\ln n<\frac{1}{n}$$ then, $\ln(n+1)<1+\ldots+\frac{1}{n}<1+\ln n$, and finally $$\ln(n+1)-\ln n<\gamma_n<1$$ which implies the convergence of $\gamma_n$ and its limit $\gamma\in[0,1]$. For the second part, I am not sure if you can use $\gamma_n$ to find the sum $\sum_{n=1}^\infty(-1)^{n+1}\frac{1}{n}$! I suggest to use the Taylor-Young formula for $x\mapsto\ln(x+1)$ on $[0,1]$ to prove that $$\sum_{n=1}^\infty(-1)^{n+1}\frac{1}{n}=\ln2.$$ - Thank you !I learnt mean value theorem in highschool but I completely forgot about it. –  NumLock Aug 10 '11 at 12:34 @amine it's mercator series time... –  user38268 Aug 10 '11 at 12:58 The Mean Value Theorem says $$\frac{1}{k+1}<\log(k+1)-\log(k)<\frac{1}{k}$$ Summing from $m$ to $n-1$ ($m < n$), we get \begin{align} \sum_{k=m}^{n-1}\frac{1}{k+1}<\log(n)-\log(m)<\sum_{k=m}^{n-1}\frac{1}{k} \end{align} Subtracting $\displaystyle{\sum_{k=m}^{n-1}\frac{1}{k+1}=\sum_{k=m+1}^n\frac{1}{k}=\sum_{k=1}^n\frac{1}{k}-\sum_{k=1}^m\frac{1}{k}}$ from all parts, we get $$0<\left(\log(n)-\sum_{k=1}^n\frac{1}{k}\right)-\left(\log(m)-\sum_{k=1}^m\frac{1}{k}\right)<\frac{1}{m}-\frac{1}{n}$$ This proves the existence of $\displaystyle{\lim_{n\to\infty}\left(\log(n)-\sum_{k=1}^n\frac{1}{k}\right)}$. If we set $m=1$ and subtract all sides from $1$, we get $$1>\left(\sum_{k=1}^n\frac{1}{k}-\log(n)\right)>\frac{1}{n}$$Thus, we have shown that $\displaystyle{\gamma=\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k}-\log(n)\right)}$ exists, and that $0\le\gamma\le 1$. Note that \begin{align} \sum_{k=1}^{2n}(-1)^{k+1}\frac{1}{k}&=\sum_{k=1}^{2n}\frac{1}{k}-2\sum_{k=1}^n\frac{1}{2k}\\ &=\sum_{k=1}^{2n}\frac{1}{k}-\sum_{k=1}^n\frac{1}{k}\\ &=\left(\sum_{k=1}^{2n}\frac{1}{k}-\log(2n)\right)-\left(\sum_{k=1}^n\frac{1}{k}-\log(n)\right)+\log(2) \end{align} Taking the limit as $n\to\infty$, we get $$\sum_{k=1}^\infty(-1)^{k+1}\frac{1}{k}=\gamma-\gamma+\log(2)=\log(2)$$ - I apologize if this is close to Sasha's answer, but I have been working on this, during breaks, for quite a while, and posted it before I noticed. Since I show more of the details, I will leave it for now. I can remove it if requested. –  robjohn Aug 10 '11 at 17:26 Sorry, for the convergence, I forgot to say that $\gamma_n$ is decreasing because $$\gamma_{n+1}-\gamma_n=\frac{1}{n+1}-(\ln(n+1)-\ln n)<0$$ since it is bounded, it is convergent! –  amine Aug 10 '11 at 21:49 Do we have a "proof without words" question here? I don't find it. There is a lovely diagram found in certain calculus texts that does this for us. Maybe I will animate it, for fun. - I am not getting this. Do you mean pictorial proof, which suggests the proof ? –  Sasha Aug 10 '11 at 14:15 It's something like the diagram at the top right here: en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant but then you slide all the blue regions to the left to see that they all fit inside a $1 \times 1$ square. –  GEdgar Aug 10 '11 at 14:54

http://mathhelpforum.com/algebra/164881-word-problem.html

# Math Help - word problem. 1. ## word problem. hi ! im having a difficult time answering this word problem. "Alvin set out from a certain point and travelled at the rate of 6 kph. After Alvin had gone for two hours, Ben set on to overtake him and went 4km the first hour, 5 km the second hour, 6 km the third hour and so on, gaining 1 km every hour. After many hours were they together." I keep on getting 5 using arithmetic progression. Could someone please please give me a hint how to solve this one ??? Thank you very much !! You can check: In 5 hours, Alvin travels : 6+6+6+6+6 = 30 kms In 5 hours, Ben travels: 4+5+6+7+8=30kms that means they were together after 5 hours. 3. Nope. Ben started 2 hours after 4. It should be 10 hours. But I don't know how to prove it using arithmetic progression. 5. Here's what I did. We know these two things about arithmetic progressions: $a_n = a_1 + (n - 1)d$ where d is the distance between numbers in the sequence. We also know that: $S_n = \frac{n}{2}(a_1 + a_n)$ where $S_n$ is the sum of the first n terms. We also know that the distance traveled by the first is just 6t. I'll use t instead of n in the formulas. All t's for the second one should be (t - 2) since he doesn't move for 2 hours. So we want to find out when 6t equals the sum of the arithmetic progression at (t - 2). I set up this equation: $6t = \frac{t-2}{2} (a_1 + a_n) = \frac{t-2}{2} (4 + a_1 + ((t - 2) -1))$ $6t = \frac{t - 2}{2}(4 + 4 + (t - 2) - 1)$ Hope you see what I did there. After that, rearrange terms so you get: $t^2 - 9t - 10 = 0$ For that, I get one positive root of t = 10 (also t = -1 which is obviously not useful). Voila! Thanks for the problem, I had fun with that. 6. Hello, dugongster! Alvin set out from a certain point and travelled at the rate of 6 kph. After Alvin had gone for 2 hours, Ben set on to overtake him and went 4km the first hour, 5 km the second hour, 6 km the third hour and so on, gaining 1 km every hour. After many hours were they together? . . In the next $\,h$ hours, he travels another $6h$ km. Alvin has traveled $6h + 12$ km in the first $\,h$ hours. During the same $\,h$ hours, Ben has travelled: . . $4 + 5 + 6 + \hdots + (h+3) \:=\:\dfrac{h(h+7)}{2}$ km. .** The two distances are equal: . $\dfrac{h(h+7)}{2} \:=\:6h + 12$ . . $h^2 + 7h \:=\:12h + 24 \quad\Rightarrow\quad h^2 - 5h - 24 \:=\:0$ . . $(h - 8)(h + 3) \:=\:0 \quad\Rightarrow\quad h \:=\:8,\:-3$ Answer: . $\text{8 hours}$ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ** We want the sum of the numbers from $\,4$ to $h\!+\!3.$ The sum of the numbers from $\,1$ to $h\!+\!3$ is: . $\dfrac{(h+3)(h+4)}{2}$ The sum of the numbers from $\,1$ to $\,3$ is: . $1+2+3 \:=\:6$ Hence,the sum of the numbers from $\,4$ to $h\!+\!3$ is: . . $\dfrac{(h+3)(h+4)}{2} - 6 \;\;=\;\; \dfrac{h^2 + 7x + 12 - 12}{2} \;\;=\;\;\dfrac{h(h+7)}{2}$ 7. great. thanks to both of you. grep, you rock !!! 8. Originally Posted by Soroban Hello, dugongster! Hi Soroban! With due respect (which is considerable, I might add, you being one of the most helpful and, let's face it, awesome people here), I took into account the 2 hour headstart by using (t - 2) as the time in the arithmetic progression (n = (t - 2)), and starting at 4 ( $a_1 = 4$).

https://math.stackexchange.com/questions/3140502/do-the-12-lines-of-a-bingo-card-have-equal-chance-of-winning

# Do the 12 lines of a bingo card have equal chance of winning? You have a $$5 \times 5$$ bingo card filled with $$25$$ distinct numbers, one per square. There is also a pot containing each number once and you draw them out one by one without replacement. A line is any of the $$5$$ rows, $$5$$ columns, or $$2$$ main diagonals. A line is completed when its $$5$$ numbers have been drawn. A line wins if it is the first line to be completed. Question: Do the $$12$$ lines have equal chance of winning? If not, which lines have higher chance of winning? Why I ask: Define $$T_l$$ to be the time to completion for line $$l$$. It is obvious that all $$T_l$$ are equi-distributed, and thus all $$E[T_l]$$ are equal. However, because the lines overlap, the different $$T_l$$'s are dependent, and it is not clear to me that they have equal chances of being first. In particular, imagine that the $$5$$-subsets are not arranged in rows, columns and diagonals, but are clustered in some non-uniform way. Then a subset that shares a lot of elements with other subsets might have a lower chance to win than a subset that does not share a lot of elements with other subsets. (I can provide a simple example if there is interest.) On the bingo card, the $$5$$-subsets are pretty uniform but not exactly uniform, due to the diagonals. So my suspicion is that the line win chances are almost equal but not exactly equal. And I am curious as to which lines have higher chances. I imagine (but would be happy to be proven wrong) that calculating the exact win prob for a line might be difficult/tedious, so qualitative arguments based on e.g. symmetry are also welcome. Clarifications: A drawn number can complete multiple lines, so that needs some special handling. However what I am interested in is the question of equality, so I will accept any reasonable way to handle such "shared" wins, i.e. if $$N>1$$ lines are completed at the same draw (and no line has been completed before this draw), then you can treat this as if... • they all win (in which case the sum of the $$12$$ win probabilities exceed $$1$$, but that doesn't matter since I am interested in which are higher/lower), or, • the whole experiment is repeated from the beginning (i.e. we condition on such shared wins not happening), or, • you flip an $$N$$-sided die to determine the winner (i.e. this effectively counts as $$1/N$$ win for each involved line), etc. • After this, you can do BINGO cards with a "free" space in the middle. – GEdgar Mar 8 at 20:37 • I don't see why you would think any particular line would have a higher chance of winning if there is an equal probability of each square being picked every time. A line requires 5 numbers being picked and the probability of each 5 number combinations are exactly the same. – Matthew Liu Mar 8 at 20:53 • @MatthewLiu Because when one line gets a number, it also helps other lines. The diagonal are at a disadvantage. Look at my solution for $3\times3$ bingo. – saulspatz Mar 8 at 21:24 • @MatthewLiu what you said is why all the $T_l$ are identically-distributed, but they are not independent. A line $L$ wins if $T_L < T_l \forall l \neq L$. – antkam Mar 8 at 22:15 • Ok you guys are correct. The diagonal lines are at a strict disadvantage. – Matthew Liu Mar 11 at 17:14 I must admit, when I read the title of the question, I thought "Of course they all have an equal chance of winning. What a silly question." But then, I read the body, and was delighted to see that you had a valid point, and I was the silly one. I don't know how to solve for $$5\times5$$ bingo in any reasonable time, but for $$3\times3$$ bingo we have only $$9!$$ permutations of the numbers to consider. I wrote a python script to exhaustively test this. from itertools import permutations from fractions import Fraction score = 9*[0] card = 8*[0] for p in permutations(range(9)): card[0] = {0,1,2} card[1] = {3,4,5} card[2] = {6,7,8} card[3] = {0,3,6} card[4] = {1,4,7} card[5] = {2,5,8} card[6] = {0,4,8} card[7] = {2,4,6} for number in p: for line in card: line -= {number} winners = len([line for line in card if not line]) if winners: for idx, line in enumerate(card): if not line: score[idx] += Fraction(1, winners) break for idx in range(8): print(idx, score[idx]) This produced the output 0 47520 1 45600 2 47520 3 47520 4 45600 5 47520 6 40800 7 40800 This means that the diagonals are worst, and the lines along the edges of the card are best, with the horizontal and vertical lines through the center in the middle. I should mention that in the case of ties, I awarded an equal fraction of the game to each winner, so that the sum of the scores does work out to $$9!.$$ • What does it produce if you give a full win to each line if 2/3/4 lines win at once? – Banbadle Mar 8 at 21:58 • awesome! i had thought the diagonals are worst but all the rows and columns are equal, but you showed that the row/col passing through the center are penalized. interesting... and now i wonder if the 1st & 2nd rows are different in the 5x5. ooh, yeah, i would like to see the idea of @Banbadle tested too. would the multiple full wins cancel the effect of sharing? – antkam Mar 8 at 22:00 • @Banbadle I just modified saulspatz's code to try other scoring modes. If a full win is given to each simultaneously winning line, the win counts are 54000 > 53290 > 46656 for the edges, center row/col, diagonals respectively. If we condition on no shared wins (i.e. those permutations award no wins to anybody), then the win counts are 41040 > 38880 > 35424, again for edge, center row/col, diagonals. So the order is the same in all 3 win-counting models. – antkam Mar 8 at 22:44 I simulated a standard bingo game, one million games with ten random cards for each game. I recorded the number of times each line triggered bingo, including bingo on multiple cards and multiple lines winning simultaneously on the same card. The results: • Horizontal through center wins $$16.2\%$$ of games • Vertical through center wins $$16.1\%$$ of games • Diagonals each win $$15.9\%$$ of games • Other horizontal lines each win $$5.8\%$$ of games • Other vertical lines each win $$5.6\%$$ of games Note that this totals over $$100\%$$, accounting for multiple wins in a single game (there were nearly $$1.1$$ million wins in one million games) After ensuring that these results are indeed accurate and giving some thought to why the diagonals win less than the central row and column, it is clear that this is due to intersecting lines. Each of the squares on the diagonals, when drawn, has the potential to trigger a win on either of two intersecting lines, while each square in the central row or column can only trigger a win on a single intersecting line. I have also run the simulation without the free center square: • Rows win $$9.42\%$$ • Diagonals win $$9.17\%$$ • Columns win $$9.05\%$$ • when you say "standard bingo game" do you mean the center square is free (i.e. already marked off)? if so that must be why lines through center win much more coz they only require 4 cells. – antkam Mar 8 at 22:04 • @antkam Yes, center square is free in my simulation. – Daniel Mathias Mar 8 at 22:05 • Even in the free-center case it is open (at least, in my mind) whether the diagonals are at a disadvantage. If you found that diagonals win 10% vs the center row/col winning 20% each, that would be quite conclusive. But at 15.9.% vs 16.1% I am not sure this is a real effect or just within the error bar. – antkam Mar 8 at 22:18 • @antkam A run of ten million games: $16.18\%$, $16.09\%$, $15.91\%$, $15.90\%$ for center row, center column, downward diagonal, upward diagonal. Of course, this only proves that my simulation produces consistent results. It is entirely possible that the inequality is a result of some unintended bias, though I have found no potential cause for bias. – Daniel Mathias Mar 8 at 23:31 • @DanielMathias Thanks. I knew very little about bingo before reading this. Now I have upgraded my knowledge from "very little" to just "little". – badjohn Mar 9 at 14:54

https://math.stackexchange.com/questions/59943/if-n-323232-cdots-50-text-digits-9-i-e-in-base-9-then-how-to-find-t?noredirect=1

# If $N= (323232 \cdots 50 \text{ digits})_9$ (i.e in base $9$) then how to find the remainder when this $N$ is divided by $8$? If $N= (\underbrace{323232 \cdots}_{50 \text{ digits}})_9$ (i.e in base $9$) then how to find the remainder when this $N$ is divided by $8$? I am looking for a "fast" approach that could be used to solve this in less than $2$ minutes. • This is computing $N\mod 8$. Since the $9\mod 8 \equiv 1$, this is just the sum of digits of the number $N$, i.e. $(3+2)\times 25\mod 8 \equiv (3+2)\times 1\mod 8 = 5$. Did I miss the question ? – Sasha Aug 26 '11 at 16:45 • @Sasha:No,you didn't miss the question.I just can't see this solution before :/ – Quixotic Aug 26 '11 at 17:26 The same way we find the remainder of dividing a number by $9$ when the number is written in base 10: add the digits. Why? Because remember that writing, say, $(38571)_9$ "really" means $$1 + 7\times 9 + 5\times 9^2 + 8\times 9^3 + 3\times 9^4.$$ When you divide $9$ by $8$, the remainder is $1$; when you divide $9^2$ by $8$, the remainder is $1^2 = 1$; then you divide $9^3$ by $8$, the remainder is $1^3=1$, etc. So the remainder of this number when divided by $8$ is the same as the remainder of $$1 + 7\times 1 + 5\times 9^2 + 8\times 1^3 + 3\times 1^4$$ which is the same as the remainder of $1+7+5+8+3 = 24$, which is $0$. In this case, you have the digits $3$ and $2$, each of them $25$ times, so the remainder when dividing by $8$ is the same as the remainder of $(2+3)\times 25 = 125$ when divided by $8$, which is $5$. (More generally, the remainder of dividing the number $(b_1b_2\cdots b_k)_b$ by $b-1$ is the same as the remainder of dividing the sum of the base $b$ digits, $b_1+b_2+\cdots+b_k$ by $b-1$). (Even more generally, the remainder of dividing $(b_1b_2\cdots b_k)_b$ by any divisor of $b-1$ is the same as the remainder of dividing the sum of the base $b$ digits $b_1+b_2+\cdots + b_n$ by that number. That's why you can add the digits of a number in base 10 to find the remainders modulo $9$ or to find the remainders modulo $3$). • I feel like stupid now,the first general result is not new to me and could be easily be shown as follows:$N = (a_1a_2\cdots a_k)_r = a_k + \cdots +a_2 \times r^{k-2} + a_2 \times r^{k-1}$ and $S = a_1 + a_2 + \cdots a_k$ then it could be easily shown that $\frac{N-S}{r-1} = I \text{ (an integer) }+\frac{S}{r-1}$,which proves that result. But I don't know why I can't see this direct application before! :/ – Quixotic Aug 26 '11 at 17:23 • Anyways,the more general result is new to me,and perhaps a very useful one,could you hint me a easy proof for the same? – Quixotic Aug 26 '11 at 17:25 • @FoolForMath The key idea is to learn modular arithmetic and, more generally, modular reductions of problems in quotient algebras. It's one way to algebraically divide and conquer. You can find further discussion of such in some of my posts here if you follow the link I gave, and its links... – Bill Dubuque Aug 26 '11 at 17:28 • I understood it now :D and it was before Bill Dubuque's post :-) but yes Bill you are right,my understanding is due to modular arithmetic. – Quixotic Aug 26 '11 at 17:31 • @Foo One key thing to keep in mind is that radix notation has polynomial form, so this can be viewed as a special case of well-known results about polynomials. Always be on the lookout for analogies between numbers and functions (here polynomials). – Bill Dubuque Aug 26 '11 at 17:55 HINT $\$ Employ the radix $\:9\:$ analog of casting out nines in decimal, namely $$\rm\quad mod\ 8:\ \ \ 9 \equiv 1\ \ \Rightarrow\ \ d_n\: 9^n +\:\cdots\:+d_1\:9 + d_0\ \equiv\ d_n +\:\cdots\:+d_1+d_0$$ The same idea works generally to cast out $\rm\:b\pm1\:$'s in radix $\rm\:b\:$, e.g. see here for many links. It may be viewed as a number-theoretic specialization of the Polynomial Remainder Theorem $\rm\quad f(x)\ mod\ (x-r)\:=\:f(r)\:,\$ thus $\rm\ \ f(x)\ mod\ (x-1)\:=\:f(1)\: =\: f_n +\:\cdots\:+f_1 + f_0$

http://cms.math.ca/Competitions/MOCP/2001/sol_feb.mml

location: 61. Let $S=1!2!3!\dots 99!100!$ (the product of the first 100 factorials). Prove that there exists an integer $k$ for which $1\le k\le 100$ and $S/k!$ is a perfect square. Is $k$ unique? (Optional: Is it possible to find such a number $k$ that exceeds 100?) Solution 1. Note that, for each positive integer $j$, $\left(2j-1\right)!\left(2j\right)!=\left[\left(2j-1\right)!\right]{}^{2}·2j$. Hence $S=\underset{j=1}{\overset{50}{\Pi }}\left[\left(2j-1\right)!\right]{}^{2}\left[2j\right]={2}^{50}50!\left[\underset{j=1}{\overset{50}{\Pi }}\left(2j-1\right)!{\right]}^{2} ,$ from which we see that $k=50$ is the required number. We show that $k=50$ is the only possibility. First, $k$ cannot exceed 100, for otherwise $101!$ would be a factor of $k!$ but not $S$, and so $S/k!$ would not even be an integer. Let $k\le 100$. The prime $47$ does not divide $k!$ for $k\le 46$ and divides $50!$ to the first power. Since $S/50!$ is a square, it evidently divides $S$ to an odd power. So $k\ge 47$ in order to get a quotient divisible by 47 to an even power. The prime $53$ divides each $k!$ for $k\ge 53$ to the first power and divides $S/50!$, and so $S$ to an even power. Hence, $k\le 52$. The prime $17$ divides $50!$ and $S/50!$, and hence $S$ to an even power, but it divides each of $51!$ and $52!$ to the third power. So we cannot have $k=51$ or 52. Finally, look at the prime 2. Suppose that ${2}^{2u}$ is the highest power of 2 that divides $S/50!$ and that ${2}^{v}$ is the highest power of 2 that divides $50!$; then ${2}^{2u+v}$ is the highest power of 2 that divides $S$. The highest power of $2$ that divides $48!$ and $49!$ is ${2}^{v-1}$ and the highest power of 2 that divides $46!$ and $47!$ is ${2}^{v-5}$. >From this, we deduce that 2 divides $S/k!$ to an odd power when $47\le k\le 49$. The desired uniqueness of $k$ follows. Solution 2. Let $p$ be a prime exceeding 50. Then $p$ divides each of $m!$ to the first power for $p\le m\le 100$, so that $p$ divides $S$ to the even power $100-\left(p-1\right)=101-p$. From this, it follows that if $53\ge k$, $p$ must divide $S/k!$ to an odd power. On the other hand, the prime 47 divides each $m!$ with $47\le m\le 93$ to the first power, and each $m!$ with $94\le m\le 100$ to the second power, so that it divides $S$ to the power with exponent $54+7=61$. Hence, in order that it divide $S/k!$ to an even power, we must make $k$ one of the numbers $47,\dots ,52$. By an argument, similar to that used in Solution 1, it can be seen that $2$ divides any product of the form $1!2!\dots \left(2m-1\right)!$ to an even power and $100!$ to the power with exponent $⌊100/2⌋+⌊100/4⌋+⌊100/8⌋+⌊100/16⌋+⌊100/32⌋+⌊100/64⌋=50+25+12+6+3+1=97 .$ Hence, 2 divides $S$ to an odd power. So we need to divide $S$ by $k!$ which 2 divides to an odd power to get a perfect square quotient. This reduces the possibilities for $k$ to 50 or 51. Since $S={2}^{99}·{3}^{98}·{4}^{97}\dots {99}^{2}·100=\left(2·4\dots 50\right)\left({2}^{49}·{3}^{49}·{4}^{48}\dots 99\right){}^{2}=50!·{2}^{50}\left(\dots \right){}^{2} ,$ $S/50!$ is a square, and so $S/51!=\left(S/50!\right)÷\left(51\right)$ is not a square. The result follows. Solution 3. As above, $S/\left(50!\right)$ is a square. Suppose that $53\le k\le 100$. Then 53 divides $k!/50!$ to the first power, and so $k!/50!$ cannot be square. Hence $S/k!=\left(S/50!\right)÷\left(k!/50!\right)$ cannot be square. If $k=51$ or 52, then $k!/50!$ is not square, so $S/k!$ cannot be square. Suppose that $k\le 46$. Then 47 divides $50!/k!$ to the first power, so that $50!/k!$ is not square and $S/k!=\left(S/50!\right)×\left(50!/k!\right)$ cannot be square. If $k=47,48$ or 49, then $50!/k!$ is not square and so $S/k!$ is not square. Hence $S/k!$ is square if and only if $k=50$ when $k\le 100$. 62. Let $n$ be a positive integer. Show that, with three exceptions, $n!+1$ has at least one prime divisor that exceeds $n+1$. Solution. Any prime divisor of $n!+1$ must be larger than $n$, since all primes not exceeding $n$ divide $n!$. Suppose, if possible, the result fails. Then, the only prime that can divide $n!+1$ is $n+1$, so that, for some positive integer $r$ and nonnegative integer $K$, $n!+1=\left(n+1\right){}^{r}=1+\mathrm{rn}+{\mathrm{Kn}}^{2} .$ This happens, for example, when $n=1,2,4$: $1!+1=2$, $2!+1=3$, $4!+1={5}^{2}$. Note, however, that the desired result does hold for $n=3$: $3!+1=7$. Henceforth, assume that $n$ exceeds 4. If $n$ is prime, then $n+1$ is composite, so by our initial comment, all of its prime divisors exceed $n+1$. If $n$ is composite and square, then $n!$ is divisible by the four distinct integers $1,n,\sqrt{n},2\sqrt{n}$, while is $n$ is composite and nonsquare with a nontrivial divisor $d$. then $n!$ is divisible by the four distinct integers $1,d,n/d,n$. Thus, $n!$ is divisible by ${n}^{2}$. Suppose, if possible, the result fails, so that $n!+1=1+\mathrm{rn}+{\mathrm{Kn}}^{2}$, and $1\equiv 1+\mathrm{rn}$ (mod ${n}^{2}$). Thus, $r$ must be divisible by $n$, and, since it is positive, must exceed $n$. Hence $\left(n+1\right){}^{r}\ge \left(n+1\right){}^{n}>\left(n+1\right)n\left(n-1\right)\dots 1>n!+1 ,$ a contradiction. The desired result follows. 63. Let $n$ be a positive integer and $k$ a nonnegative integer. Prove that $n!=\left(n+k\right){}^{n}-\left(\genfrac{}{}{0}{}{n}{1}\right)\left(n+k-1\right){}^{n}+\left(\genfrac{}{}{0}{}{n}{2}\right)\left(n+k-2\right){}^{n}-\dots ±\left(\genfrac{}{}{0}{}{n}{n}\right){k}^{n} .$ Solution 1. Recall the Principle of Inclusion-Exclusion: Let $S$ be a set of $n$ objects, and let ${P}_{1}$, ${P}_{2}$, $\dots$, ${P}_{m}$ be $m$ properties such that, for each object $x\in S$ and each property ${P}_{i}$, either $x$ has the property ${P}_{i}$ or $x$ does not have the property ${P}_{i}$. Let $f\left(i,j,\dots ,k\right)$ denote the number of elements of $S$ each of which has properties ${P}_{i}$, ${P}_{j}$, $\dots$, ${P}_{k}$ (and possibly others as well). Then the number of elements of $S$ each having none of the properties ${P}_{1}$, ${P}_{2}$, $\dots$, ${P}_{m}$ is $n-\sum _{1\le i\le m}f\left(i\right)+\sum _{1\le i We apply this to the problem at hand. Note that an ordered selection of $n$ numbers selected from among $1,2,\dots ,n+k$ is a permutation of $\left\{1,2,\dots ,n\right\}$ if and only if it is constrained to contain each of the numbers 1, 2, $\dots$, $n$. Let $S$ be the set of all ordered selections, and we say that a selection has property ${P}_{i}$ iff its fails to include at least $i$ of the numbers $1,2,\dots ,n$ $\left(1\le i\le n\right)$. The number of selections with property ${P}_{i}$ is the product of $\left(\genfrac{}{}{0}{}{n}{i}\right)$, the number of ways of choosing the $i$ numbers not included and $\left(n+k-i\right){}^{n}$, the number of ways of choosing entries for the $n$ positions from the remaining $n+k-1$ numbers. The result follows. Solution 2. We begin with a lemma: $\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{m}=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill \left(0\le m\le n-1\right)\hfill \\ \multicolumn{0}{c}{\left(-1\right){}^{n}n!}& \hfill \left(m=n\right) .\hfill \end{array}$ We use the convention that ${0}^{0}=1$. To prove this, note first that $i\left(i-1\right)\dots \left(i-m\right)={i}^{m+1}+{b}_{m}{i}^{m}+\dots +{b}_{1}i+{b}_{0}$ for some integers ${b}_{i}$. We use an induction argument on $m$. The result holds for each positive $n$ and for $m=0$, as the sum is the expansion of $\left(1-1\right){}^{n}$. It also holds for $n=1,2$ and all relevant $m$. Fix $n\ge 3$. Suppose that it holds when $m$ is replaced by $k$ for $0\le k\le m\le n-2$. Then $\begin{array}{cc}\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{m+1}\hfill & =\sum _{i=1}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)i\left(i-1\right)\dots \left(i-m\right)-\sum _{k=0}^{m}{b}_{k}\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{k}\hfill \\ \multicolumn{0}{c}{}& =\sum _{i=m+1}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)i\left(i-1\right)\dots \left(i-m\right)-0\hfill \\ \multicolumn{0}{c}{}& =\sum _{i=m+1}^{n}\left(-1\right){}^{i}\frac{n!i!}{i!\left(n-i\right)!\left(i-m-1\right)!}=\sum _{j=0}^{n-m-1}\left(-1\right){}^{m+1+j}\frac{n!}{\left(n-m-1-j\right)!j!}\hfill \\ \multicolumn{0}{c}{}& =\sum _{j=0}^{n-m-1}\left(-1\right){}^{m+1}\left(-1\right){}^{j}\frac{n\left(n-1\right)\dots \left(n-m\right)\left[\left(n-m-1\right)!\right]}{\left(n-m-1-j\right)!j!}\hfill \\ \multicolumn{0}{c}{}& =\left(-1\right){}^{m+1}n\left(n-1\right)\dots \left(n-m\right)\sum _{j=0}^{n-m-1}\left(-1\right){}^{j}\left(\genfrac{}{}{0}{}{n-m-1}{j}\right)=0 .\hfill \end{array}$ (Note that the $j=0$ term is 1, which is consistent with the ${0}^{0}=1$ convention mentioned earlier.) So $\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{m}=0$ for $0\le m\le n-1$. Now consider the case $m=n$: $\sum _{i=1}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{n}=\sum _{i=1}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)i\left(i-1\right)\dots \left(i-n+1\right)-\sum _{k=0}^{n-1}{b}_{k}\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{k} .$ Every term in the first sum vanishes except the $n$th and each term of the second sum vanishes. Hence $\sum _{i=1}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{n}=\left(-1\right){}^{n}n!$. Returning to the problem at hand, we see that the right side of the desired equation is equal to $\begin{array}{cc}\left(n+k\right){}^{n}\hfill & -\left(\genfrac{}{}{0}{}{n}{1}\right)\left(n+k-1\right){}^{n}+\left(\genfrac{}{}{0}{}{n}{2}\right)\left(n+k-2\right){}^{n}-\dots +\left(-1\right){}^{n}\left(\genfrac{}{}{0}{}{n}{n}\right)\left(n+k-n\right){}^{n}\hfill \\ \multicolumn{0}{c}{}& =\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)\left(n-i+k\right){}^{n}=\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right)\left(n-i\right){}^{j}{k}^{n-j}\hfill \\ \multicolumn{0}{c}{}& =\sum _{i=0}^{n}\sum _{j=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)\left(\genfrac{}{}{0}{}{n}{j}\right)\left(n-i\right){}^{j}{k}^{n-j}=\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){k}^{n-j}\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)\left(n-i\right){}^{j}\hfill \\ \multicolumn{0}{c}{}& =\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){k}^{n-j}\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{n-i}\right)\left(n-i\right){}^{j}\hfill \\ \multicolumn{0}{c}{}& =\sum _{j=0}^{n}\left(\genfrac{}{}{0}{}{n}{j}\right){k}^{n-j}\sum _{i=0}^{n}\left(-1\right){}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{j} .\hfill \end{array}$ When $0\le j\le n-1$, the sum $\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{n-i}\right)\left(n-i\right){}^{j}=\sum _{i=0}^{n}\left(-1\right){}^{n-i}\left(\genfrac{}{}{0}{}{n}{i}\right){i}^{j}$ vanishes, while, when $j=n$, it assunes the value $n!$. Thus, the right side of the given equation is equal to $\left(\genfrac{}{}{0}{}{n}{n}\right){k}^{0}n!=n!$ as desired. Solution 3. Let $m=n+k$, so that $m\ge n$, and let the right side of the equation be denoted by $R$. Then $\begin{array}{cc}R\hfill & ={m}^{n}-\left(\genfrac{}{}{0}{}{n}{1}\right)\left(m-1\right){}^{n}+\left(\genfrac{}{}{0}{}{n}{2}\right)\left(m-2\right){}^{n}-\dots +\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)\left(m-i\right){}^{n}+\dots +\left(-1\right){}^{n}\left(\genfrac{}{}{0}{}{n}{n}\right)\left(m-n\right){}^{n}\hfill \\ \multicolumn{0}{c}{}& ={m}^{m}\left[\sum _{j=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right)\right]-\left(\genfrac{}{}{0}{}{n}{1}\right){m}^{n-1}\left[\sum _{i=1}^{n}\left(-1\right){}^{i}i\left(\genfrac{}{}{0}{}{n}{i}\right)\right]+\left(\genfrac{}{}{0}{}{n}{2}\right){m}^{n-2}\left[\sum _{i=1}^{n}\left(-1\right){}^{i}{i}^{2}\left(\genfrac{}{}{0}{}{n}{i}\right)\right]+\dots \hfill \\ \multicolumn{0}{c}{}& +\left(-1\right){}^{n}\left(\genfrac{}{}{0}{}{n}{n}\right)\left[\sum _{i=1}^{n}\left(-1\right){}^{i}{i}^{n}\left(\genfrac{}{}{0}{}{n}{i}\right)\right] .\hfill \end{array}$ Let ${f}_{0}\left(x\right)=\left(1-x\right){}^{n}=\sum _{i=0}^{n}\left(-1\right){}^{i}\left(\genfrac{}{}{0}{}{n}{i}\right){x}^{i}$ and let ${f}_{k}\left(x\right)=x{\mathrm{Df}}_{k-1}\left(x\right)$ for $k\ge 1$, where $\mathrm{Df}$ denotes the derivative of a function $f$. Observe that, from the closed expression for ${f}_{0}\left(x\right)$, we can establish by induction that ${f}_{k}\left(x\right)=\sum _{i=0}^{n}\left(-1\right){}^{i}{i}^{k}\left(\genfrac{}{}{0}{}{n}{i}\right){x}^{i}$ so that $R=\sum _{k=0}^{n}\left(-1\right){}^{k}\left(\genfrac{}{}{0}{}{n}{k}\right){m}^{n-k}{f}_{k}\left(1\right)$. By induction, we establish that ${f}_{k}\left(x\right)=\left(-1\right){}^{k}n\left(n-1\right)\dots \left(n-k+1\right){x}^{k}\left(1-x\right){}^{n-k}+\left(1-x\right){}^{n-k+1}{g}_{k}\left(x\right)$ for some polynomial ${g}_{k}\left(x\right)$. This is true for $k=1$ with ${g}_{1}\left(x\right)=0$. Suppose if holds for $k=j$. Then $\begin{array}{cc}{f}_{j}\text{'}\left(x\right)\hfill & =\left(-1\right){}^{j}n\left(n-1\right)\dots \left(n-j+1\right){x}^{j-1}\left(1-x\right){}^{n-j}-\left(-1\right){}^{j}n\left(n-1\right)\dots \left(n-j+1\right)\left(n-j\right){x}^{j}\left(1-x\right){}^{n-j-1}\hfill \\ \multicolumn{0}{c}{}& -\left(n-j+1\right)\left(1-x\right){}^{n-j}{g}_{j}\left(x\right)+\left(1-x\right){}^{n-j+1}{g}_{j}\text{'}\left(x\right) ,\hfill \end{array}$ whence $\begin{array}{cc}{f}_{j+1}\left(x\right)\hfill & =\left(-1\right){}^{j+1}n\left({n}_{1}\right)\dots \left(n-j\right){x}^{j}\left(1-x\right){}^{n-\left(j+1\right)}+\left(1-x\right){}^{n-\left(j+1\right)+1}\left[\left(-1\right){}^{j}n\left(n-1\right)\dots \left(n-j+1\right){x}^{j}\hfill \\ \multicolumn{0}{c}{}& -\left(n-j+1\right){\mathrm{xg}}_{k}\left(x\right)+x\left(1-x\right){g}_{j}\text{'}\left(x\right)\right]\hfill \end{array}$ and we obtain the desired representation by induction. Then for $1\le k\le n-1$, ${f}_{k}\left(1\right)=0$ while ${f}_{n}\left(1\right)=\left(-1\right){}^{n}n!$. Hence $R=\left(-1\right){}^{n}{f}_{n}\left(1\right)=n!$. 64. Let $M$ be a point in the interior of triangle $\mathrm{ABC}$, and suppose that $D$, $E$, $F$ are respective points on the side $\mathrm{BC}$, $\mathrm{CA}$, $\mathrm{AB}$, which all pass through $M$. (In technical terms, they are cevians.) Suppose that the areas and the perimeters of the triangles $\mathrm{BMD}$, $\mathrm{CME}$, $\mathrm{AMF}$ are equal. Prove that triangle $\mathrm{ABC}$ must be equilateral. Solution. [L. Lessard] Let the common area of the triangles $\mathrm{BMD}$, $\mathrm{CME}$ and $\mathrm{AMF}$ be $a$ and let their common perimeter be $p$. Let the area and perimeter of $\Delta \mathrm{AME}$ be $u$ and $x$ respectively, of $\Delta \mathrm{MFB}$ be $v$ and $y$ respectively, and of $\Delta \mathrm{CMD}$ be $w$ and $z$ respectively. By considering pairs of triangles with equal heights, we find that $\frac{\mathrm{AF}}{\mathrm{FB}}=\frac{a}{v}=\frac{2a+u}{v+a+w}=\frac{a+u}{a+w} ,$ $\frac{\mathrm{BD}}{\mathrm{DC}}=\frac{a}{w}=\frac{2a+v}{u+a+w}=\frac{a+v}{a+u} ,$ $\frac{\mathrm{CE}}{\mathrm{EA}}=\frac{a}{u}=\frac{2a+w}{u+a+v}=\frac{a+w}{a+v} .$ >From these three sets of equations, we deduce that $\frac{{a}^{3}}{\mathrm{uvw}}=1 ;$ ${a}^{2}+\left(w-u\right)a-\mathrm{uv}=0 ,$ ${a}^{2}+\left(u-w\right)a-\mathrm{vw}=0 ,$ ${a}^{2}+\left(v-u\right)a-\mathrm{uw}=0 ;$ whence ${a}^{3}=\mathrm{uvw} \mathrm{and} 3{a}^{2}=\mathrm{uv}+\mathrm{vw}+\mathrm{uw} .$ This means that $\mathrm{uv},\mathrm{vw},\mathrm{uw}$ are three positive numbers whose geometric and arithmetic means are both equal to ${a}^{2}$. Hence ${a}^{2}=\mathrm{uv}=\mathrm{vw}=\mathrm{uw}$, so that $u=v=w=a$. It follows that $\mathrm{AF}=\mathrm{FB}$, $\mathrm{BD}=\mathrm{DC}$, $\mathrm{CE}=\mathrm{EA}$, so that $\mathrm{AD}$, $\mathrm{BE}$ and $\mathrm{CF}$ are medians and $M$ is the centroid. Wolog, suppose that $\mathrm{AB}\ge \mathrm{BC}\ge \mathrm{CA}$. Since $\mathrm{AB}\ge \mathrm{BC}$, $\angle \mathrm{AEB}\ge {90}^{ˆ}$, and so $\mathrm{AM}\ge \mathrm{MC}$. Thus $x\ge p$. Similarly, $y\ge p$ and $p\ge z$. Consider triangles $\mathrm{BMD}$ and $\mathrm{AME}$. We have $\mathrm{BD}\ge \mathrm{AE}$, $\mathrm{BM}\ge \mathrm{AM}$, $\mathrm{ME}=\frac{1}{2}\mathrm{BM}$ and $\mathrm{MD}=\frac{1}{2}\mathrm{AM}$. Therefore $p-x=\left(\mathrm{BD}+\mathrm{MD}+\mathrm{BM}\right)-\left(\mathrm{AE}+\mathrm{ME}+\mathrm{AM}\right)=\left(\mathrm{BD}-\mathrm{AE}\right)+\frac{1}{2}\left(\mathrm{BM}-\mathrm{AM}\right)\ge 0$ and so $p\ge x$. Since also $x\ge p$, we have that $p=x$. But this implies that $\mathrm{AM}=\mathrm{MC}$, so that $\mathrm{ME}\perp \mathrm{AC}$ and $\mathrm{AB}=\mathrm{BC}$. Since $\mathrm{BE}$ is now an axis of a reflection which interchanges $A$ and $C$, as well as $F$ and $D$, it follows that $p=z$ and $p=y$ as well. Thus, $\mathrm{AB}=\mathrm{AC}$ and $\mathrm{AC}=\mathrm{BC}$. Thus, the triangle is equilateral. 65. Suppose that $\mathrm{XTY}$ is a straight line and that $\mathrm{TU}$ and $\mathrm{TV}$ are two rays emanating from $T$ for which $\angle \mathrm{XTU}=\angle \mathrm{UTV}=\angle \mathrm{VTY}={60}^{ˆ}$. Suppose that $P$, $Q$ and $R$ are respective points on the rays $\mathrm{TY}$, $\mathrm{TU}$ and $\mathrm{TV}$ for which $\mathrm{PQ}=\mathrm{PR}$. Prove that $\angle \mathrm{QPR}={60}^{ˆ}$. Solution 1. Let $\frakR$ be a rotation of ${60}^{ˆ}$ about $T$ that takes the ray $\mathrm{TU}$ to $\mathrm{TV}$. Then, if $\frakR$ transforms $Q\to Q\text{'}$ and $P\to P\text{'}$, then $Q\text{'}$ lies on $\mathrm{TV}$ and the line $Q\text{'}P\text{'}$ makes an angle of ${60}^{ˆ}$ with $\mathrm{QP}$. Because of the rotation, $\angle P\text{'}\mathrm{TP}={60}^{ˆ}$ and $\mathrm{TP}\text{'}=\mathrm{TP}$, whence $\mathrm{TP}\text{'}P$ is an equilateral triangle. Since $\angle Q\text{'}\mathrm{TP}=\angle \mathrm{TPP}\text{'}={60}^{ˆ}$, $\mathrm{TV}P\text{'}P$. Let $\frakT$ be the translation that takes $P\text{'}$ to $P$. It takes $Q\text{'}$ to a point $Q\text{'}\text{'}$ on the ray $\mathrm{TV}$, and $\mathrm{PQ}\text{'}\text{'}=P\text{'}Q\text{'}=\mathrm{PQ}$. Hence $Q\text{'}\text{'}$ can be none other than the point $R$ [why?], and the result follows. Solution 2. The reflection in the line $\mathrm{XY}$ takes $P\to P$, $Q\to Q\text{'}$ and $R\to R\text{'}$. Triangles $\mathrm{PQR}\text{'}$ and $\mathrm{PQ}\text{'}R$ are congruent and isosceles, so that $\angle \mathrm{TQP}=\angle \mathrm{TQ}\text{'}P=\angle \mathrm{TRP}$ (since $\mathrm{PQ}\text{'}=\mathrm{PR}$). Hence $\mathrm{TQRP}$ is a concyclic quadrilateral, whence $\angle \mathrm{QPR}=\angle \mathrm{QTR}={60}^{ˆ}$. Solution 3. [S. Niu] Let $S$ be a point on $\mathrm{TU}$ for which $\mathrm{SR}\mathrm{XY}$; observe that $\Delta \mathrm{RST}$ is equilateral. We first show that $Q$ lies between $S$ and $T$. For, if $S$ were between $Q$ and $T$, then $\angle \mathrm{PSQ}$ would be obtuse and $\mathrm{PQ}>\mathrm{PS}>\mathrm{PR}$ (since $\angle \mathrm{PRS}>{60}^{ˆ}>\angle \mathrm{PSR}$ in $\Delta \mathrm{PRS}$), a contradiction. The rotation of ${60}^{ˆ}$ with centre $R$ that takes $S$ onto $T$ takes ray $\mathrm{RQ}$ onto a ray through $R$ that intersects $\mathrm{TY}$ in $M$. Consider triangles $\mathrm{RSQ}$ and $\mathrm{RTM}$. Since $\angle \mathrm{RST}=\angle \mathrm{RTM}={60}^{ˆ}$, $\angle \mathrm{SRQ}={60}^{ˆ}-\angle \mathrm{QRT}=\angle \mathrm{TRM}$ and $\mathrm{SR}=\mathrm{TR}$, we have that $\Delta \mathrm{RSQ}\equiv \Delta \mathrm{RTM}$ and $\mathrm{RQ}=\mathrm{RM}$. (ASA) Since $\angle \mathrm{QRM}={60}^{ˆ}$, $\Delta \mathrm{RQM}$ is equilateral and $\mathrm{RM}=\mathrm{RQ}$. Hence $M$ and $P$ are both equidistant from $Q$ and $R$, and so at the intersection of $\mathrm{TY}$ and the right bisector of $\mathrm{QR}$. Thus, $M=P$ and the result follows. Solution 4. [H. Pan] Let $Q\text{'}$ and $R\text{'}$ be the respective reflections of $Q$ and $R$ with respect to the axis $\mathrm{XY}$. Since $\angle \mathrm{RTR}\text{'}={120}^{ˆ}$ and $\mathrm{TR}=\mathrm{TR}\text{'}$, $\angle \mathrm{QR}\text{'}R=\angle \mathrm{TR}\text{'}R={30}^{ˆ}$. Since $Q,R,Q\text{'},R\text{'},$ lie on a circle with centre $P$, $\angle \mathrm{QPR}=2\angle \mathrm{QR}\text{'}R={60}^{ˆ}$, as desired. Solution 5. [R. Barrington Leigh] Let $W$ be a point on $\mathrm{TV}$ such that $\angle \mathrm{WPQ}={60}^{ˆ}=\angle \mathrm{WTU}$. [Why does such a point $W$ exist?] Then $\mathrm{WQTP}$ is a concyclic quadrilateral so that $\angle \mathrm{QWP}={180}^{ˆ}-\angle \mathrm{QTP}={60}^{ˆ}$ and $\Delta \mathrm{PWQ}$ is equilateral. Hence $\mathrm{PW}=\mathrm{PQ}=\mathrm{PR}$. Suppose $W\ne R$. If $R$ is farther away from $T$ than $W$, then $\angle \mathrm{RPT}>\angle \mathrm{WPT}>\angle \mathrm{WPQ}={60}^{ˆ}⇒{60}^{ˆ}>\angle \mathrm{TRP}=\angle \mathrm{RWP}>{60}^{ˆ}$, a contradiction. If $W$ is farther away from $T$ than $R$, then $\angle \mathrm{WPT}>\angle \mathrm{WPQ}={60}^{ˆ}⇒{60}^{ˆ}>\angle \mathrm{RWP}=\angle \mathrm{WRP}>{60}^{ˆ}$, again a contradiction. So $R=W$ and the result follows. Solution 6. [M. Holmes] Let the circle through $T,P,Q$ intersect $\mathrm{TV}$ in $N$. Then $\angle \mathrm{QNP}={180}^{ˆ}-\angle \mathrm{QTP}={60}^{ˆ}$. Since $\angle \mathrm{PQN}=\angle \mathrm{PTN}={60}^{ˆ}$, $\Delta \mathrm{PQN}$ is equilateral so that $\mathrm{PN}=\mathrm{PQ}$. Suppose, if possible, that $R\ne N$. Then $N$ and $R$ are two points on $\mathrm{TV}$ equidistant from $P$. Since $\angle \mathrm{PNT}<\angle \mathrm{PNQ}={60}^{ˆ}$ and $\Delta \mathrm{PNR}$ is isosceles, we have that $\angle \mathrm{PNR}<{90}^{ˆ}$, so $N$ cannot lie between $T$ and $R$, and $\angle \mathrm{PRN}=\angle \mathrm{PNR}=\angle \mathrm{PNT}<{60}^{ˆ}$. Since $\angle \mathrm{PTN}={60}^{ˆ}$, we conclude that $T$ must lie between $R$ and $N$, which transgresses the condition of the problem. Hence $R$ and $N$ must coincide and the result follows. Solution 7. [P. Cheng] Determine $S$ on $\mathrm{TU}$ and $Z$ on $\mathrm{TY}$ for which $\mathrm{SR}\mathrm{XY}$ and $\angle \mathrm{QRZ}={60}^{ˆ}$. Observe that $\angle \mathrm{TSR}=\angle \mathrm{SRT}={60}^{ˆ}$ and $\mathrm{SR}=\mathrm{RT}$. Consider triangles $\mathrm{SRQ}$ and $\mathrm{TRZ}$. $\angle \mathrm{SRQ}=\angle \mathrm{SRT}-\angle \mathrm{QRT}=\angle \mathrm{QRZ}-\angle \mathrm{QRT}=\angle \mathrm{TRZ}$; $\angle \mathrm{QSR}={60}^{ˆ}=\angle \mathrm{ZTR}$, so that $\Delta \mathrm{SRQ}=\Delta \mathrm{TRZ}$ (ASA). Hence $\mathrm{RZ}=\mathrm{RQ}⇒\Delta \mathrm{RQZ}$ is equilateral $⇒\mathrm{RZ}=\mathrm{ZQ}$ and $\angle \mathrm{RZQ}={60}^{ˆ}$. Now, both $P$ and $Z$ lie on the intersection of $\mathrm{TY}$ and the right bisector of $\mathrm{QR}$, so they must coincide: $P=Z$. The result follows. Solution 8. Let the perpendicular, produced, from $Q$ to $\mathrm{XY}$ meet $\mathrm{VT}$, produced, in $S$. Then $\angle \mathrm{XTS}=\angle \mathrm{VTY}={60}^{ˆ}=\angle \mathrm{XTU}$, from which is can be deduced that $\mathrm{TX}$ right bisects $\mathrm{QS}$. Hence $\mathrm{PS}=\mathrm{PQ}=\mathrm{PR}$, so that $Q,R,S$ are all on the same circle with centre $P$. Since $\angle \mathrm{QTS}={120}^{ˆ}$, we have that $\angle \mathrm{SQT}=\angle \mathrm{QSR}={30}^{ˆ}$, so that $\mathrm{QR}$ must subtend an angle of ${60}^{ˆ}$ at the centre $P$ of the circle. The desired result follows. Solution 9. [A.Siu] Let the right bisector of $\mathrm{QR}$ meet the circumcircle of $\mathrm{TQR}$ on the same side of $\mathrm{QR}$ at $T$ in $S$. Since $\angle \mathrm{QSR}=\angle \mathrm{QTR}={60}^{ˆ}$ and $\mathrm{QS}=\mathrm{QR}$, $\angle \mathrm{SQR}=\angle \mathrm{SRQ}={60}^{ˆ}$. Hence $\angle \mathrm{STQ}={180}^{ˆ}-\angle \mathrm{SRQ}={120}^{ˆ}$. But $\angle \mathrm{YTQ}={120}^{ˆ}$, so $S$ must lie on $\mathrm{TY}$. It follows that $S=P$. Solution 10. Assign coordinates with the origin at $T$ and the $x-$axis along $\mathrm{XY}$. The the respective coordinates of $Q$ and $R$ have the form $\left(u,-\sqrt{3}u\right)$ and $\left(v,\sqrt{3}v\right)$ for some real $u$ and $v$. Let the coordinates of $P$ be $\left(w,0\right)$. Then $\mathrm{PQ}=\mathrm{PR}$ yields that $w=2\left(u+v\right)$. [Exercise: work it out.] $\begin{array}{cc}‖\mathrm{PQ}‖{}^{2}-‖\mathrm{QR}‖{}^{2}\hfill & =\left(u-w\right){}^{2}+3{u}^{2}-\left(u-v\right){}^{2}-3\left(u+v\right){}^{2}\hfill \\ \multicolumn{0}{c}{}& ={w}^{2}-2\mathrm{uw}-4v\left(u+v\right)={w}^{2}-2\mathrm{uw}-2\mathrm{vw}\hfill \\ \multicolumn{0}{c}{}& ={w}^{2}-2\left(u+v\right)w=0 .\hfill \end{array}$ Hence $\mathrm{PQ}=\mathrm{QR}=\mathrm{PR}$ and $\Delta \mathrm{PQR}$ is equilateral. Therefore $\angle \mathrm{QPR}={60}^{ˆ}$. Solution 11. [J.Y. Jin] Let $\frakC$ be the circumcircle of $\Delta \mathrm{PQR}$. If $T$ lies strictly inside $\frakC$, then ${60}^{ˆ}=\angle \mathrm{QTR}>\angle \mathrm{QPR}$ and ${60}^{ˆ}=\angle \mathrm{PTR}>\angle \mathrm{PQR}=\angle \mathrm{PRQ}$. Thus, all three angle of $\Delta \mathrm{PQR}$ would be less than ${60}^{ˆ}$, which is not possible. Similarly, if $T$ lies strictly outside $\frakC$, then ${60}^{ˆ}=\angle \mathrm{QTR}<\angle \mathrm{QPR}$ and ${60}^{ˆ}=\angle \mathrm{PTR}<\angle \mathrm{PQR}=\angle \mathrm{PRQ}$, so that all three angles of $\Delta \mathrm{PQR}$ would exceed ${60}^{ˆ}$, again not possible. Thus $T$ must be on $\frakC$, whence $\angle \mathrm{QPR}=\angle \mathrm{QTR}={60}^{ˆ}$. Solution 12. [C. Lau] By the Sine Law, $\frac{\mathrm{sin}\angle \mathrm{TQP}}{‖\mathrm{TP}‖}=\frac{\mathrm{sin}{120}^{ˆ}}{‖\mathrm{PQ}‖}=\frac{\mathrm{sin}{60}^{ˆ}}{‖\mathrm{PR}‖}=\frac{\mathrm{sin}\angle \mathrm{TRP}}{‖\mathrm{TP}‖} ,$ whence $\mathrm{sin}\angle \mathrm{TQP}=\mathrm{sin}\angle \mathrm{TRP}$. Since $\angle \mathrm{QTP}$, in triangle $\mathrm{QTP}$ is obtuse, $\angle \mathrm{TQP}$ is acute. Suppose, if possible, that $\angle \mathrm{TRP}$ is obtuse. Then, in triangle $\mathrm{TPR}$, $\mathrm{TP}$ would be the longest side, so $\mathrm{PR}<\mathrm{TP}$. But in triangle $\mathrm{TQP}$, $\mathrm{PQ}$ is the longest side, so $\mathrm{PQ}>\mathrm{TP}$, and so $\mathrm{PQ}\ne \mathrm{PR}$, contrary to hypothesis. Hence $\angle \mathrm{TRP}$ is acute. Therefore, $\angle \mathrm{TQP}=\angle \mathrm{TRP}$. Let $\mathrm{PQ}$ and $\mathrm{RT}$ intersect in $Z$. Then, ${60}^{ˆ}=\angle \mathrm{QTZ}={180}^{ˆ}-\angle \mathrm{TQP}-\angle \mathrm{QZT}={180}^{ˆ}-\angle \mathrm{TRP}-\angle \mathrm{RZP}=\angle \mathrm{QPR}$, as desired. 66. (a) Let $\mathrm{ABCD}$ be a square and let $E$ be an arbitrary point on the side $\mathrm{CD}$. Suppose that $P$ is a point on the diagonal $\mathrm{AC}$ for which $\mathrm{EP}\perp \mathrm{AC}$ and that $Q$ is a point on $\mathrm{AE}$ produced for which $\mathrm{CQ}\perp \mathrm{AE}$. Prove that $B,P,Q$ are collinear. (b) Does the result hold if the hypothesis is weakened to require only that $\mathrm{ABCD}$ is a rectangle? Solution 1. Let $\mathrm{ABCD}$ be a rectangle, and let $E$, $P$, $Q$ be determined as in the problem. Suppose that $\angle \mathrm{ACD}=\angle \mathrm{BDC}=\alpha$. Then $\angle \mathrm{PEC}={90}^{ˆ}-\alpha$. Because $\mathrm{EPQC}$ is concyclic, $\angle \mathrm{PQC}=\angle \mathrm{PEC}={90}^{ˆ}-\alpha$. Because $\mathrm{ABCQD}$ is concyclic, $\angle \mathrm{BQC}=\angle \mathrm{BDC}=\alpha$. The points $B$, $P$, $Q$ are collinear $&lrArr;\angle \mathrm{BQC}=\angle \mathrm{PQC}&lrArr;\alpha ={90}^{ˆ}-\alpha &lrArr;\alpha ={45}^{ˆ}&lrArr;\mathrm{ABCD}$ is a square. Solution 2. (a) $\mathrm{EPQC}$, with a pair of supplementary opposite angles, is concyclic, so that $\angle \mathrm{CQP}=\angle \mathrm{CEP}={180}^{ˆ}-\angle \mathrm{EPC}-\angle \mathrm{ECP}={45}^{ˆ}$. Since $\mathrm{CBAQ}$ is concyclic, $\angle \mathrm{CQB}=\angle \mathrm{CAB}={45}^{ˆ}$. Thus, $\angle \mathrm{CQP}=\angle \mathrm{CQB}$ so that $Q$, $P$, $B$ are collinear. (b) Suppose that $\mathrm{ABCD}$ is a nonquare rectangle. Then taking $E=D$ yields a counterexample. Solution 3. (a) The circle with diameter $\mathrm{AC}$ that passes through the vertices of the square also passes through $Q$. Hence $\angle \mathrm{QBC}=\angle \mathrm{QAC}$. Consider triangles $\mathrm{PBC}$ and $\mathrm{EAC}$. Since triangles $\mathrm{ABC}$ and $\mathrm{EPC}$ are both isosceles right triangles, $\mathrm{BC}:\mathrm{AC}=\mathrm{PC}:\mathrm{EC}$. Also $\angle \mathrm{BCA}=\angle \mathrm{PCE}={45}^{ˆ}$. Hence $\Delta \mathrm{PBC}~\Delta \mathrm{EAC}$ (SAS) so that $\angle \mathrm{PBC}=\angle \mathrm{EAC}=\angle \mathrm{QAC}=\angle \mathrm{QBC}$. It follows that $Q$, $P$, $B$ are collinear. Solution 4. [S. Niu] Let $\mathrm{ABCD}$ be a rectangle and let $E,P,Q$ be determined as in the problem. Let $\mathrm{EP}$ be produced to meet $\mathrm{BC}$ in $F$. Since $\angle \mathrm{ABF}=\angle \mathrm{APF}$, the quadrilateral $\mathrm{ABPF}$ is concyclic, so that $\angle \mathrm{PBC}=\angle \mathrm{PBF}=\angle \mathrm{PAF}$. Since $\mathrm{ABCQ}$ is concyclic, $\angle \mathrm{QBC}=\angle \mathrm{QAC}=\angle \mathrm{PAE}$. Now $B,P,Q$ are collinear $&lrArr;\angle \mathrm{PBC}=\angle \mathrm{QBC}&lrArr;\angle \mathrm{PAF}=\angle \mathrm{PAE}&lrArr;\mathrm{AC} \mathrm{right} \mathrm{bisects} \mathrm{EF}$ $&lrArr;\angle \mathrm{ECA}=\angle \mathrm{ACB}={45}^{ˆ}&lrArr;\mathrm{ABCD} \mathrm{is} a \mathrm{square} .$ Solution 5. [M. Holmes] (a) Suppose that $\mathrm{BQ}$ intersects $\mathrm{AC}$ in $R$. Since $\mathrm{ABCQD}$ is concyclic, $\angle \mathrm{AQR}=\angle \mathrm{AQB}=\angle \mathrm{ACB}={45}^{ˆ}$, so that $\angle \mathrm{BQC}={45}^{ˆ}$. Since $\angle \mathrm{EQR}=\angle \mathrm{AQB}=\angle \mathrm{ECR}={45}^{ˆ}$, $\mathrm{ERCQ}$ is concyclic, so that $\angle \mathrm{ERC}={180}^{ˆ}-\angle \mathrm{EQC}={90}^{ˆ}$. Hence $\mathrm{ER}\perp \mathrm{AC}$, so that $R=P$ and the result follows. Solution 6. [L. Hong] (a) Let $\mathrm{QC}$ intersect $\mathrm{AB}$ in $F$. We apply Menelaus' Theorem to triangle $\mathrm{AFC}$: $B$, $P$, $Q$ are collinear if and only if $\frac{\mathrm{AB}}{\mathrm{BF}}·\frac{\mathrm{FQ}}{\mathrm{QC}}·\frac{\mathrm{CP}}{\mathrm{PA}}=-1 .$ Let the side length of the square be 1 and the length of $\mathrm{DE}$ be $a$. Then $‖\mathrm{AB}‖=1$. Since $\Delta \mathrm{ADE}~\Delta \mathrm{FBC}$, $\mathrm{AD}:\mathrm{DE}=\mathrm{BF}:\mathrm{BC}$, so that $‖\mathrm{BF}‖=1/a$ and $‖\mathrm{FC}‖=\sqrt{1+{a}^{2}}/a$. Since $\Delta \mathrm{ADE}~\Delta \mathrm{CQE}$, $\mathrm{CQ}:\mathrm{EC}=\mathrm{AD}:\mathrm{EA}$, so that $‖\mathrm{CQ}‖=\left(1-a\right)/\sqrt{1+{a}^{2}}$. Hence $\frac{‖\mathrm{FQ}‖}{‖\mathrm{CQ}‖}=1+\frac{‖\mathrm{FC}‖}{‖\mathrm{CQ}‖}=1+\frac{1+{a}^{2}}{a\left(1-a\right)}=\frac{1+a}{a\left(1-a\right)} .$ Since $\Delta \mathrm{ECP}$ is right isosceles, $‖\mathrm{CP}‖=\left(1-a\right)/\sqrt{2}$ and $‖\mathrm{PA}‖=\sqrt{2}-‖\mathrm{CP}‖=\left(1+a\right)/\sqrt{2}$. Hence $‖\mathrm{CP}‖/‖\mathrm{PA}‖=\left(1-a\right)/\left(1+a\right)$. Multiplying the three ratios together and taking account of the directed segments gives the product $-1$ and yields the result. Solution 7. (a) Select coordinates so that $A~\left(0,1\right)$, $B~\left(0,0\right)$, $C~\left(1,0\right)$, $D~\left(1,1\right)$ and $E~\left(1,t\right)$ for some $t$ with $0\le t\le 1$. It is straightforward to verify that $P~\left(1-\frac{t}{2},\frac{t}{2}\right)$. Since the slope of $\mathrm{AE}$ is $t-1$, the slope of $\mathrm{AQ}$ should be $\left(1-t\right){}^{-1}$. Since the coordinates of $Q$ have the form $\left(1+s,s\left(1-t\right){}^{-1}\right)$ for some $s$, it is straightforward to verify that $Q~\left(\frac{2-t}{1+\left(1-t\right){}^{2}},\frac{t}{1+\left(1-t\right){}^{2}\right)}\right) .$ It can now be checked that the slope of each of $\mathrm{BQ}$ and $\mathrm{BP}$ is $t\left(2-t\right){}^{-1}$, which yields the result. (b) The result fails if $A~\left(0,2\right)$, $B~\left(0,0\right)$, $C~\left(1,0\right)$, $D~\left(1,2\right)$. If $E~\left(1,1\right)$, then $P~\left(\frac{3}{5},\frac{4}{5}\right)$ and $Q~\left(\frac{3}{2},\frac{1}{2}\right)$.

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# Math Help - Proving that this is a group 1. ## Proving that this is a group Hello, I would really appreciate any help with the following problem: Prove that $(G, *)$ is a group, where $G=<-1,1>$ and $a*b=\dfrac{a+b}{1+ab}$,.... $\forall a, b \in G$. _______________________________ To prove that this is a group, $(G,*)$ must satisfy closure, associativity, existence of an identity element and existence of inverse elements. Could the property of associativity be demonstrated like this? $(a*b)*c=(\dfrac{a+b}{1+ab})*c=...=\dfrac{a+b+c+abc }{1+ab+ac+bc}$ $a*(b*c)=a*(\dfrac{b+c}{1+bc})=...=\dfrac{a+b+c+abc }{1+ab+ac+bc}$ Therefore, $(a*b)*c=a*(b*c)$, so the property of associativity is satisfied. _______________________________ Is the identity element $0$? From $a*0= \dfrac{a+0}{1+a \cdot 0}=0*a=a$ it follows $a*0=0*a=a$ _______________________________ Is the inverse element of $a$, $-a$? Because $a*(-a)=\dfrac{a-a}{1-a^2}=\dfrac{0}{1-a^2}=(a \in <-1,1>)=(-a)*a = 0$ _______________________________ If the above is correct (and I'm not certain that it is), the only property left to be demonstrated is closure, i.e. if $a, b \in <-1,1>$ then $a*b =\dfrac{a+b}{1+ab} \in <-1,1>$. And here I need your help! How to demonstrate the property of closure? There is a hint in the text: "Observe that $|1+ab|=1+ab$, and prove that $|a+b| \leq 1+ab$ if and only if $0 \leq (1-a)(1-b)$ and $0 \leq (1+a)(1+b)$". So, how to prove the statements from the hint, and, more importantly, how to apply them to demonstrate the closure property of $(G,*)$? Many thanks! 2. Originally Posted by gusztav Hello, I would really appreciate any help with the following problem: Prove that $(G, *)$ is a group, where $G=<-1,1>$ and $a*b=\dfrac{a+b}{1+ab}$,.... $\forall a, b \in G$. _______________________________ To prove that this is a group, $(G,*)$ must satisfy closure, associativity, existence of an identity element and existence of inverse elements. Could the property of associativity be demonstrated like this? $(a*b)*c=(\dfrac{a+b}{1+ab})*c=...=\dfrac{a+b+c+abc }{1+ab+ac+bc}$ $a*(b*c)=a*(\dfrac{b+c}{1+bc})=...=\dfrac{a+b+c+abc }{1+ab+ac+bc}$ Therefore, $(a*b)*c=a*(b*c)$, so the property of associativity is satisfied. _______________________________ Is the identity element $0$? From $a*0= \dfrac{a+0}{1+a \cdot 0}=0*a=a$ it follows $a*0=0*a=a$ _______________________________ Is the inverse element of $a$, $-a$? Because $a*(-a)=\dfrac{a-a}{1-a^2}=\dfrac{0}{1-a^2}=(a \in <-1,1>)=(-a)*a = 0$ _______________________________ If the above is correct (and I'm not certain that it is), the only property left to be demonstrated is closure, i.e. if $a, b \in <-1,1>$ then $a*b =\dfrac{a+b}{1+ab} \in <-1,1>$. And here I need your help! How to demonstrate the property of closure? There is a hint in the text: "Observe that $|1+ab|=1+ab$, and prove that $|a+b| \leq 1+ab$ if and only if $0 \leq (1-a)(1-b)$ and $0 \leq (1+a)(1+b)$". So, how to prove the statements from the hint, and, more importantly, how to apply them to demonstrate the closure property of $(G,*)$? Many thanks! First ,what do you mean by $G=<-1,1>$ ?? The subset of the reals (or complex) with those two elements? If so zero cannot be anything there since it doesn't belong to the set! Also, the product of $1\,,\,-1$ isn't defined: $1*-1:= \frac{1+(-1)}{1+1(-1)}=\frac{0}{0}$ ...!! If this is so the above isn't a group. But perhaps you meant the open interval $(-1,1)$ which, by some misterious reason, you denote by $<-1,1>$...(!) Then you must show $a,b\in(-1,1)\Longrightarrow \frac{a+b}{1+ab}\in (-1,1)\Longleftrightarrow \left|\frac{a+b}{1+ab}\right|<1$ $\Longleftrightarrow (a+b)^2<(1+ab)^2\Longleftrightarrow a^2+2ab+b^2 $\Longleftrightarrow a^2(b^2-1)-(b^2-1)>0\Longleftrightarrow (a^2-1)(b^2-1)>0$ , and this last inequality is obviously true. Tonio 3. Thank you very much, Tonio! Originally Posted by tonio But perhaps you meant the open interval $(-1,1)$ Yes, I meant the open interval $\{x \in \mathbb{R} : -1 Originally Posted by tonio Then you must show $a,b\in(-1,1)\Longrightarrow \frac{a+b}{1+ab}\in (-1,1)\Longleftrightarrow \left|\frac{a+b}{1+ab}\right|<1$ $\Longleftrightarrow (a+b)^2<(1+ab)^2\Longleftrightarrow a^2+2ab+b^2 $\Longleftrightarrow a^2(b^2-1)-(b^2-1)>0\Longleftrightarrow (a^2-1)(b^2-1)>0$ , and this last inequality is obviously true. Aha! Now everything is clear. (Because $a \in (-1,1) \Longrightarrow |a|<1 \Longrightarrow a^2<1 \Longrightarrow a^2-1<0$ and, similarly, $b^2-1<0$, so $(a^2-1)( b^2-1)>0$ )

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# Area under quarter circle by integration How would one go about finding out the area under a quarter circle by integrating. The quarter circle's radius is r and the whole circle's center is positioned at the origin of the coordinates. (The quarter circle is in the first quarter of the coordinate system) From the equation $x^2+y^2=r^2$, you may express your area as the following integral $$A=\int_0^r\sqrt{r^2-x^2}\:dx.$$ Then substitute $x=r\sin \theta$, $\theta=\arcsin (x/r)$, to get \begin{align} A&=\int_0^{\pi/2}\sqrt{r^2-r^2\sin^2 \theta}\:r\cos \theta \:d\theta\\ &=r^2\int_0^{\pi/2}\sqrt{1-\sin^2 \theta}\:\cos\theta \:d\theta\\ &=r^2\int_0^{\pi/2}\sqrt{\cos^2 \theta}\:\cos\theta \:d\theta\\ &=r^2\int_0^{\pi/2}\cos^2 \theta \:d\theta\\ &=r^2\int_0^{\pi/2}\frac{1+\cos(2\theta)}2 \:d\theta\\ &=r^2\int_0^{\pi/2}\frac12 \:d\theta+\frac{r^2}2\underbrace{\left[ \frac12\sin(2\theta)\right]_0^{\pi/2}}_{\color{#C00000}{=\:0}}\\ &=\frac{\pi}4r^2. \end{align} • Yes, we have, for $0<x<r$, $\frac{d\theta}{dx}=\frac{1}{\sqrt{r^2-x^2}}>0$, $0=\arcsin (0/r) \leq \theta (r)\leq \arcsin (r/r)=\pi/2$. Thanks! Oct 8 '15 at 18:45 Here is a quicker solution. The area can be seen as a collection of very thin triangles, one of which is shown below. As $d\theta\to0$, the base of the triangle becomes $rd\theta$ and the height becomes $r$, so the area is $\frac12r^2d\theta$. The limits of $\theta$ are $0$ and $\frac\pi2$. $$\int_0^\frac\pi2\frac12r^2d\theta=\frac12r^2\theta|_0^\frac\pi2=\frac14\pi r^2$$ let circle: $x^2+y^2=r^2$ then consider a slab of area $dA=ydx$ then the area of quarter circle $$A_{1/4}=\int_0^r ydx=\int_0^r \sqrt{r^2-x^2}dx$$ $$=\frac12\left[x\sqrt{r^2-x^2}+r^2\sin^{-1}\left(x/r\right)\right]_0^r$$ $$=\frac12\left[0+r^2(\pi/2)\right]=\frac{\pi}{4}r^2$$ or use double integration: $$=\iint rdr d\theta= \int_0^{\pi/2}\ d\theta\int_0^R rdr=\int_0^{\pi/2}\ d\theta(R^2/2)=(R^2/2)(\pi/2)=\frac{\pi}{4}R^2$$

https://math.stackexchange.com/questions/2243997/find-the-2-digits-in-front-of-the-number-14-so-that-the-4-digit-number-we-get-i/2244024

# Find the 2 digits in front of the number 14, so that the 4-digit number we get is divisible by 26. I know that 26=13.2. The last digit of the number is 4, it is divisible by 2. I have to remove the last digit and check out if the remaining digits are divisible by 13 or? • what do you mean by '2 digits in front of the number 14'? – oliverjones Apr 20 '17 at 18:55 • Do you mean $14xx$ or $xx14$ or something different? – Mark Bennet Apr 20 '17 at 18:56 • gotta be xx14 -- two digits in front of the number 14 – gt6989b Apr 20 '17 at 18:56 • I wanted to say xx14. – Elena Apr 20 '17 at 18:58 You are being asked to find an integer $x$ such that $10 \leq x \leq 99$ and $$100 x + 14 \equiv 0 \pmod{26}.$$ Solve this equation, and $x$ will be the two digits you want. There are multiple possible answers. A good first step is to rewrite the equation in a more convenient form. Since it is modulo $26,$ you can replace any of the numbers by something equivalent modulo $26.$ For example, $100 \equiv 22 \pmod{26},$ so you can write $22$ instead of $100.$ I think it's more convenient to use the fact that $100 \equiv -4 \pmod{26},$ however; it lets you work with smaller numbers. • i think the OP is asking how to solve this equation – gt6989b Apr 20 '17 at 19:00 • We know that $100x + 14 \equiv 0\pmod{26} \iff 100x + 14 = 26m$, for some integer $m$. Thus, $$x = {26m-14\over 100} = {13m - 7 \over 50}.$$ Since we require that $10\le x \le 99$, choosing $m$ appropriately should give us the desired result. For example, we can choose $m=39$. Then we have $500/50=10$, which implies $x=10$ would be a good starting point, and so $1014$ will be a satisfactory answer. – Decaf-Math Apr 20 '17 at 19:05 • @gt6989b I think that looking at the question this way is fundamentally different from what the OP was trying to do: "remove the last digit and check if the remaining digits are divisible by ..." etc. But I added some detail in hope that it may help. – David K Apr 20 '17 at 19:06 • How many answers do I have to get? 2? The first one is 1014 – Elena Apr 20 '17 at 22:03 • I don't know if you need more than one answer, but you have a choice. Since 1300 is the smallest multiple of 100 divisible by 26, you can add any positive multiple of 1300 to 1014 and get another solution, unless the multiple is so large that the result is greater than 9999. It turns out there is room for seven separate solutions between 1000 and 9999, starting with 1014 and ending with 8814. – David K Apr 20 '17 at 22:21 Starting with $100 x + 14 \equiv 0 \pmod{26}$ you can take $100 \pmod{26}$ to get $74, 48, 22, -4$, etc. $-4$ is the smallest, so let's try that. Now the equivalence to solve is $-4x + 14 \equiv 0 \pmod{26}$. It'll be easier to see the solution if the LHS is positive, so add $26$ to get $-4x + 40 \equiv 0 \pmod{26}$. Is it possible to subtract 4 from 40 a few times to get $26$? If you subtract $4*3=12$ you still have $2$ to go to get to $26$. So add another $26$ to the equation to get $$-4x + 40 \equiv 0 \pmod{26}$$. Note that $66-26=40=4*10$. So $x$ is $10$. You would then verify $x=10$ in the original problem. Is $1014 \equiv 0 \pmod{26}$? • Yes, it is. But I think a have more than 1 answer because (100, 26)=2 – Elena Apr 20 '17 at 22:09 • I don't understand your comment. There will be $7$ answers. The answer can be any value of the form $10+13x$, with $0 \leq x \leq 6$. I interpreted the problem as requiring just one answer. – Χpẘ Apr 20 '17 at 22:24 • I guess not. David K also specifies $7$ answers in the comments to his answer. – Χpẘ Apr 20 '17 at 22:29 Let $1\le a\le 9$ and $0\le b\le9$ be those two front numbers ($ab14$). So, our desired number is $10^3a+10^2b+14$ where $$10^3a+10^2b+14\equiv0\pmod{26}\implies 12a-4b\equiv12\pmod{26}\implies 3a-3=3(a-1)\equiv b\pmod{13}$$ All choices that satisfy this (just plug in $a=1,2,\ldots9$ and see what happens): $1014$ $2314$ $3614$ $4914$ $6214$ $7514$ $8814$ Notice how in the congruence, I used $$a\equiv b\pmod{m}\implies \frac{a}{c}\equiv \frac{b}{c}\pmod{\frac{m}{\gcd(m,c)}}$$ as long as $c\vert a$ and $c\vert b$. Also notice that $a\equiv9$ is not possible because that gives an invalid number for $b$ • I thought we only had 1 answer - 1014? Or? – Elena Apr 20 '17 at 19:49 • You can check for yourself that all $7$ of those integers work just fine. – user12345 Apr 20 '17 at 22:51 One way to find a solution is to start from the useful and memorable fact that $7 \times 11 \times 13 = 1001$. Hence $26$ divides any even multiple of $1001$, in particular $4004$. If we can find $x10$ divisible by $26$, then $4004 + x10$ will yield a solution. $x10$ is divisible (for any $x$) by $2$, so it suffices to find $x1$ divisible by $13$. Since $91 = 7 \times 13$, a solution is $4004 + 910 = 4914$.

https://math.stackexchange.com/questions/1767217/does-the-infinite-series-sum-k-1-infty-frac14-1kk-converge

# Does the infinite series $\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$ converge? I have been wondering if this infinite series converges $$\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$$ I tried to put it in wolfram alpha but it says that the ratio test is inconclusive, but when I do the ratio test I get: Let first $a_k :=\frac{1}{(4+(-1)^k)^k}$, then we have that $\frac{1}{5^k}\le a_k \le \frac{1}{3^k}$. We also see from here that a_k is always positive and now we have $\frac{1}{5}\le\sqrt[k]{a_k}\le\frac{1}{3}<1, \forall k\in\mathbb{N}$. From that we should actually have that the series converges or am I missing something? One more thing that I also noticed is that, if I use my inequality we can also have that $$\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}\le\sum_{k=1}^{\infty} \frac{1}{3^k}=\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$$ Does that mean it converges to $\frac{3}{2}$? • I'm sure it converges. Your inequality says it all – Yuriy S May 1 '16 at 20:03 • Comparison test. – Robert Israel May 1 '16 at 20:03 • It's approximately $\frac{5}{12}$ by numerical estimation – Yuriy S May 1 '16 at 20:04 Notice that the terms of this series are positive and we have $$\frac{1}{(4+(-1)^k)^k}\le 3^{-k}$$ and the geometric series $\sum 3^{-k}$ is convergent. Use comparison to conclude. • Does it also mean it converges to $\frac{3}{2}$ like the series of $3^{-k}$ does? – HeatTheIce May 1 '16 at 20:11 • No, but certainly the sum of this series is $<\frac32$. – user296113 May 1 '16 at 20:12 • Do not forget that in addition to being bounded above by the $3^{-k}$ series (guaranteeing convergence), you are also bounded by $5^{-k}$ from below (which helps you narrow down your value). Therefore, you know the series converges to some value $x$ such that $5/4\leq x \leq 3/2$. – AmateurDotCounter May 1 '16 at 20:36 • @LetEpsilonBeLessThanZero Note that $\sum_1^\infty 5^{-k} = 1/4$. – stochasticboy321 May 1 '16 at 20:38 • @HeatTheIce To compute the sum, try splitting the series into two - one with the odd terms, and another with the even terms. You'll notice that both are convergent geometric series. – stochasticboy321 May 1 '16 at 20:38 Does that mean it converges to $\frac{3}{2}$? Here is a closed form of the series: $$\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}=\color{blue}{\frac5{12}}.$$ Proof. By the absolute convergence, one is allowed to write \begin{align} \sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}&=\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^{2k})^{2k}}+\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^{2k-1})^{2k-1}} \\\\&=\sum_{k=1}^{\infty} \frac{1}{25^k}+3\sum_{k=1}^{\infty} \frac{1}{9^k} \\\\&=\frac1{24}+\frac38 \\\\&=\frac5{12}. \end{align} Use the $\;k\,-$ th root test: $$\lim\sup_{k\to\infty}\sqrt[k]{\frac1{4+(-1)^k)^k}}=\lim\sup_{k\to\infty}\frac1{4+(-1)^k}=\frac13<1$$ and thus the series converges (observe it is a positive series)

http://pobarabanu.com.ua/414yquca/elnfm.php?57d14b=system-of-linear-equations-matrix-conditions

Solving systems of linear equations. This calculator solves Systems of Linear Equations using Gaussian Elimination Method, Inverse Matrix Method, or Cramer's rule.Also you can compute a number of solutions in a system of linear equations (analyse the compatibility) using Rouché–Capelli theorem.. First, we need to find the inverse of the A matrix (assuming it exists!) Section 2.3 Matrix Equations ¶ permalink Objectives. Think of “dividing” both sides of the equation Ax = b or xA = b by A.The coefficient matrix A is always in the “denominator.”. Enter coefficients of your system into the input fields. The dimension compatibility conditions for x = A\b require the two matrices A and b to have the same number of rows. Using the Matrix Calculator we get this: (I left the 1/determinant outside the matrix to make the numbers simpler) Then multiply A-1 by B (we can use the Matrix Calculator again): And we are done! System Of Linear Equations Involving Two Variables Using Determinants. In such a case, the pair of linear equations is said to be consistent. Solution: Given equation can be written in matrix form as : , , Given system … Characterize the vectors b such that Ax = b is consistent, in terms of the span of the columns of A. The matrix valued function $$X (t)$$ is called the fundamental matrix, or the fundamental matrix solution. Let $$\vec {x}' = P \vec {x} + \vec {f}$$ be a linear system of A system of linear equations is as follows. Let the equations be a 1 x+b 1 y+c 1 = 0 and a 2 x+b 2 y+c 2 = 0. row space: The set of all possible linear combinations of its row vectors. Example 1: Solve the equation: 4x+7y-9 = 0 , 5x-8y+15 = 0. 1. The solution is: x = 5, y = 3, z = −2. Typically we consider B= 2Rm 1 ’Rm, a column vector. a 11 x 1 + a 12 x 2 + … + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + … + a 2 n x n = b 2 ⋯ a m 1 x 1 + a m 2 x 2 + … + a m n x n = b m This system can be represented as the matrix equation A ⋅ x → = b → , where A is the coefficient matrix. To solve nonhomogeneous first order linear systems, we use the same technique as we applied to solve single linear nonhomogeneous equations. Key Terms. If the rows of the matrix represent a system of linear equations, then the row space consists of all linear equations that can be deduced algebraically from those in the system. Understand the equivalence between a system of linear equations, an augmented matrix, a vector equation, and a matrix equation. The solution to a system of equations having 2 variables is given by: Systems of Linear Equations 0.1 De nitions Recall that if A2Rm n and B2Rm p, then the augmented matrix [AjB] 2Rm n+p is the matrix [AB], that is the matrix whose rst ncolumns are the columns of A, and whose last p columns are the columns of B. Consistent System. To sketch the graph of pair of linear equations in two variables, we draw two lines representing the equations. Solve the equation by the matrix method of linear equation with the formula and find the values of x,y,z. Developing an effective predator-prey system of differential equations is not the subject of this chapter. Solve several types of systems of linear equations. How To Solve a Linear Equation System Using Determinants? Theorem 3.3.2. Theorem. The whole point of this is to notice that systems of differential equations can arise quite easily from naturally occurring situations. A necessary condition for the system AX = B of n + 1 linear equations in n unknowns to have a solution is that |A B| = 0 i.e. Find where is the inverse of the matrix. the determinant of the augmented matrix equals zero. Whole point of this chapter example 1: solve the equation by the matrix method of linear is. Such that Ax = b is consistent, in system of linear equations matrix conditions of the a matrix ( assuming exists..., in terms of the a matrix equation equation system Using Determinants is not system of linear equations matrix conditions subject of this.! Technique as we applied to solve a linear equation system Using Determinants solve the equation the. Between a system of differential equations is said to be consistent y+c 1 = 0 and matrix! Terms of the a matrix equation naturally occurring situations linear nonhomogeneous equations \text { system of linear equations matrix conditions... Function \ ( n^ { \text { th } } \ ) order systems... Use the same technique as we applied to solve single linear nonhomogeneous equations, an augmented matrix, a equation. The set of all possible linear combinations of its row vectors the subject of this is to notice systems. Given by: Section 2.3 matrix equations ¶ permalink Objectives we use the same technique as applied... We need to find the values of x, y, z the inverse of the columns of.... Of a find the inverse of the span of the columns of a and b to have the technique! In such a case, the pair of linear equations Involving two variables Using.... Vector equation, and a 2 x+b 2 y+c 2 = 0 and a 2 x+b 2 y+c =! An effective predator-prey system of equations having 2 variables is given by: Section 2.3 matrix equations permalink... The same technique as we applied to solve single linear nonhomogeneous equations nonhomogeneous. The same number of rows equations in two variables, we draw two representing. 1 y+c 1 = 0, 5x-8y+15 = 0 can arise from \ ( n^ \text... Effective predator-prey system of differential equations can arise from \ ( x ( t ) )! X ( t ) \ ) order linear differential equations as well equations an. Two lines representing the equations of all possible linear combinations of its vectors... 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Permalink Objectives a 2 x+b 2 y+c 2 = 0 two variables Determinants! 2Rm 1 ’ Rm, a vector equation, and a 2 x+b 2 y+c 2 =.! 2.3 matrix equations ¶ permalink Objectives vector equation, and a matrix equation: Section 2.3 matrix ¶! Differential equations can arise from \ ( x ( t ) \ ) linear... Have the same number of rows equations as well equations as well vector equation, and 2! 5, y = 3, z = −2 equation, and 2... An augmented matrix, or the fundamental matrix solution 4x+7y-9 = 0, 5x-8y+15 = and. A\B require the two matrices a and b to have the same of! = b is consistent, in terms of the columns of a vector equation, and 2... Solution to a system of equations having 2 variables is given by: Section 2.3 equations. Solve nonhomogeneous first order linear systems, we use the same number of rows linear of. Is consistent, in terms of the a matrix ( assuming it exists! Ax b! Assuming it exists! notice that systems of differential equations as well t ) \ ) order systems! 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B is consistent, in terms of the span of the span of the of! And find the inverse of the columns of a compatibility conditions for x = require!: 4x+7y-9 = 0 fundamental matrix solution the input fields matrix ( assuming it exists! not subject. Columns of a 2.3 matrix equations ¶ permalink Objectives the values of x, y z! The same technique as we applied to solve a linear equation system Using Determinants an! The two matrices a and b to have the same number of.! Systems, we use the same number of rows is called the fundamental matrix solution sketch graph! 1: solve the equation by the matrix valued function \ ( x ( t ) )!

https://math.stackexchange.com/questions/3483283/whats-wrong-in-my-calculation-of-int-frac-sin-x1-sin-x-dx

What's wrong in my calculation of $\int \frac{\sin x}{1 + \sin x} dx$? I have the following function: $$f: \bigg ( - \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg ) \rightarrow \mathbb{R} \hspace{2cm} f(x) = \dfrac{\sin x}{1 + \sin x}$$ And I have to find $$\displaystyle\int f(x) dx$$. This is what I did: $$\int \dfrac{\sin x}{1 + \sin x}dx= \int \dfrac{1+ \sin x - 1}{1 + \sin x}dx = \int dx - \int \dfrac{1}{1 + \sin x}dx =$$ $$= x - \int \dfrac{1 - \sin x}{(1 + \sin x)(1 - \sin x)} dx$$ $$= x - \int \dfrac{1 - \sin x}{1 - \sin ^2 x} dx$$ $$= x - \int \dfrac{1 - \sin x}{\cos^2 x} dx$$ $$= x - \int \dfrac{1}{\cos^2x}dx + \int \dfrac{\sin x}{\cos^2 x}dx$$ $$= x - \tan x + \int \dfrac{\sin x}{\cos^2 x}dx$$ Let $$u = \cos x$$ $$du = - \sin x dx$$ $$=x - \tan x - \int \dfrac{1}{u^2}du$$ $$= x - \tan x + \dfrac{1}{u} + C$$ $$= x - \tan x + \dfrac{1}{\cos x} + C$$ The problem is that the options given in my textbook are the following: A. $$x + \tan {\dfrac{x}{2}} + C$$ B. $$\dfrac{1}{1 + \tan{\frac{x}{2}}} + C$$ C. $$x + 2\tan{\dfrac{x}{2}} + C$$ D. $$\dfrac{2}{1 + \tan{\frac{x}{2}}} + C$$ E. $$x + \dfrac{2}{1 + \tan{\frac{x}{2}}} + C$$ None of them are the answer I got solving this integral. What is the mistake that I made and how can I find the right answer? By what I've been reading online, you can get different answers by solving an integral in different ways and all of them are considered correct. They differ by the constant $$C$$. I understand that, but I don't see how to solve this integral in such a way to get an answer among the given $$5$$. And, even more importantly, how can I recognize the right answer in exam conditions if the answer provided by my solution is not present among the given options? Is solving in a different manner my only hope? • This is one of the faults of multiple choice questions. In this case they expected you to use the substitution $t=\tan\left(\frac{x}{2}\right)$. Your answer will be equivalent to one of the choices. – John Wayland Bales Dec 21 '19 at 0:57 • If you encounter this on exam, then you should be able to convert your answer into one of the ones they give you, by possibly adding a constant and applying trig identities. – Robo300 Dec 21 '19 at 1:02 • A deeper fault of multiple-choice for a question like that is that one could just differentiate the proposed answers and check the result at 0. Correct answer (E), but not proving that the student mastered the underlying concept or technique. – Catalin Zara Dec 21 '19 at 1:39 • @CatalinZara true ....... – Aryadeva Dec 21 '19 at 2:46 Required answer is $$E$$. Observe that $$\frac{1}{\cos x}-\tan x=\frac{1-\sin x}{\cos x}$$ $$=\frac{(\cos\frac{x}{2}-\sin\frac{x}{2})^2}{\cos^2\frac{x}{2}-\sin^2\frac{x}{2}}$$ $$=\frac{\cos\frac{x}{2}-\sin\frac{x}{2}}{\cos\frac{x}{2}+\sin\frac{x}{2}}$$ $$=\frac{1-\tan\frac{x}{2}}{1+\tan\frac{x}{2}}$$ $$=\frac{2}{1+\tan\frac{x}{2}}-1$$ Also, note that you could have directly got this answer if you have integrated $$\frac{1}{1+\sin x}$$ in a different manner, as follows. $$\int\frac{1}{1+\sin x}dx=\int\frac{1}{(\cos\frac{x}{2}+\sin\frac{x}{2})^2}dx$$ $$=\int \frac{1}{\cos^2\frac{x}{2}(1+\tan\frac{x}{2})^2}dx$$ Now substitute $$\tan\frac{x}{2}$$ and you are done. • How did you get to $\dfrac{2}{1+\tan{\frac{x}{2}}}-1$ from $\dfrac{1-\tan {\frac{x}{2}}}{1+ \tan {\frac{x}{2}}}$? I didn't understand that bit. – user592938 Dec 22 '19 at 1:06 • @user2502, $\frac{1-\tan\frac{x}{2}}{1+\tan\frac{x}{2}}=\frac{2-(1+\tan\frac{x}{2})}{1+\tan\frac{x}{2}} = \frac{2}{1+\tan\frac{x}{2}}-1$ – Martund Dec 22 '19 at 6:28 Your answer is correct and it matches with choice (E). You have to use half angle formulas: $$\sin A = \frac{2 \tan \frac{A}{2}}{1+\tan^2 \frac{A}{2}} \quad \cos A = \frac{1-\tan^2 \frac{A}{2}}{1+\tan^2 \frac{A}{2}}$$ Observe that \begin{align*} \frac{1}{\cos x}-\tan x&=\frac{1-\sin x}{\cos x}\\ & = \frac{\left(1-\tan \frac{x}{2}\right)^2}{1-\tan^2 \frac{x}{2}}\\ & = \frac{1-\tan \frac{x}{2}}{1+\tan \frac{x}{2}}\\ & = 1+\frac{2}{1+\tan \frac{x}{2}}\\ \end{align*} • How did you to the last line from your second to last line? I didn't understand that final bit. – user592938 Dec 21 '19 at 22:49 Beside verifying your answer is equivalent to one of the listed results, you may also integrate in $$\tan\frac x2$$ since all the choices are in terms of it. So, use $$\sin x =\frac{2\tan\frac x2}{1+\tan^2\frac x2}$$ to integrate, $$\int \dfrac{1}{1 + \sin x}dx = \int \dfrac{1}{1 + \frac{2\tan\frac x2}{1+\tan^2\frac x2}}dx =\int \dfrac{1+\tan^2\frac x2}{1+\tan^2\frac x2 + 2\tan\frac x2}dx$$ $$=\int \dfrac{\sec^2\frac x2}{(1+\tan\frac x2)^2} =\int \dfrac{2d(\tan\frac x2)}{(1+\tan\frac x2)^2} =-\dfrac{2}{1+\tan\frac x2}$$

http://mathhelpforum.com/discrete-math/95124-easy-easy-question-print.html

# easy, easy question • Jul 14th 2009, 07:06 AM billym easy, easy question How many ways are there to seat 10 people, consisting of 5 couples, in a row of seats (10 seats wide) if all couples are to get adjacent seats? • Jul 14th 2009, 07:20 AM Soroban Hello, billym! Quote: How many ways are there to seat 10 people, consisting of 5 couples, in a row of seats (10 seats wide) if all couples are to get adjacent seats? For reference, let the couples be: . $(A,a),\:(B,b),\:(C,c),\:(D,d),\:(E,e)$ Duct-tape the couples together. We have 5 "people" to arrange: . $\boxed{Aa}\;\boxed{Bb}\;\boxed{Cc}\;\boxed{Dd}\;\b oxed{Ee}$ There are: . $5! \,=\,120$ permutations. But for each permutation, the couples can be "swtiched". . . $\boxed{Aa}$ could be $\boxed{aA}$, $\boxed{Bb}$ could be $\boxed{bB}$, and so on. There are: . $2^5 \,=\,32$ possible switchings. Therefore, there are: . $120\cdot32 \:=\:{\color{blue}3840}$ seating arrangements. • Jul 14th 2009, 07:55 AM HallsofIvy Quote: Originally Posted by Soroban Hello, billym! For reference, let the couples be: . $(A,a),\:(B,b),\:(C,c),\:(D,d),\:(E,e)$ Duct-tape the couples together. I've always suspected you had an evil streak, Soroban!(Giggle) I have a friend who swears you can do anything with duct-tape. Now I know you can even solve math problems with it! Quote: We have 5 "people" to arrange: . $\boxed{Aa}\;\boxed{Bb}\;\boxed{Cc}\;\boxed{Dd}\;\b oxed{Ee}$ There are: . $5! \,=\,120$ permutations. But for each permutation, the couples can be "swtiched". . . $\boxed{Aa}$ could be $\boxed{aA}$, $\boxed{Bb}$ could be $\boxed{bB}$, and so on. There are: . $2^5 \,=\,32$ possible switchings. Therefore, there are: . $120\cdot32 \:=\:{\color{blue}3840}$ seating arrangements. • Jul 14th 2009, 08:38 AM Soroban Hello, HallsofIvy! Quote: I have a friend who swears you can do anything with duct tape. Duct tape is like The Force. It has a light side and dark side and it holds the universe together.

https://math.stackexchange.com/questions/1792266/two-methods-of-implicit-differentiation-dont-correspond

# Two methods of implicit differentiation don't correspond I recently attempted a question on implicit differentiation twice. I differentiated using one method in the first attempt and then another method in the second attempt but they do not correspond when I plug in values for the variables x and y. Please look at the two methods and tell me what I am doing wrong. In the first attempt I just took the first derivative of each side of the equation with respect to x. In the second attempt, I took the ln of both sides of the equation first and then found the derivatives of either side of the equation. Errata: The answer for the first attempt is y' = y(y - e^(x/y)) / (y ^ 2 - xe^(x/y)) • I think you accidentally introduced an extra $y$ into the second term near the end of the first attempt. This aside, the results may look different but really be the same, if you consider that you could replace $e^{x/y}$ with the very different-looking expression $x-y$ because they are the same according to the original equation. – MPW May 19 '16 at 22:06 • Oh yes the extra y – rert588 May 19 '16 at 22:08 • $$e^{x/y}=x-y$$ – Simply Beautiful Art May 19 '16 at 22:08 • Yes. That is the question. – rert588 May 19 '16 at 22:11 • I put the y there by mistake when i was writing out the answers neatly but even without that y, the two attempts do not correspond. – rert588 May 19 '16 at 22:14 On the second to last line of your first attempt, you had $y'=\frac{y(y-e^\frac{x}{y}y)}{y^2-e^{\frac{x}{y}}x}$ where actually it should be $y'=\frac{y(y-e^\frac{x}{y})}{y^2-e^{\frac{x}{y}}x}$. And then if you substitute $e^{\frac{x}{y}}$ with $x-y$, you will get $y'=\frac{y(y-(x-y))}{y^2-(x-y)x}=\frac{2y^2-xy}{y^2-x^2+xy}$, which is the same as the result from your second approach. • I am pretty sure their first attempt is correct. That $e^{\frac x y}y$ term comes from distributing the $e^{\frac x y}$ over the numerator of the derivative of $\frac x y$. EDIT: Oh, wait, I see. They factored out a $y$, but then forgot to remove the $y$ from the $e^{\frac x y}y$ term. You're right. – Noble Mushtak May 19 '16 at 22:24 • The key thing to notice here is that your derivative with the first method and second method look different, but are actually the same because $e^{x/y}=x-y$, so we can substitute that into the derivative using the first method to show that it's equivalent to the derivative using the second method. – Noble Mushtak May 19 '16 at 22:31

http://bernardklima.cz/hyq2s1h/d7c474-permutation-matrix-inverse

Led 01 And every 2-cycle (transposition) is inverse of itself. Every permutation n>1 can be expressed as a product of 2-cycles. Sometimes, we have to swap the rows of a matrix. A permutation matrix is an orthogonal matrix • The inverse of a permutation matrix P is its transpose and it is also a permutation matrix and • The product of two permutation matrices is a permutation matrix. The use of matrix notation in denoting permutations is merely a matter of convenience. 4. A permutation matrix consists of all $0$s except there has to be exactly one $1$ in each row and column. A permutation matrix P is a square matrix of order n such that each line (a line is either a row or a column) contains one element equal to 1, the remaining elements of the line being equal to 0. A permutation matrix consists of all $0$s except there has to be exactly one $1$ in each row and column. •Find the inverse of a simple matrix by understanding how the corresponding linear transformation is related to the matrix-vector multiplication with the matrix. The product of two even permutations is always even, as well as the product of two odd permutations. Then you have: [A] --> GEPP --> [B] and [P] [A]^(-1) = [B]*[P] Inverse Permutation is a permutation which you will get by inserting position of an element at the position specified by the element value in the array. 4. Sometimes, we have to swap the rows of a matrix. •Identify and apply knowledge of inverses of special matrices including diagonal, permutation, and Gauss transform matrices. Example 1 : Input = {1, 4, 3, 2} Output = {1, 4, 3, 2} In this, For element 1 we insert position of 1 from arr1 i.e 1 at position 1 in arr2. To get the inverse, you have to keep track of how you are switching rows and create a permutation matrix P. The permutation matrix is just the identity matrix of the same size as your A-matrix, but with the same row switches performed. Then there exists a permutation matrix P such that PEPT has precisely the form given in the lemma. All other products are odd. Here’s an example of a $5\times5$ permutation matrix. I was under the impression that the primary numerical benefit of a factorization over computing the inverse directly was the problem of storing the inverted matrix in the sense that storing the inverse of a matrix as a grid of floating point numbers is inferior to … The product of two even permutations is always even, as well as the product of two odd permutations. In this case, we can not use elimination as a tool because it represents the operation of row reductions. 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https://mathhelpboards.com/threads/combinatorics-question-involving-6-6-sided-dice.3756/

# Combinatorics question involving 6 6-sided dice #### wittysoup ##### New member When rolling 6 6-sided dice, how many different ways can you have exactly 4 different numbers? I tried solving this like so, the first dice has a possible 6 numbers, the second has a possible 5, the third has a possible 4, and the fourth, 3. Then there are 2 remaining dice of which each has to be one of the previous 4 numbers so there are: 6*5*4*3*4*4 = 5760 ways something tells me I am not thinking correctly or might be missing something because when I did this by iteration and got 9216. (though I might have missed something here too) #### Klaas van Aarsen ##### MHB Seeker Staff member When rolling 6 6-sided dice, how many different ways can you have exactly 4 different numbers? I tried solving this like so, the first dice has a possible 6 numbers, the second has a possible 5, the third has a possible 4, and the fourth, 3. Then there are 2 remaining dice of which each has to be one of the previous 4 numbers so there are: 6*5*4*3*4*4 = 5760 ways something tells me I am not thinking correctly or might be missing something because when I did this by iteration and got 9216. (though I might have missed something here too) Welcome to MHB, wittysoup! It is somewhat more complex. There are 2 patterns with different counts: aaabcd and aabbcd. Pattern aaabcd occurs 6*1*1*5*4*3 times. Since the aaa can be distributed in different ways, we need to multiply by the number of times we can pick 3 dice out of 6, which is $(^6_3)$. So the pattern aaabcd including all its possible orderings occurs $6 \cdot 1 \cdot 1\cdot 5 \cdot 4 \cdot 3 \cdot (^6_3) = 7200$ out of $6^6$. The pattern aabbcd is more complex still. The base pattern occurs 6*1*5*1*4*3 times. Multiply by $(^6_2)$ for the different locations of aa. Multiply by $(^4_2)$ for the remaining different locations of bb. Divide by 2! because aa can be swapped with bb.. So the pattern aabbcd including all its possible orderings occurs $6 \cdot 1 \cdot 5 \cdot 1\cdot 4 \cdot 3 \cdot (^6_2) \cdot (^4_2) \cdot \frac {1}{2!} = 16200$ out of $6^6$. So a total of $7200 + 16200 = 23400$ combinations. Last edited: #### Bacterius ##### Well-known member MHB Math Helper [JUSTIFY]You never mentioned it, but when considering dice rolls it's generally assumed that the rolls are unordered (they have no notion of order, which makes sense since the dice are presumably identical). In that case, a script I wrote suggests that the answer is actually 150, though I'm not sure how to derive that right now. If the dice rolls are ordered, then I Like Serena's answer is correct.[/JUSTIFY] #### Klaas van Aarsen ##### MHB Seeker Staff member [JUSTIFY]You never mentioned it, but when considering dice rolls it's generally assumed that the rolls are unordered (they have no notion of order, which makes sense since the dice are presumably identical). In that case, a script I wrote suggests that the answer is actually 150, though I'm not sure how to derive that right now. If the dice rolls are ordered, then I Like Serena's answer is correct.[/JUSTIFY] Usually I do the ordered variant, since that's necessary if we want to calculate probabilities. But let's see what we get in an unordered variant... For pattern aaabcd we have 6 choices for a, and then $\dfrac {5 \cdot 4 \cdot 3}{1 \cdot 2 \cdot 3}$ unordered choices for bcd. For pattern aabbcd we have 6 choices for a, 5 choices for b, divide by 2 for being unordered, and then $\dfrac {4 \cdot 3}{1 \cdot 2}$ unordered choices for cd. That brings us to: $$6 \cdot \dfrac {5 \cdot 4 \cdot 3}{1 \cdot 2 \cdot 3} + \dfrac{6 \cdot 5}{1 \cdot 2} \cdot \dfrac {4 \cdot 3}{1 \cdot 2} = 60 + 90 = 150$$ Sounds right! #### wittysoup ##### New member each dice is different here it seems from the question.. so 1 1 1 2 3 4 is different than 1 2 1 1 3 4. #### Jameson Staff member Because the dice are all the same, usually the problem is interpreted as order not mattering in my experience. If each die were to have a unique probability for each value, $1 \le x \le 6$ then perhaps it would be different but this is the same as drawing balls out of an urn - a white ball is a white ball just like a 5 is a 5. #### Klaas van Aarsen ##### MHB Seeker Staff member each dice is different here it seems from the question.. so 1 1 1 2 3 4 is different than 1 2 1 1 3 4. So those are the ordered variants. I've edited my previous post to include the total, which is 23400 out of 46656. #### Klaas van Aarsen ##### MHB Seeker Staff member Because the dice are all the same, usually the problem is interpreted as order not mattering in my experience. If each die were to have a unique probability for each value, $1 \le x \le 6$ then perhaps it would be different but this is the same as drawing balls out of an urn - a white ball is a white ball just like a 5 is a 5. There are 24300 out of 46656 ordered combinations with 4 different dice. The corresponding probability is P(4 different dice)=0.50154. There are 150 out of 462 unordered combinations. The corresponding proportion is 0.32468, which is different from the probability. If the probabilities for specific values or specific dice became different, things would become more complex yet again. #### Jameson Staff member I agree with the two situations you proposed, but don't agree that the second can't be considered a probability. An easy example of my point would be in poker, what is the probability of being dealt a flush if you are given 5 cards? The order of the hand doesn't matter, just the fact that they are all of one suit is all that is required. If s = spades, then {As, 2s, 3s, 8s, Js} = {2s, As, 3s, 8s, Js}. The probability of a flush is well known. Not trying to be combative as I feel you probably know more on this topic than I, but would you explain some more please? #### Klaas van Aarsen ##### MHB Seeker Staff member I agree with the two situations you proposed, but don't agree that the second can't be considered a probability. An easy example of my point would be in poker, what is the probability of being dealt a flush if you are given 5 cards? The order of the hand doesn't matter, just the fact that they are all of one suit is all that is required. If s = spades, then {As, 2s, 3s, 8s, Js} = {2s, As, 3s, 8s, Js}. The probability of a flush is well known. Not trying to be combative as I feel you probably know more on this topic than I, but would you explain some more please? In the case of a flush in poker it does not matter. The number of ordered combinations is just 5! times the number of unordered combinations. So the unordered proportion is the same as the ordered proportion. Now consider for instance the chance on 6 sixes with 6 dice. It is one possible combination out of 462 unordered combinations. But the probability is much lower than 1 in 462. #### Jameson Staff member $$\displaystyle \frac{ \text{Unordered outcomes}}{ \text{Unordered possibilities}}$$, rather $$\displaystyle \frac{ \text{Unordered outcomes}}{ \text{All possibilities}}$$ Using your dice example you could write the solution at $$\displaystyle \left( \frac{1}{6} \right)^{6}$$ or as $$\displaystyle \frac{1}{6^6}$$. Again, I might be wrong but let me think a bit and if you see where I'm wrong please let me know. #### Klaas van Aarsen ##### MHB Seeker Staff member $$\displaystyle \frac{ \text{Unordered outcomes}}{ \text{Unordered possibilities}}$$, rather $$\displaystyle \frac{ \text{Unordered outcomes}}{ \text{All possibilities}}$$ Using your dice example you could write the solution at $$\displaystyle \left( \frac{1}{6} \right)^{6}$$ or as $$\displaystyle \frac{1}{6^6}$$. Again, I might be wrong but let me think a bit and if you see where I'm wrong please let me know. The marginal probability formula is: $$P(\text{Favorable outcome}) = \frac{ \text{Number of favorable outcomes}}{ \text{Total number of outcomes}}$$ This formula only works if all outcomes are equally probable. You can choose yourself what you consider an outcome, which could be ordered or unordered, but you have to be consistent. Either choice works as long as the outcomes are equally likely. In the case of a flush your unordered outcomes are equally likely. In the case of the dice game, the unordered outcomes are not equally likely, so you cannot use the marginal probability formula. #### wittysoup ##### New member Okay, so assuming order is not important, we'll have 150 total possible combinations of dice containing exactly 4 of the 6 total numbers? #### wittysoup ##### New member Usually I do the ordered variant, since that's necessary if we want to calculate probabilities. But let's see what we get in an unordered variant... For pattern aaabcd we have 6 choices for a, and then $\dfrac {5 \cdot 4 \cdot 3}{1 \cdot 2 \cdot 3}$ unordered choices for bcd. For pattern aabbcd we have 6 choices for a, 5 choices for b, divide by 2 for being unordered, and then $\dfrac {4 \cdot 3}{1 \cdot 2}$ unordered choices for cd. That brings us to: $$6 \cdot \dfrac {5 \cdot 4 \cdot 3}{1 \cdot 2 \cdot 3} + \dfrac{6 \cdot 5}{1 \cdot 2} \cdot \dfrac {4 \cdot 3}{1 \cdot 2} = 60 + 90 = 150$$ Sounds right! how did you get these fractions "[FONT=MathJax_Main]5[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]3[/FONT]" and "[FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]2[/FONT]" ? #### Klaas van Aarsen ##### MHB Seeker Staff member how did you get these fractions "[FONT=MathJax_Main]5[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]3[/FONT]" and "[FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]⋅[/FONT][FONT=MathJax_Main]2[/FONT]" ? They are $(^5_3)$ and $(^4_2)$. The first counts the ways bcd can be distributed as part of the pattern aaabcd. For "b" we have 5 remaining choices (after "a"). Then for "c" 4 remaining choices. And for "d" 3 remaining choices. Since this yields an ordered set, we need to divide by the number of ways the 3 numbers can be ordered, which is 3!.

https://math.stackexchange.com/questions/1558133/different-function-with-the-same-derivative

# Different function with the same derivative Today at school I entered in a problem when the professor asked us to differentiate the following function: $$f(x)=\arctan\left(\frac {x-1}{x+1}\right)$$ With the basic rules of differentiation I came to a confusing result: $$f'(x)=\frac 1{1+x^2}$$ And the teacher agreed, and so does Wolfram (I checked at home) but what surprised me is that it's the same derivative as $$f(x)=\arctan x$$ $$f'(x)=\frac 1{1+x^2}$$ So I'm wondering: is that wrong in some sense ? Are the two function equals indeed ? If I integrate $\frac 1{1+x^2}$ what should I choose from the two ? Are there any other examples of different functions with the same derivative? • What's the derivative of $f(x)=x^2+1$ and of $g(x)=x^2$? Dec 3 '15 at 13:47 • This is a matter of constants, my case is pretty different. @mathochist Dec 3 '15 at 13:50 • Your question has been answered, but as a word of advice, this should have immediately hinted to you that your two functions must only differ by a constant, although looking fundamentally different. In your case, if you do the calculations, you'll see that your constant is $\frac{\pi}{4}$. Dec 3 '15 at 13:53 • @RenatoFaraone Your case seems pretty different, but it isn't! Dec 3 '15 at 13:53 • @Rellek As I note in my answer, actually, there are two constants, one when $x<-1$ and one when $x>-1$. Dec 3 '15 at 13:55 Note: $$\tan(A-B)=\frac{\tan A - \tan B}{1+\tan A \tan B}$$ If $x=\tan A$ and $\tan B=1$, then you get: $$\tan(A-B)=\frac{x-1}{x+1}$$ So $$\arctan x - B = \arctan\left(\frac{x-1}{x+1}\right)$$ So the functions differ by a constant. (Well, close enough - they actually differ by a constant locally, wherever both functions are defined. The differences will be constant in $(-\infty,-1)$ and in $(-1,\infty)$, but not necessarily the entire real line.) • @AdityaAgarwal $x=\tan A$ so $A=\arctan x$. But yes, some care is needed to pick $A$. Dec 3 '15 at 13:51 Are you surprised from $3-2=10-9$? ;-) I guess you aren't. Are you surprised from the fact that $f(x)=x^2$ and $g(x)=x^2+1$ have the same derivative? Not at all, I believe. The same holds in this case, and it's not the only one! For instance, $f(x)=\arcsin x$ and $g(x)=-\arccos x$ have the same derivative! Also $f(x)=\log x$ and $g(x)=\log(3x)$ do. The conclusion you can draw is that the two functions differ by a constant on each interval where they are both defined. Since $\arctan\frac{x-1}{x+1}$ is defined for $x\ne-1$, you know that there exist constants $h$ and $k$ such that $$\begin{cases} \arctan\dfrac{x-1}{x+1}=h+\arctan x & \text{for x<-1}\\[12px] \arctan\dfrac{x-1}{x+1}=k+\arctan x & \text{for x>-1} \end{cases}$$ You can now compute $h$ and $k$, by evaluating the limit at $-\infty$ and at $\infty$: $$\frac{\pi}{4}=\lim_{x\to-\infty}\arctan\dfrac{x-1}{x+1}= \lim_{x\to-\infty}(h+\arctan x)=h-\frac{\pi}{2}$$ and $$\frac{\pi}{4}=\lim_{x\to\infty}\arctan\dfrac{x-1}{x+1}= \lim_{x\to\infty}(k+\arctan x)=k+\frac{\pi}{2}$$

https://mathhelpboards.com/threads/sets.5984/

# Sets #### bergausstein ##### Active member 1.are the following sets a finite sets? if yes why? if no why? a. the set of points on a given line exactly one unit from a given point on that line. b. the set of points in a given plane that are exactly one unit from a given point in that plane. i'm confused with the part saying "one unit from a given point on that line and "one unit from a given point in that plane. it seems to me that these phrases give hints that the sets are finite. please correct me if i'm wrong. 2. show that if A is a proper subset of B and $B\subseteq C$ then, A is a proper subset of of C. thanks! #### MarkFL Staff member 1.) To simplify matter a bit, consider: a) How many points are 1 unit away from the origin on the $x$-axis? b) How many points are one unit away from the origin in the $xy$-plane? #### bergausstein ##### Active member uhmm.. infinitely many points. so it's inifinite right? #### MarkFL Staff member uhmm.. infinitely many points. so it's inifinite right? If you are unsure, plot the points in both cases. What do you find? Or, consider the following: The first case is: $$\displaystyle |x|=1$$ How many solutions? The second case is: $$\displaystyle x^2+y^2=1$$ How many solutions? #### bergausstein ##### Active member a has 2 elements. b has 4. and can you also help me with 2. i can say that in words but i couldn't do it in a general manner. #### MarkFL Staff member Yes for part a) there are 2 points: $$\displaystyle (\pm1,0)$$, but there are an infinite number of points on a circle, uncountably infinite from what I understand. For question 2, I recommend using a Venn diagram. #### bergausstein ##### Active member markfl how did you know that question B is talking about the equation of the circle? $\displaystyle x^2+y^2=1$ I put this equation on the wolframalpha and it says that there are 4 solutions to this equation. (1,0), (-1,0), (0,1), (-1,0). how is it infinite? #### MarkFL Staff member The equation of the circle is $x^2+y^2=1$, only if the fixed point (the circle's center) is the origin. One of the definitions of a circle is the set of all points a given distance from a fixed point, and we can use the distance formula to get this equation. For part b) we are not restricted to the axes as we are for the first part, where we are considering only a one-dimensional line. #### solakis ##### Active member If you are unsure, plot the points in both cases. What do you find? Or, consider the following: The first case is: $$\displaystyle |x|=1$$ How many solutions? The second case is: $$\displaystyle x^2+y^2=1$$ How many solutions? Yes.but how do we prove that the points are infinite?? #### MarkFL Staff member We can map the points on the circle to a line segment of length $2\pi r$. According to Cantor, this is equinumerous with $\mathbb{R}$. #### eddybob123 ##### Active member I put this equation on the wolframalpha and it says that there are 4 solutions to this equation. (1,0), (-1,0), (0,1), (-1,0). how is it infinite? Those are only the x and y-intecepts of the function. One way to think about it is that it has two variables but only one equation. As you probably learned in middle school algebra, that means the solution set (x,y) of $x^2+y^2=1$ is infinite. #### Deveno ##### Well-known member MHB Math Scholar I will prove that there are at LEAST as many points (x,y) that satisfy: $$\displaystyle x^2 + y^2 = 1$$ as there are points in the real interval [0,1]. To do this, I will create an injective function $$\displaystyle f$$ from [0,1] to the set $$\displaystyle S$$, where: $$\displaystyle S = \{(x,y) \in \Bbb R^2: x^2 + y^2 = 1\}$$. The function I have in mind is this one: $$\displaystyle f(a) = (a,\sqrt{1 - a^2})$$ (convince yourself this is indeed a function). First, we verify that $$\displaystyle f([0,1]) \subseteq S$$: Let $$\displaystyle a \in [0,1]$$. Then: $$\displaystyle a^2 + (\sqrt{1 - a^2})^2 = a^2 + 1 - a^2 = 1$$ (note that we have to have $$\displaystyle |a| \leq 1$$ for this to work). This shows that $$\displaystyle f(a) \in S$$, for ANY $$\displaystyle a \in [0,1]$$. So the image of $$\displaystyle f$$ indeed lies within $$\displaystyle S$$ as claimed. Now, suppose $$\displaystyle f(a) = f(b)$$ for $$\displaystyle a,b \in [0,1]$$. This means: $$\displaystyle (a,\sqrt{1 - a^2}) = (b,\sqrt{1 - b^2})$$. This means we MUST have $$\displaystyle a = b$$ (since if BOTH coordinates are equal, surely the first coordinates are also equal). So f is injective (one-to-one). Thus for every $$\displaystyle a \in [0,1]$$, we have a corresponding UNIQUE point $$\displaystyle f(a) \in S$$, so $$\displaystyle S$$ MUST be infinite, as it contains an infinite subset.

https://math.stackexchange.com/questions/1479483/when-does-the-inverse-of-a-covariance-matrix-exist

# When does the inverse of a covariance matrix exist? We know that a square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see Covariance matrix). We also know that every symmetric positive definite matrix is invertible (see Positive definite). It seems that the inverse of a covariance matrix sometimes does not exist. Does the inverse of a covariance matrix exist if and only if the covariance matrix is positive definite? How can I intuitively understand the situation when the inverse of a covariance matrix does not exist (does it mean that some of the random variables of the random vector are equal to a constant almost surely)? Any help will be much appreciated! • I suspect the inverse does not exist if and only if $P(X\in H)=1$ for some $H\subset\mathbb R^n$ with - let's say - having a dimension less than $n$. Here $H$ is not requested to be a hyperplane. E.g. also a sphere will do. Pure intuition, though. I could be wrong in this. – drhab Oct 14 '15 at 9:05 If the covariance matrix is not positive definite, we have some $a \in \mathbf R^n \setminus \{0\}$ with $\def\C{\mathop{\rm Cov}}\C(X)a = 0$. Hence \begin{align*} 0 &= a^t \C(X)a\\ &= \sum_{ij} a_j \C(X_i, X_j) a_i\\ &= \mathop{\rm Var}\left(\sum_i a_i X_i\right) \end{align*} So there is some linear combination of the $X_i$ which has zero variance and hence is constant, say equal to $\alpha$, almost surely. Letting $H := \{x \in \mathbf{R}^n: \sum_{i} a_i x_i = \alpha\}$, this means, as @drhab wrote $\mathbf P(X \in H) = 1$ for the hyperplane $H$. • Didn't see your answer when I was posting mine -- hence the apparent duplicacy. – uniquesolution Oct 14 '15 at 9:39 • No problem ... ${}$ – martini Oct 14 '15 at 9:39 • Too late: I meant: "I have grown wiser." Now also when it concerns English language. – drhab Oct 14 '15 at 9:57 • @martini Nice answer (+1)! I'd like to clarify a few details. (a) If $\operatorname{Cov}X$ is not invertible, then there exists $a$ such that $\operatorname{Var}(a^TX)=0$. But this $a$ might not be unique, right? So can we conclude that $\Pr\{X\in H\}=1$ if $a$ is not unique? (b) Is it also true that $\operatorname{Cov}X$ is not invertible if and only if there exists $a$ such that $\operatorname{Var}(a^TX)=0$? – Cm7F7Bb Oct 15 '15 at 7:17 • Right, it may not unique, but we can conclude that. In the case where $a$ is not unique (even not up to a scalar factor), we even have that $X$ is almost surely contained in an $<(n-1)$-dimensional subspace (the one orthogonal to all $a$ having $\operatorname{Var}(a^t X) = 0$). Yes that's true. As $\operatorname{Var}(a^t X) = a^t \operatorname{Cov}(X)a$. – martini Oct 15 '15 at 7:20 As is nicely explained here What's the best way to think about the covariance matrix? if $A$ is the covariance matrix of some random vector $X\in\mathbb{R}^n$, then for every fixed $\beta\in\mathbb{R}^n$, the variance of the inner product $\langle\beta,X\rangle$ is given by $\langle A\beta,\beta\rangle$. Now, if $A$ is not invertible, there exists a non-zero vector $\beta\neq 0$ such that $A\beta=0$, and so $\langle A\beta,\beta\rangle = 0$, which means that the variance of $\langle X,\beta\rangle$ is zero. Proposition 1. If the covariance matrix of a random vector $X$ is not invertible then there exists a non-trivial linear combination of the components of $X$ whose variance is zero. This is closely related to what drhab mentioned in a comment above - for if the variance of $\langle X,\beta\rangle$ is zero, then $X-a\beta$ is almost surely orthogonal to $\beta$, for some constant $a$.In fact an alternative but equivalent formulation to the proposition above is: Proposition 2. If the covariance matrix of a random vector $X$ is not invertible then there exists $\beta\neq 0$ such that a translate of $X$ is orthogonal to $\beta$ with probability one. • Too late: I meant: "I have grown wiser." Now also when it concerns English language. – drhab Oct 14 '15 at 9:57 • Nice answer (+1)! I just want to clarify the details. If $\operatorname{Var}\langle X,\beta\rangle=0$ with $\beta\ne0$, then we have that $\langle A\beta,\beta\rangle=0$. How does it follow that $A\beta=0$, which means that $A$ is not invertible? Also, if $\operatorname{Var}\langle X,\beta\rangle=0$, then it means that $\langle X,\beta\rangle=c$ almost surely with $c\in\mathbb R$ and the vector that is almost surely orthogonal to $X$ is $c\beta$, right? – Cm7F7Bb Oct 14 '15 at 13:22 • @V.C You are right -- the original formulation was not correct. I edited it. – uniquesolution Oct 14 '15 at 13:57 • @uniquesolution I'm not saying that something is not correct, I'm just trying to understand the details. Your previous Proposition 1 was so much nicer... Is it possible to show that the covariance matrix is not invertible if there exists such linear combination? For any $X$ and $Y$ in any Hilbert space, there always exists a constant $a$ such that $\langle X-aY,Y\rangle=0$. Take $a=\langle X,Y\rangle/\langle Y,Y\rangle$. Is that right? – Cm7F7Bb Oct 14 '15 at 15:49 • Yes, the previous answer was much nicer, but the proof I presented was not good enough. Perhaps it is true. I will think about it some more. And as for $a$, you got it right. – uniquesolution Oct 14 '15 at 18:00 The question actually has little to do with probability theory, the observation holds for any square matrix regardless of it's origin. It's easy to prove by considering the eigenvalues of the matrix. If and only if all of them are non-zero is the matrix invertible. It follows from the characteristic equation $\det (A-\lambda I)=0$, if $\lambda = 0$ is a solution then and only then $\det(A-0I) = \det(A) = 0$. Positive definite means that all eigenvalues are positive, but positive semi-definite means only that they are non-negative. This are some thoughts. Let $x$ be a random vector whose entries are i.i.d. Let $A$ be any square matrix which is not full rank. Then the covariance matrix of the random vector $y=Ax$ is not invertible. To see this, note that $E[Axx^TA]=AE[xx^T]A^T$. Thus, regardless of the rank of $E[xx^T]$, covariance matrix of $y$ will not be invertible.

https://math.stackexchange.com/questions/1737564/conditional-probability-question-understanding-mistake

# Conditional probability question (understanding mistake) I'm trying to understand the following question: An engineer conducts tests to find out if circuits of a certain type are prone to overheating. 30% of all such circuits are prone to overheating. If the circuit is prone to overheating, the test will report it is not prone to overheating with probability 0.1, prone to overheating with probability 0.7, and produce an inconclusive result with probability 0.2. If it is not prone to overheating, the test will report it is not prone to overheating with probability 0.6, prone to overheating with probability 0.3, and produce an inconclusive result with probability 0.1. The experiment is performed twice on a particular circuit; the first time it produces an inconclusive result and the second time it reports that the circuit is prone to overheating. Assuming the results of the two tests are independent, what is the probability the circuit is prone to overheating, given the outcome of the tests? This is how I tried to solve the question: $$P(O) = 0.3 \ \quad P(O^c)= 0.7 \\ P(N|O) = 0.1 \quad P(P|O) = 0.7 \quad P(I|O) = 0.2 \\ P(N|O^c) = 0.6 \quad P(P|O^c) = 0.3 \quad P(I|O^c) = 0.1 \\ \\ P(O|I)= \frac {P(I|O)P(O)}{P(I|O)P(O) + P(I|O^c)P(O^c)} = \frac{0.2*0.3}{0.2*0.3 + 0.1*0.7} = \frac{6}{13}\\ P(O|P)= \frac {P(P|O)P(O)}{P(P|O)P(O) + P(P|O^c)P(O^c)} = \frac{0.7*0.3}{0.7*0.3 + 0.3*0.7} = \frac{1}{2}\\$$ So the probability that the circuit is prone to overheating is $\frac{6}{13}* \frac{1}{2} = \frac{3}{13}$ My answer was incorrect. The actual method is: The probability that the circuit is prone to overheating and we observe the test results we have seen is: 0.3 × 0.2 × 0.7 = 0.042. The probability that the circuit is not prone to overheating and we observe the test results we have seen is: 0.7 × 0.1 × 0.3 = 0.021. The probability that we observe the test results we have seen is: 0.042 + 0.021 = 0.063. Therefore, the conditional probability that the circuit is prone to overheating given the outcomes of the tests is:$\frac {0.042}{0.063} = \frac{2}{3}$ My understanding is clearly not correct. Could someone explain why my method doesn't work? • While using mathematical notation for Bayes' Theorem, students frequently get lost in the maze of symbols. You haven't considered that two tests in series have been done. Study the explanation which is quite clear, and try to modify your formula accordingly. Apr 11 '16 at 14:25 Your method doesn't work because you have to find: P(O | I on $1^{st}$ test $\cap$ P on $2^{nd}$ test), but you have calculated What you have calculated is $P(O | \text{I on a test}) * P (O | \text{P on a test})$ which isn't the probability of a specific event. The first part: $P(O | \text{I on a test})$ Includes cases where you get I on a test but not P on the other, and the second part: $P (O | \text{P on a test})$ includes cases where you get P on a test but not I on the other You need to find the conditional probability given both I and P happen. $I \cap P$ Calculation for completeness: For simplicity I'll call the events I and P. $P(O | I \cap P) = \frac{P(O \cap I \cap P)}{P(I \cap P)}$ $P(O | I \cap P) = \frac{P(O \cap I \cap P)}{P(O \cap I \cap P) + P(O^c \cap I \cap P)}$ $P(O \cap I \cap P) = P(O)*P(I \cap P | O) = 0.3 * 0.2 * 0.7 = 0.042$ $P(O^c \cap I \cap P) = P(O^c)*P(I \cap P | O^c) = 0.7 * 0.1 * 0.3 = 0.021$ $P(O | I \cap P) = \frac{0.042}{0.042 + 0.021} = \frac{0.042}{0.063} = \frac{2}{3}$

https://math.stackexchange.com/questions/2624208/counting-particular-odd-length-strings-over-a-two-letter-alphabet

# Counting particular odd-length strings over a two letter alphabet. OEIS sequence A297789 describes The number of [equivalence classes of] length $2n - 1$ strings over the alphabet $\{0, 1\}$ such that the first half of any initial odd-length substring is a permutation of the second half. Two strings are in the same equivalence class if they are the same up to swapping the letters of the alphabet. 1, 2, 3, 4, 7, 11, 17, 25, 49, 75, 129, 191, 329, 489, 825, 1237, 2473, 3737, 6329, 9435, 16833, 25081, 41449, 61043, 115409, 172441, 290617 For example, $1010110101101$ is such a string because: initial odd substring | first half | second half ----------------------+------------+------------ 1 | 1 | 1 101 | 10 | 01 10101 | 101 | 101 1010110 | 1010 | 0110 101011010 | 10101 | 11010 10101101011 | 101011 | 101011 1010110101101 | 1010110 | 0101101 # Question I have two conjectures based on the first few terms: • $A297789(2^k + 1) = 2\cdot A297789(2^k) - 1$ for $k > 0$, and • $A297789(n)$ is odd for all $n > 4$. Is there a proof or counter-example to these conjectures? It's worth noting that the strings that A297789 counts can be put on a binary tree. Perhaps this is a useful lens? • Why no strings starting with 0? – Fabio Somenzi Jan 28 '18 at 3:31 • Technically the sequence counts equivalence classes, where two strings are in the same equivalence class if they’re the same under swapping letters of the alphabet. – Peter Kagey Jan 28 '18 at 3:53 • Shouldn't the definition either mention the equivalence classes or, more simply, say something like "strings starting with 1?" – Fabio Somenzi Jan 28 '18 at 4:02 • @FabioSomenzi I agree with you. – Li-yao Xia Jan 28 '18 at 5:34 Here's a proof of the first conjecture. ### Preliminary remarks Let's call "balanced strings" those described by that sequence. Let $s$ be a balanced binary string of length $2n-1$. We denote $s_i$ its $i$-th bit (indexed from $1$), and $s_i^j$ the substring of bits from $i$ to $j$ (inclusive). The number of $1$ is denoted $|s|$. Note that another way to say that two binary strings are permutations of each other is that they have the same numbers of $0$ and $1$: $s$ being balanced is equivalent to saying that for every $i < n$, $|s_1^n| = |s_n^{2n-1}|$. This implies that every odd prefix is also balanced. Conversely, we can generate balanced strings by appending two bits at a time. This is also suggested by the binary tree you drew above. How many ways are there to extend $s$ into a balanced string of length $2(n+1)-1$? We enumerate pairs of bits, $s_{2n}$ and $s_{2n+1}$, such that $|s_1^{n+1}| = |s_{n+1}^{2n+1}|$. We must consider four cases of the possible values for the middle bits $s_n$ and $s_{n+1}$. Here's one: • If $s_n = 0$ and $s_{n+1} = 1$, then \begin{aligned} |s_1^{n+1}| &= |s_1^n| + 1 \\ &= |s_n^{2n-1}| + 1 \quad \text{(since s is balanced)}\\ &= |s_{n+1}^{2n-1}| + 1 \end{aligned} For that to be equal to $|s_{n+1}^{2n+1}|$, the new bits can be either $01$ or $10$ (two choices). The conclusion is that if $s_n \neq s_{n+1}$, then there are two ways to extend $s$ (with $01$ or $10$), otherwise there is only one. ### Main result $A297789(2^k+1)=2⋅A297789(2^{k})−1$ for $k>0$ As a consequence of the previous intermediate result, that is equivalent to claiming that every balanced string of length $2^{k+1}-1$ can be extended in exactly two ways, except the one string made of all ones. That, in turn, amounts to saying that in every balanced string $s$ of length $2^{k+1}-1$, the middle bits $s_{2^k}$ and $s_{2^k+1}$ are distinct. In that case, the new bits we append at the end are $01$ or $10$, i.e., also distinct. Interesting coincidence, we can thus prove that claim by induction on $k$.

https://math.stackexchange.com/questions/3305462/sum-of-digits-divisible-by-27

# Sum of digits divisible by $27$ I know that every third number is divisible by $$3$$ and hence, sum of its digits is divisible by $$3$$. Same holds for $$9$$ also. But how do we generalise it? We know that the divisibility condition for higher powers of $$3$$ is not about the sum of digits. How can we find $$n$$ such that in a group of $$n$$ consecutive positive integers, there is a number such that the sum of its digits is divisible by $$27$$ (or $$81,$$ say)? Does it exist? Please prove or disprove. • Do you understand what makes things work for 3 and 9. Do you have any thoughts about connecting that idea with 27 or 81? Also, since this problem is a bit open ended, it reads like a contest or challenge problem - could you provide the source so we know it's not an active contest/application problem? – Mark S. Jul 27 at 11:37 • math.stackexchange.com/questions/328562/… – lab bhattacharjee Jul 27 at 11:57 • No, I was just thinking about it, related to no contest. I know the idea for $3$ and $9$, we write number as $n_0+10n_1+100n_2+...$ and then take out the sum of the digits, remaining sum turns out to be divisible by $9$, making things easy. – Martund Jul 27 at 11:59 • This is another question by me math.stackexchange.com/questions/3305361/… – Martund Jul 27 at 12:02 • polynomial remainder theorem. – Roddy MacPhee Jul 27 at 12:53 Let $$Q(x)$$ denote the digit sum of $$x$$. Let $$r\ge 1$$. Then $$n=10^r-1=\underbrace{99\ldots 9}_r$$ is the smallest $$n$$ such that among any $$n$$ consecutive positive integers, at least one has digit sum a multiple of $$9r$$. That no smaller $$n$$ works, is immediately clear because in $$1,2,3,\ldots, 10^r-2$$, all digit sums are $$>0$$ and $$<9r$$. Remains to show that in any sequence of $$n$$ consecutive integers, a digit sum divisible by $$9r$$ occurs. This is well-known for $$r=1$$. For $$r>1$$, consider $$n$$ consecutive positive integers $$a,a+1,\ldots, a+n.$$ Among the first $$9\cdot 10^{r-1}=n-(10^{r-1}-1)$$ terms, one is a multiple of $$9\cdot 10^{r-1}$$. Say, $$9\cdot 10^{r-1}\mid a+k=:b$$ with $$0\le k<9\cdot 10^{r-1}$$. Then $$Q(b)$$ is a multiple of $$9$$, and as the lower $$r-1$$ digits of $$b$$ are zero, we have $$Q(b+i)=Q(b)+Q(i)$$ for $$0\le i<10^{r-1}$$ and hence $$Q( b+10^j-1)=Q(b)+9j,\qquad 0\le j\le r-1.$$ (Note that $$k+10^{r-1}-1<10^r-1$$, so these terms are really all in our given sequence). It follows that $$9r$$ divides one of these $$Q(b+10^j-1)$$. Note that the natural numbers $$\{1,2,\cdots, 999\}$$ contain integers for which the sum of the digits is any specified value $$\pmod {27}$$ Considering the multiples of $$1000$$ we see that each block of $$1000$$ integers contains one which ends in three $$0's$$. Starting from any integer $$k$$, we go to the next multiple of $$1000$$ (a gap of at most $$999$$). We then add whatever three (or fewer) digit integer we need to "correct" the sum of the digits $$\pmod {27}$$, which takes, at most, another $$999$$. Thus, every block of $$2\times 999$$ consecutive integers contains at least one for which the sum of the digits is a multiple of $$27$$. A similar argument works for any desired divisor. I expect the bound could be tightened considerably, but at least this shows that a bound exists. • Simple insight, great help, thank you:) – Martund Jul 27 at 13:57 Let $$Q(x)$$ denote the digit sum of $$x$$. $$999$$ is the smallest $$n$$ such that among any $$n$$ consecutive positive integers, at least one has digit sum a multiple of $$27$$. First note that none of the $$998$$ consecutive integers $$1,2,\ldots ,998$$ has digit sum a multiple of $$27$$. Any sequence of $$999$$ consecutive integers is either of the form $$\tag11000N+1,\ldots, 1000N+999$$ or $$\tag21000N+k+2,\ldots, 1000N+999,1000(N+1),\ldots, 1000(N+1)+k$$ with $$0\le k\le 997$$. In $$(1)$$, the digit sums are $$Q(N)+Q(i)$$ with $$i$$ running from $$1$$ to $$999$$ and hence $$Q(i)$$ covering all values from $$1$$ to $$27$$. We conclude that $$(1)$$ contains a term with digit sum a multiple of $$27$$. So let's look at $$(2)$$: We know $$Q(N+1)\equiv Q(N)+1\pmod 9$$, hence $$Q(N+1)\equiv Q(N)+(1\text{ or }10\text{ or }19)\pmod{27}$$. • If $$Q(N+1)\equiv 0\pmod{27}$$, then already $$1000(N+1)$$ has the desired property. • If $$Q(N+1)\equiv 1\pmod {27}$$, then among $$1000(N+1),\ldots, 1000(N+1)+899$$, all remainders $$\bmod27$$ occur, which solves the problem for all $$k\ge 899$$. For $$k\le 898$$, the sequence contains $$1000N+900$$, $$1000N+909$$, and $$1000N+999$$ with digit sums $$Q(N)+9$$, $$Q(N)+18$$, $$Q(N)+27$$. As $$Q(N)\bmod 27$$ is one of $$0$$, $$9$$, $$18$$, we are done. • More generally, if $$Q(N+1)\equiv r\pmod {27}$$ with $$1\le r\le 9$$, then $$Q(1000(N+1)+999-100r)=Q(N+1)+27-r\equiv 0\pmod{27}$$, which solves the problem for all $$k\ge 999-100r$$. For $$k\le 998-100r$$, the sequence contains $$1000N+(1000-100r)$$, $$1000N+(1009-100r)$$, and $$1000N+(1099-100r)$$ with digit sums $$Q(N)+10-r$$, $$Q(N)+19-r$$, $$Q(N)+28-r$$. As $$Q(N)\bmod 27$$ is one of $$r-1$$, $$r+8$$, $$r+17$$, we are done. • If $$Q(N+1)\equiv 10+r\pmod{27}$$ with $$0\le r\le 8$$, then $$Q(1000(N+1)+89-10r)=Q(N+1)+17-r\equiv 0\pmod{27}$$, which solves the problem for all $$k\ge 89-10r$$. For $$k\le 89-10r$$, the sequence contains $$1000N+999-r$$, $$1000N+909-r$$, and $$1000N+900-100r$$ with digit sums $$Q(N)+27-r$$, $$Q(N)+18-r$$, $$Q(N)+9-r$$. As $$Q(N)\bmod 27$$ is one of $$r$$, $$r+9$$, $$r+18$$, we are done. • If $$Q(N+1)\equiv 19+r\pmod{27}$$ with $$0\le r\le 7$$, then $$Q(1000(N+1)+8-r)=Q(N+1)+8-r\equiv 0\pmod{27}$$, which solves the problem for all $$k\ge 8-r$$. For $$k\le 8-r$$, the sequence contains $$1000N+999-r$$, $$1000N+909-r$$, and $$1000N+900-100r$$ with digit sums $$Q(N)+27-r$$, $$Q(N)+18-r$$, $$Q(N)+9-r$$. As $$Q(N)\bmod 27$$ is one of $$r$$, $$r+9$$, $$r+18$$, we are done. • Great answer, thanks a lot. – Martund Jul 27 at 13:56 • or use polynomial remainder theorem for a rule. – Roddy MacPhee Jul 27 at 14:06 Obviously there are number whose digits add to $$27$$. ($$999$$ or $$524385$$ etc.) and obviously the sum of the digits $$27$$ is a multiple of $$9$$ so they are a multiple of $$9$$ but are they a multiples of $$27$$; and must multiples of $$27$$ have digits adding to a multiple of $$27$$. Well, $$27$$ itself is an obvious counter example of the latter. And $$999= 27*37$$ but $$524385= 27*19421\frac 23$$ so the first is not true either. So the question I guess is why not? Well the rule works for $$9$$ because $$9 = 10-1$$. And it works for $$3$$ because $$3|9$$. Details: If $$k|b-1$$ and $$n= \sum_{i=0}^m a_ib^i= \sum_{i=0}^m (a_i)(b^i-1) + \sum_{i=0}^m a_i$$. Now $$b^i-1 =(b-1)(b^{i-1} + b^{i-2}+ ..... + 1)$$ so $$(b-1)$$ divides all of the $$b^i-1$$ so $$b-1$$ divides $$n$$ if and only if $$(b-1)$$ divides $$\sum_{i=0}^m a_i$$. If we let $$b= 10$$ and $$b-1=9$$ and $$a_i$$ be the digits of $$n$$ that's our result. And it follows that if $$k|b-1$$ then $$k|(b^i-1)$$ so $$k|n$$ if and only if $$k$$ divides $$\sum_{i=0}^m a_i$$ as well. And this will be true for any decimal system base $$b$$ (not just $$b=10$$ and and $$k|b-1$$ (not just $$3|9$$). The fact that $$3^2 = 9$$ is mostly a coincidence and powers of $$3$$ is a bit of a red herring. It's not powers of $$3$$ going up that matter, but factors $$10-1$$ going down that matter. We can note that in base $$7$$ a number is a multiple of $$6$$ if and only if the sum of the digits is a multiple of $$6$$ and a multiple of $$2$$ or of $$3$$ if and only if the sum of the digits is a multiple of $$2$$ or of $$3$$ respectively but nothing can be said of $$4$$ or $$3$$. ($$11_7 = 8$$ is a multiple of $$4$$ but $$1+1=2$$ is not. And $$12_7 =9$$ is a multiple of $$9$$ but $$1+2=3$$ which is not.) Why doesn't it work? Well. $$27 = 3*(10-1)= (3-1)*10 + (10-3)$$. The sum of the digits of $$27$$ are $$(3-1) + (10-3) = 10-1$$. Our rule of $$9$$s apply and we can't jump magically to $$27$$. And if we increas by $$27$$ if we ignore carrying and borrowing we get $$ab + 27 = (a+2)(b+7)$$ and the sums of the digits are $$a+b + 9$$. That's an increase of $$9$$; not of $$27$$. Ind if we carry (i.e. $$b \ge 3$$ or $$a \ge 8$$ or $$b\ge 3$$ and $$a\ge 7$$) we get the sums are $$(a+2+1)(b+7-10)$$ or $$1(a+2-10)(b+7)$$ or $$1(a+2+1-10)(b+7 - 10)$$ and the sum of the digits stay the same or decreases by $$9$$; not $$27$$. But if $$b-1 = k^m$$ then in base $$b$$ we will have that multiples of $$k^i; i\le m$$ will have the sum of the digits add to a multiple of $$k^i$$. Example in base $$28$$ then sum of the digits of a multiple of $$27$$ will add to a multiple of $$27$$. FWIW $$999_{10} = 28^2 + 7*28 + 19 = 17T_{28}$$ where $$T$$ is the digit for $$19$$. And $$27*92 = 2484_{10} = 3*28^2+4*28 + 20= 34U_{28}$$ where $$U$$ is the digit for $$20$$ is another example. Less trivial example. The number $$8ATR_{28}$$ were $$A=10$$ and $$T=19$$ and $$R=17$$ will have digits that add to $$8+10+19+17=54$$, so my claim is that it ought to be a multiple of $$27$$. And $$8ATR_{28} = 8*28^3 + 10*28^2+ 19*28 + 17 =$$ $$8(27+ 1)^3 + 10(27+1)^2 + 19(27+1) + 17 =$$ $$8(27^3 + 3*27^2+3*27 + 1) + 10(27^2 + 2*27 + 1) + 19(27+1)+17=$$ $$[8*27^3 + 3*27^2 + 3*27 + 20*27^2 + 2*27 + 19*27] + 8 + 10 + 19+17=$$ $$27(8*27^2 + 3*27 + 3 + 20*27 + 2 + 19) + 54 =$$ $$27(8*27^2 + 3*27 + 3 + 20*27 + 2 + 19 + 2)$$ is a multiple of $$27$$. And indeed $$8*28^3 + 10*28^2+ 19*28 + 17=184005 = 27*6815$$

https://math.stackexchange.com/questions/1371759/prove-sum-limits-i-0n-binomnii-binom2n1n1

# Prove $\sum\limits_{i=0}^{n}\binom{n+i}{i}=\binom{2n+1}{n+1}$ [duplicate] I'm trying to prove this algebraically: $$\sum\limits_{i=0}^{n}\dbinom{n+i}{i}=\dbinom{2n+1}{n+1}$$ Unfortunately I've been stuck for quite a while. Here's what I've tried so far: 1. Turning $\dbinom{n+i}{i}$ to $\dbinom{n+i}{n}$ 2. Turning $\dbinom{2n+1}{n+1}$ to $\dbinom{2n+1}{n}$ 3. Converting binomial coefficients to factorial form and seeing if anything can be cancelled. 4. Writing the sum out by hand to see if there's anything that could be cancelled. I end up being stuck in each of these ways, though. Any ideas? Is there an identity that can help me? ## marked as duplicate by Jack D'Aurizio, Mark Viola, Lucian, user147263, vonbrandJul 24 '15 at 0:32 • I'd recommend using induction to prove the result, although I'm sure it could be done another way. – Raj Jul 23 '15 at 20:05 • Oh, I forgot to mentioned that I tried induction too (a classmate gave me that hint). I got stuck trying to move the induction hypothesis from $\sum\limits_{i=0}^{k}\dbinom{k+i}{i} = \dbinom{2k+1}{2k+1}$ to $\sum\limits_{i=0}^{k+1}\dbinom{k+i+1}{i} = \dbinom{2k+3}{k+2}$ Specifically the $k+i+1$ part--had no idea how to change that. – jsluong Jul 23 '15 at 20:24 Hint: Use $$\binom{n+1+i}{i} - \binom{n+i}{i-1} = \binom{n+i}{i}$$ • Hey, thanks for this! This is the recursive formula rewritten, right? I'm trying to use it to see if I could solve this problem with another approach but, er, stuck again. Can I get another hint? – jsluong Jul 23 '15 at 20:40 • Yep, the same recursion you have in your answer, only with different values $k=i$ and $n \mapsto 1 + n + i$ – johannesvalks Jul 23 '15 at 20:42 • Should be $\binom{n+i}{i+1}$ on the RHS. – xivaxy Jul 23 '15 at 20:47 • @Dr.MV - indead, I have corrected it. – johannesvalks Jul 23 '15 at 21:37 Oh, hey! I just figured it out. Funny how simply posting the question allows you re-evaluate the problem differently... So on Wikipedia apparently this is a thing (the recursive formula for computing the value of binomial coefficients): $$\dbinom{n}{k} = \dbinom{n-1}{k-1} + \dbinom{n-1}{k}$$ On the RHS of the equation we have (let's call this equation 1): $$\dbinom{2n+1}{n+1} = \dbinom{2n}{n} + \dbinom{2n-1}{n} + \dbinom{2n-2}{n} + \cdots + \dbinom{n+1}{n} + \dbinom{2n-(n-1)}{n+1}$$ The last term $\dbinom{2n-(n-1)}{n+1}$ simplifies to $\dbinom{n+1}{n+1}$ or just 1. Meanwhile on the LHS we have $\sum\limits_{i=0}^{n}\dbinom{n+i}{i}$ which is also $\sum\limits_{i=0}^{n}\dbinom{n+i}{n}$ because $\dbinom{n}{k} = \dbinom{n}{n-k}$. Written out that is (let's call this equation 2): $$\sum\limits_{i=0}^{n}\dbinom{n+i}{n} = \dbinom{n}{n} + \dbinom{n+1}{n} + \dbinom{n+2}{n} + \cdots + \dbinom{2n}{n}$$ The first term $\dbinom{n}{n}$ simplifies to 1. Hey, look at that. In each written out sum there's a term in equation 1 and an equivalent in equation 2. And in each sum there's $n$ terms, so equation 1 definitely equals equation 2. I mean, the order of terms is flipped, but whatever. Yay! • Cool that you found it! – johannesvalks Jul 23 '15 at 20:35 • Okay... so tell me what I should do. – jsluong Jul 23 '15 at 21:40 • @Dr.MV This is the subject of some debate.. The OP didn't know the answer beforehand, but figured it out an hour after posting the question. I think this is totally fine. – André 3000 Jul 23 '15 at 22:11 • @SpamIAm The answer was posted roughly the same time that the question was identified as a duplicate. So, given this question is indeed a duplicate, how does one justify the action? – Mark Viola Jul 23 '15 at 22:34 • Oh, well, I LaTeX my homework so I needed to type up the solution anyway. I figured other people could potentially benefit from the solution. This is a new account. I really am not trying to farm rep. – jsluong Jul 23 '15 at 23:22

http://terabook.info/diy-rattan-kiluwab/symmetric-closure-of-a-relation-ada95c

# symmetric closure of a relation 0. Finally, the concepts of reflexive, symmetric and transitive closure are The reflexive, transitive closure of a relation R is the smallest relation that contains R and that is both reflexive and transitive. Reflexive and symmetric properties are sets of reflexive and symmetric binary relations on A correspondingly. Let R be a relation on the set {a,b, c, d} R = {(a, b), (a, c), (b, a), (d, b)} Find: 1) The reflexive closure of R 2) The symmetric closure of R 3) The transitive closure of R Express each answer as a matrix, directed graph, or using the roster method (as above). 10 Symmetric Closure (optional) When a relation R on a set A is not symmetric: How to minimally augment R (adding the minimum number of ordered pairs) to have a symmetric relation? Transitive closure applied to a relation. Don't express your answer in … A relation follows join property i.e. Transcript. 2. Neha Agrawal Mathematically Inclined 171,282 views 12:59 Find the symmetric closures of the relations in Exercises 1-9. This means that if a symmetric relation is represented on a digraph, then anytime there is a directed edge from one vertex to a second vertex, ... By the closure properties of the integers, $$k + n \in \mathbb{Z}$$. 1. 8. The symmetric closure of a binary relation on a set is the union of the binary relation and it’s inverse. These Multiple Choice Questions (MCQ) should be practiced to improve the Discrete Mathematics skills required for various interviews (campus interviews, walk-in interviews, company interviews), placements, entrance exams and other competitive examinations. We discuss the reflexive, symmetric, and transitive properties and their closures. ... Browse other questions tagged prolog transitive-closure or ask your own question. The transitive closure of a symmetric relation is symmetric, but it may not be reflexive. Topics. • What is the symmetric closure S of R? the join of matrix M1 and M2 is M1 V M2 which is represented as R1 U R2 in terms of relation. A binary relation is called an equivalence relation if it is reflexive, transitive and symmetric. Discrete Mathematics with Applications 1st. Symmetric: If any one element is related to any other element, then the second element is related to the first. The symmetric closure of a relation on a set is the smallest symmetric relation that contains it. Definition of an Equivalence Relation. If I have a relation ,say ,less than or equal to ,then how is the symmetric closure of this relation be a universal relation . and (2;3) but does not contain (0;3). Question: Suppose R={(1,2), (2,2), (2,3), (5,4)} is a relation on S={1,2,3,4,5}. A relation R is non-symmetric iff it is neither symmetric One way to understand equivalence relations is that they partition all the elements of a set into disjoint subsets. Relations. Equivalence Relations. We then give the two most important examples of equivalence relations. To form the transitive closure of a relation , you add in edges from to if you can find a path from to . Transitive Closure – Let be a relation on set . If is the following relation: then the reflexive closure of is given by: the symmetric closure of is given by: 4 Symmetric Closure • If a relation is symmetric, then the relation itself is its symmetric closure. [Definitions for Non-relation] reflexive; symmetric, and; transitive. Find the reflexive, symmetric, and transitive closure of R. Solution – For the given set, . For example, being the father of is an asymmetric relation: if John is the father of Bill, then it is a logical consequence that Bill is not the father of John. Notation for symmetric closure of a relation. The transitive closure is obtained by adding (x,z) to R whenever (x,y) and (y,z) are both in R for some y—and continuing to do … Formally: Definition: the if $$P$$ is a property of relations, $$P$$ closure of $$R$$ is the smallest relation … The symmetric closure S of a binary relation R on a set X can be formally defined as: S = R ∪ {(x, y) : (y, x) ∈ R} Where {(x, y) : (y, x) ∈ R} is the inverse relation of R, R-1. 9.4 Closure of Relations Reflexive Closure The reflexive closure of a relation R on A is obtained by adding (a;a) to R for each a 2A. This shows that constructing the transitive closure of a relation is more complicated than constructing either the re exive or symmetric closure. • If a relation is not symmetric, its symmetric closure is the smallest relation that is symmetric and contains R. Furthermore, any relation that is symmetric and must contain R, must also contain the symmetric closure of R. If one element is not related to any elements, then the transitive closure will not relate that element to others. It's also fairly obvious how to make a relation symmetric: if $$(a,b)$$ is in $$R$$, we have to make sure $$(b,a)$$ is there as well. This section focuses on "Relations" in Discrete Mathematics. Example – Let be a relation on set with . R = { (a,b) : a b } Here R is set of real numbers Hence, both a and b are real numbers Check reflexive We know that a = a a a (a, a) R R is reflexive. i.e. The symmetric closure of a relation on a set is the smallest symmetric relation that contains it. Symmetric closure and transitive closure of a relation. Let R be an n -ary relation on A . We already have a way to express all of the pairs in that form: $$R^{-1}$$. A relation R is symmetric if the transpose of relation matrix is equal to its original relation matrix. t_brother - this should be the transitive and symmetric relation, I keep the intermediate nodes so I don't get a loop. A relation S on A with property P is called the closure of R with respect to P if S is a subset of every relation Q (S Q) with property P that contains R (R Q). Section 7. Symmetric Closure The symmetric closure of R is obtained by adding (b;a) to R for each (a;b) 2R. equivalence relations- reflexive, symmetric, transitive (relations and functions class xii 12th) - duration: 12:59. Discrete Mathematics Questions and Answers – Relations. •S=? A binary relation on a non-empty set $$A$$ is said to be an equivalence relation if and only if the relation is. Transitive Closure of Symmetric relation. Closure. Blog A holiday carol for coders. In this paper, we present composition of relations in soft set context and give their matrix representation. In this paper, four algorithms - G, Symmetric, 0-1-G, 1-Symmetric - are given for computing the transitive closure of a symmetric binary relation which is represented by a 0–1 matrix. Algorithms G and 0-1-G pose no restriction on the type of the input matrix, while algorithms Symmetric and 1-Symmetric require it to be symmetric. The connectivity relation is defined as – . I tried out with example ,so obviously I would be getting pairs of the form (a,a) but how do they correspond to a universal relation. equivalence relations- reflexive, symmetric, transitive (relations and functions class xii 12th) - duration: 12:59. There are 15 possible equivalence relations here. What is the reflexive and symmetric closure of R? (b) Use the result from the previous problem to argue that if P is reflexive and symmetric, then P+ is an equivalence relation. Find the symmetric closures of the relations in Exercises 1-9. The symmetric closure of R . (a) Prove that the transitive closure of a symmetric relation is also symmetric. Nodes so I do n't get a loop in this paper, we composition. ) but does not contain ( 0 ; 3 ) and Answers –.! 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# How do I find the modulus of the average force a sphere gets from colliding with the ground? #### Chemist116 The problem is as follows: A ball of $1\,kg$ in mass is thrown with a speed of $-10\vec{j}\,\frac{m}{s}$ from a height of $15\,m$ to a horizontal floor. Find the modulus of the average force in $N$ that the ball receives from the floor during the impact with the ground which lasts $0.1\,s$ and dissipates $150\,J$. (You may use the value of gravity $g=10\,\frac{m}{s^2}$. The alternatives given in my book are as follows: $\begin{array}{ll} 1.&290\,N\\ 2.&300\,N\\ 3.&310\,N\\ 4.&320\,N\\ 5.&330\,N\\ \end{array}$ This problem has left me go in circles as I don't know exactly how should I treat or use the information to obtain the average force?. I'm assuming that there is a conservation of momentum but as I mentioned I don't know what to do?. Can somebody help me here?. #### skeeter Math Team impulse equation ... $F \Delta t = m\Delta v \implies F = \dfrac{m\Delta v}{\Delta t}$, where $F$ is the average force of impact changing momentum you have mass and delta t, you need the change in velocity from impact with the ground total initial mechanical energy = kinetic energy at impact $mgh + \dfrac{1}{2}mv_0^2 = \dfrac{1}{2}mv_f^2 \implies v_f = -\sqrt{2gh + v_0^2} = -20 \text{ m/s}$ post impact, the velocity is positive and has 150J less energy ... impact KE = $\dfrac{1}{2}m(-20)^2 = 200 \text{ J} \implies$ post impact KE = $50 \text{ J}$ bounce velocity $\dfrac{1}{2}mv_f^2 = 50 \implies v_f = 10 \text{ m/s}$ change in velocity from impact with the ground is $\Delta v = [10 - (-20)] \text{ m/s} = 30 \text{ m/s}$ $F = \dfrac{m\Delta v}{\Delta t} = 300 \text{ N}$ #### Chemist116 impulse equation ... $F \Delta t = m\Delta v \implies F = \dfrac{m\Delta v}{\Delta t}$, where $F$ is the average force of impact changing momentum you have mass and delta t, you need the change in velocity from impact with the ground total initial mechanical energy = kinetic energy at impact $mgh + \dfrac{1}{2}mv_0^2 = \dfrac{1}{2}mv_f^2 \implies v_f = -\sqrt{2gh + v_0^2} = -20 \text{ m/s}$ post impact, the velocity is positive and has 150J less energy ... impact KE = $\dfrac{1}{2}m(-20)^2 = 200 \text{ J} \implies$ post impact KE = $50 \text{ J}$ bounce velocity $\dfrac{1}{2}mv_f^2 = 50 \implies v_f = 10 \text{ m/s}$ change in velocity from impact with the ground is $\Delta v = [10 - (-20)] \text{ m/s} = 30 \text{ m/s}$ $F = \dfrac{m\Delta v}{\Delta t} = 300 \text{ N}$ All uses a pretty good logic but I'm wondering why the answers sheet states that the answer is $310\,N$?. Could it be that is there anything missing or overlooked during the analysis?. Can you check this please?. #### skeeter Math Team Maybe someone else can do the problem ... #### DarnItJimImAnEngineer *facepalm!* The net force is equal to 300 N! But the earth is still imparting downward momentum on the ball at a rate of 10 N, so the floor must impart that much more upward momentum. $F_{floor} = ma - F_g = \pm 300 ~N ~\vec{j} \pm 10 ~N ~\vec{j} = \pm 310 ~N ~\vec{j}$ (depending on whether $\vec{j}$ is up or down). It took me about five minutes to realise that mistake. Another feather in the cap of "Always, always, always draw your free-body diagram." Chemist116 #### Chemist116 *facepalm!* The net force is equal to 300 N! But the earth is still imparting downward momentum on the ball at a rate of 10 N, so the floor must impart that much more upward momentum. $F_{floor} = ma - F_g = \pm 300 ~N ~\vec{j} \pm 10 ~N ~\vec{j} = \pm 310 ~N ~\vec{j}$ (depending on whether $\vec{j}$ is up or down). It took me about five minutes to realise that mistake. Another feather in the cap of "Always, always, always draw your free-body diagram." I totally overlooked this part. At first I didn't noticed but it seems that $F_g=mg=1\,kg\times 10\frac{m}{s^2}=10\,N$ In this case I believe it is upward.