Questions tagged [pi]
The number $\pi$ is the ratio of a circle's circumference to its diameter. Understanding its various properties and computing its numerical value drove the study of much mathematics throughout history. Questions regarding this special number and its properties fit in here.
1,660 questions
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What makes people say $\pi^{{\pi}^{{\pi}^{\pi}}}$ is an integer? [closed]
I know that $\pi^{{\pi}^{{\pi}^{\pi}}}$ being an integer is just a theory, but where does the theory come from? What evidence is there that tells us it could be an integer?
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Getting area between a lens and an ellipse to approximate $\pi$
I have here a desmos graph. You see a tiny bit of space in between the lens and the ellipse, some of which I have approximated using a triangle. Now, I need to calculate the rest of the space between ...
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Approximation of $\pi$ using an ellipse and a triangle
Here's how my approximation goes:
Take a point on the unit circle (in the first quadrant) and join it to the origin. Now, the area of the sector formed by the line and the x-axis is $\frac{\theta}{2}$....
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Is there a lower bound of $| \pi/4 - a/b|$ dependent only on b?
I was wondering if there exists a lower bound for $| \pi/4 -a/b|$, where $b$ is an integer and $a$ is chosen such that $a/b$ is the best approximation to $\pi/4$, which only depends on $b$. I am not ...
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Is $\frac{\pi}{\left(-\frac{x}{\pi}\right)!\left(\frac{x}{\pi}-1\right)!}$ equal to sin(x)? [closed]
Is the following true?
$sin(x)=\frac{\pi}{\left(-\frac{x}{\pi}\right)!\left(\frac{x}{\pi}-1\right)!}$
(Noted that factorial here refers to the Gamma function extension)
I can't find anything on it. If ...
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Question about irrationality of $\pi$
Based on my limited knowledge of math, there's two types of irrational numbers: 1) The 'normal' ones, which consist of a radical sign and a positive integer and 2) the 'fancy' ones, like $\pi$ or $e$
...
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Is there an explanation for this approximate calculation of pi? [closed]
Once upon a time, I wrote down for myself a simplified calculation of $\pi$.
$$\pi \approx \frac{3}{5}(3 + \sqrt{5})$$
Is there an explanation for this?
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Integrals for $\pi-\frac{333}{106}$ and $\frac{355}{113}-\pi$
Two integrals related to convergents of $\pi$ are given by
$$128\int_0^1 \frac{x^2(1-x)^2(x-\frac{1}{2})^2}{\sqrt{1-x^2}}dx = 106\pi - 333 + \frac{1}{5}$$
$$64\int_0^1 \frac{x(1-x)^3(x-\frac{1}{2})^2}{...
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Representing Pi in binary
Say we detect music or video encoded in radio signals from Alpha Centauri. We want to aim a large radio telescope there and blast them with something that's going to get their attention, something ...
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Approximating $\pi$ as a fraction
Given $n \in \mathbb{N}$, let $p(n) \in \mathbb{N}$ be such that $n/p(n)$ is the best approximation of $\pi$ (denoted as $\tilde{\pi}_n$). I have two main questions:
Is the sequence $\{p(n)\}$ a ...
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About the formula $\;\pi=\sum_{n=0}^\infty \frac{a_n}{2n+1}\cos^{2n+1}x-\sin2x+2x$, with $a_0=4$, $a_1=-2$, and $a_n=\frac{3-2n}{n!\,2^{n-2}}$
Here's something I found while playing around with Newton's approach to calculate $\pi$. This converges faster for $x$ close to $\frac{\pi}{2}$. $$\pi = \sum_{n=0}^\infty \frac{a_n}{2n+1}\cos ^{2n+1}x-...
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Pi using Newton's approximation
Instead of taking the angle that the revolving line makes with the x-axis to be 60 degrees, if we take it to be 75 degrees, it should converge faster using the expansion for (1 - x^2)^0.5. But, when I ...
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Finding a circle's circumference via $\lim_{n \to \infty} n r \sqrt{2 \left(1 - \cos(360^\circ/n)\right)}$, without using $\pi$
Say I have an equal-sided n-polygon inside a circle. To calculate the side length, I can see it as part of an isosceles triangle with vertex angle = 360/n and two legs' length = r
So, by using law of ...
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Inconsistency in subtracting area of unit square with unit circle for pi approximate calculation using Monte Carlo simulation
I have run into problems involving the fact that areas cannot be subtracted or it will lead to contradiction and inconsistency. Area division leads to no issues in $\pi = 3.14$. Can you help me see ...
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Two questions on two-term Machin-like formulae
It is known that there are exactly four two-term Machin-like formulae that allow for the estimation of $\pi$:
$$
\begin{aligned}
{\tfrac {\pi }{4}}&=\arctan {\tfrac {1}{2}}+\arctan {\tfrac {1}{3}}\...
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On a formula for $\pi e$.
I'm looking for ideas on how to represent $\pi e$ as an infinite series of rational numbers.
It is easy to build meaningless formulas using double summations by distributing a series for $\pi$ over a ...
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Exact value of $\lim_{x\to\infty}\frac{\sqrt2+1}{x\left(8-16\sum_{n=1}^x\frac1{64n^2-1}-\pi\sqrt2-\pi\right)}$, as derived from simple series for pi.
I was reading Euler's E041 ("Concerning the sums of series of reciprocals.") and found some sequences for $\pi$. I believe his derivations are not properly justified. I combined
$$\sqrt2\...
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Is there a good reason to expect $\pi^2$ here?
I was playing around with generalized hypergeometric functions in Wolfram Mathematica and noticed that:
$$\begin{eqnarray}
{}_{3} F_2\left(\begin{matrix}1 ~~~ 1 ~~~ 1 ~~~\\\frac32 ~~~ 2\end{matrix} \...
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Calculating $\int{\sqrt{1-x^2}dx}$ [duplicate]
So, I wanted to calculate the $\pi$, and I know that $\pi$ is the area of a circle its radius is 1, so I used this equation : $x^2 + y^2 = 1$ to get the half circle function : $f(x)=\sqrt{1-x^2}$ and ...
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Gauss's method and Todd's process for generating Machin-like formulas
I've recently taken an interest in Machin-like formulas, i.e., formulas of the form:
$$\frac{\pi}{4} = \sum_{k=0}^{n} q_i \arctan{\frac{a_i}{b_i}}$$
Where $q_k\in \mathbb Q$, $a_k,b_k\in \mathbb N \...
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Modifying Ramanujan's 1/$\pi$ formula to include $K(k_{58})$ and fundamental units
In this 2010 post, I pointed out that given the fundamental unit $U_{29}= \frac{5+\sqrt{29}}2$, and fundamental solutions to Pell equations $x^2-29y^2 = \pm1$,
$$\big(U_{29}\big)^3=70+13\sqrt{29},\...
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Why does my chosen method for finding the cube roots of -1 only work for two of the roots? [closed]
The problem:
Find the three cube roots of -1.
The answer:
$-1, \frac{1}{2}\pm\frac{\sqrt3}{2}i$
What I have tried:
I KNOW there are other posts on this forum with this exact problem, but my issue with ...
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Solving for $z\cdot w$ and $\frac{z}{w}$ using the polar form of z and w [closed]
The problem:
Express each of the complex numbers $z=-1+i$ and $w=3i$ in polar form. Use these expressions to calculate $z\cdot w$ and $\frac{z}{w}$.
The answer:
$z\cdot w = -3-3i, \frac{z}{w} = \frac{...
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What is $Arg(\frac{z}{w})$ if $Arg(z) = -\frac{5\pi}{6}$ and $Arg(w) = \frac{\pi}{4}$?
The problem:
If $Arg(z) = -\frac{5\pi}{6}$ and $Arg(w) = \frac{\pi}{4}$, find $Arg(\frac{z}{w})$.
The answer:
$\frac{11\pi}{12}$
What I have tried:
$Arg(\frac{z}{w}) = Arg(z) - Arg(w) = -\frac{5\pi}{6}...
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Deriving π from stereographic projection
Is it possible to derive pi from iterated stereographic projection of some kind? I ask because the link below shows an animation of pi 'unrolling' from a turning wheel, and it seems like there ought ...
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Product formula for $\dfrac{4}{\pi}$
Can someone give me a hint on how to prove the following formula:
$$\frac{4}{\pi}=\left(\prod_{p \equiv 1 \pmod{5} \atop p \in \mathbb{P} } \frac{p}{p+1}\right) \cdot \left(\prod_{p \equiv 2 \pmod{5} \...
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Why $\sum_{n=0}^{\infty} \frac{1}{(6n+1)^3}=\frac{1}{6^3}(91\zeta(3)+2\sqrt3\pi^3)$ is just the tip of the iceberg (Part 2)
(Continued from this post. Update: Section III below now includes the Glaisher H-numbers.)
I. Definitions
As before, given the Hurwitz zeta function,
$$\zeta(s,a) = \sum_{k=0}^\infty \frac1{(k+a)^s}$$
...
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Why $\sum_{n=0}^{\infty} \frac{1}{(6n+1)^3}=\frac{1}{6^3}(91\zeta(3)+2\sqrt3\pi^3)$ is just the tip of the iceberg (Part 1)
While trying to solve this post (an identity also mentioned by Ramanujan), I noticed it had two parts of interest. This is the first one.
I. Definitions
Given the Hurwitz zeta function,
$$\zeta(s,a) = ...
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Prove that the limits of these two recurring sequences is pi.
I'm trying to prove that these two sequences $a_n$ and $b_n$ converge to $\pi$, but cannot find a method of doing so. The two sequences are defined as such:
$a_0=2\sqrt3, b_0=3$
$a_n= \frac{2a_{n-1}b_{...
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Are there books with exercises on $\pi$ and $e$?
$\pi$ and $e$ are constants which we stumble on everytime we are doing math. I wonder if there is some book which exploits this fact and contains exercises whose solution involved these constants.
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What is the intuition behind calculation of the area of circle?
I am learning about approximating the area of circle ($\pi r^2$) by the area of polygons ($\frac{1}{2}$ of number of sides * the distance between side and center * length of the side).
My question is ...
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Intuitive explanation for why $\pi$ turns up in $\binom{2m}m \sim \frac{4^m}{\sqrt{m\pi}}$ [closed]
I am aware of the following approximation:
$$\dbinom{2m}m \sim \dfrac{4^m}{\sqrt{m\pi}}$$
Which is equivalent to saying the follwing (as $m \to \infty$):
Given $m$ objects, the number of ways to pick ...
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Prof. Knuth lecture about $ \pi $ and random maps
In this video, Prof. Knuth talks about an interesting combinatorial problem:
suppose you have a random map $ f\colon \{ 1, 2, 3,\ldots, n \} \rightarrow \{ 1, 2, 3,\ldots, n \}$. If you consider the ...
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How to find the roots of sin(x) using series theory
If we define $$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n \ x^{2n+1}}{(2n+1)!}$$
How to find the roots of $\sin(x)$, i.e.
$$\pi =4 \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$$
satisfies $\sin (\pi)=0$
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Is the new series for 𝜋 a Big (or even Medium) Deal?
There's been some oohing and ahhing in the science press recently over the discovery of a new formula for $\pi$ obtained as a side effect of computing amplitudes in string theory:
$$\pi=4+\sum_{n=1}^\...
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If I have x and y coordinates of a point along the arc, how do I convert that to a percentage of PI?
I am using Javascript to create shapes in canvas. I am creating an arc where you specify the start and end angle of the arc to show where along a circle the arc begins and ends. They are initially set ...
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Proving $ \sum_{k=0}^{\infty} \frac{k! \, (2k)! \, (25k - 3)}{(3k)! \, 2^k} = \frac{\pi}{2} $ [duplicate]
Wikipedia claims that
$$
\sum_{k=0}^{\infty} \frac{k! \, (2k)! \, (25k - 3)}{(3k)! \, 2^k} = \frac{\pi}{2}
$$
but does not give a citation. How can this be proved?
Edit: This was correctly closed as a ...
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What are some relations between the golden number or ratio $\phi$, and $\pi$?
What are some relations between the golden number or ratio $\phi$, and $\pi$?
For example,
by considering this answer https://math.stackexchange.com/a/744196/ ; by Steve Lewis.
Now taking the point at
...
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Finding Number Sequences in the Digits of Pi?
I understand that π (pi) has an infinite number of digits, and this means any finite sequence of numbers can theoretically be found within its digits. For example, if you're looking for the sequence &...
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Bauer's series for $\frac{1}{\pi}$
Recently, someone asked a question involving the expression
$$ \sum_{n=0}^{\infty} (-1)^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3 $$
At first glance, I knew that the expression was the value of ...
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How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$
The formula
$$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$
(in older notation) appears as eq. 38 in Ramanujan's paper Modular equations ...
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PI Approximation With Arcsine Infinite Series
While Playing with numbers last night I stumbled on an approximation of pi that was quite exciting. I'm sure it has been discovered before, but it was a fun journey nevertheless and I wanted to share ...
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Why is the definition of $\pi$ as integral by Weierstrass "inverted"?
Reading https://en.wikipedia.org/wiki/Pi#Definition I stumpled upon the following definition as an integral, presumably given by Weierstrass:
$$
\pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}
$$
However I ...
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Proof of an approximation of $\pi$
Here is an approximation of $\pi$:
$$\prod_{k=1}^{+\infty} \frac{4k^2}{4k^2-1} = \frac{\pi}{2}$$
Proof
$$\prod_{k=1}^{+\infty} \frac{4k^2}{4k^2-1} = \frac{4(1^2)}{4(1^2)-1}*\frac{4(2^2)}{4(2^2)-1}*\...
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Why is $3 + \sin(3) + \sin(3+\sin(3))$ near $\pi$?
$3 + \sin(3) + \sin(3+\sin(3)) = 3.1415926535721955587...$
$3 + \sin(3) + \sin(3+\sin(3)) + \sin(3 + \sin(3) + \sin(3+\sin(3)) ) = 3.1415926535897932384626433832795019...$
$\pi = 3....
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Efficient division by 99 by hand to compute digits of $\pi$ [closed]
I want to compute some digits of $\pi$ by hand. There are several formula which have been used in pre-computer time, among them:
\begin{align}
\frac{\pi}{4}&=4\arctan(\frac{1}{5})-\arctan(\frac{1}{...
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Approximation of $\pi$ by integral and rational number
Via WolframAlpha, I observed that
$$\int_0^1\frac{x^{4n}(1-x^2)}{1+x^2}dx=\frac\pi2-\frac pq \to 0$$
when the integer $n\to\infty$. This gives an approximation of $\pi$ by a rational number. It is not ...
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Why no one uses the product formula for sine function to calculate $\pi$?
$$\sin(\pi x)=\pi x \prod_{n \ge 1}\left(1-\frac{x^2}{n^2}\right)$$
$$\pi = \frac{\sin(\pi x)}{x\prod_{n \ge 1}\left(1-\frac{x^2}{n^2}\right)}$$
Let $x=\frac{1}{2}$
$$\pi = \frac{2}{\prod_{n \ge 1}\...
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Might there be an $n^{\text{th}}$ digit of $\pi$ where the sequence becomes palindromic?
Assuming $n>1$, would it be reasonable to think there is an $n^{\text{th}}$ digit of $\pi$ where stopping there would yield a palindromic number $(3.14159...951413)$?
Would it be more likely that ...
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How to calculate the ratio of convergence for Euler's, Gauss' and Viète's approximation of $\pi$?
Let $\sqrt{6\sum_{k=1}^\infty{\frac{1}{k^2}}}$ be Euler's approximation of $\pi$; $\lim_{n\rightarrow\infty}\frac{2}{g_n}$ Gauss approximation of $\pi$; and $2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\...
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