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.

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Does a set of all decimal expansions of $\pi$ contains $\pi?$ [duplicate]

Let's say there is a set containing all finite decimal expansions of $\pi$: $$A = \{3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... \}$$ Does this set contains $\pi$? I see that it is probably not true ...
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Why is this proof wrong? ( I proved the irrationality of $\pi$ using the most basic techniques so I suspect that it must have gone wrong somewhere)

So it starts off by way of contradiction, supposing $\pi\in \mathbb Q$, then by De-Moivre's theorem for rational powers: $$\left((\cos(2k\pi)+i\sin(2k\pi)\right)^{\pi}= \cos(2k\pi^2)+i\sin(2k\pi^2)=1^{...
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Position of specific value

Let's assume a have an arbitrarily long number, take π for example. Since we know π is infinite, there will at some point be a group of numbers like "2015201620172018...", correct? My ...
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General formula for the upper bound of pi involving nested square roots (circumscribed perimeters of regular polygons)

The formula for the lower bound of pi involving nested square roots looks like this: $p_{2^m} = 2^m\sqrt{2-\sqrt{2+\sqrt{2+ \sqrt{2+...}}}}$ where there are $m-1$ nested square roots. For example, ...
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What if $\pi e$ is algebraic? [duplicate]

Today I heard that $\pi$ and $e$ are transcendental. I don't think I have ever seen a proof of this, but I see this is available online. I also see that it is not know whether $\pi e$ is algebraic or ...
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How to detect witch direction would be the closest between 2 angles

I want to get the fastest direction (+ or -) to move into a new angle. I tried this : (newAngle - oldAngle) % PI If the value is positive, then I have to use ...
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1 vote
1 answer
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William Shanks incomplete algorithm for computing $\pi$ with different initial angle

These days, I'd watched some youtube movies regarding manual computation of $\pi$ and find out about 239, 1/5 and their inverse tangent functions ($arctan$/$arctg$/$\tan^{-1}$) and so on. Today, I've ...
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Are these numbers $\xi_1$ and $\xi_2$ writable exploiting other mathematical constants?

Let's consider the number \begin{equation} \xi_1 = \sum_{n=1}^\infty \frac{(-1)^{n^\textrm{th}\textrm{ digit of $\pi$ in base $2$}}}{n} = -1 -\frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \...
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4 votes
1 answer
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How to prove that $k^{\pi}$ is not an integer for any integer $k\geq 2$?

I strongly suspect that $k^{\pi}$ is not an integer for any integer $k\geq 2$ (for otherwise this would be a famous result of which I am not aware). But how does one prove this? The answer to this ...
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10 votes
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What is the strange pattern in the behaviour of this approximation of pi?

I have been playing around with this approximation of pi recently: $$\lim_{n\to\infty} \sum_{i=0}^{n-1} \frac{n}{n^2+i^2} = \frac{\pi}{4}$$ and although I am perfectly aware that as far as ...
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Is $5^\pi$ an integer?

I saw this very short math problem on Twitter: Is $5^\pi$ an integer? It isn't (it's 156.992545309…), but is there some technique to prove this without a calculator? My first and only idea so far is ...
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Estimate: $\pi \to e \to \log(2) \to G$ by sampling uniform distribution

Successively: $\pi \to e \to \log(2) \to G$ were calculated/estimated by sampling uniform distributions. Method: With a normal distribution $\pi$ can be calculated with help of the PDF (probability ...
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4 votes
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Ratio between circumference and "radius" of a polygon

Given some polygon $P$ in two-dimensional Euclidean space, I want to define the radius of $P$ as the average of the radii of the smallest outer circle and the largest inner circle. An outer circle has ...
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Prove that $\pi < \sqrt2 + \sqrt3$ [duplicate]

A square inscribed inside of a circle with radius $1$ must have a perimeter $4\sqrt2$. A regular hexagon circumscribed about that same circle has a perimeter $4\sqrt3$. Since we know the circle’s ...
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Estimating Pi by Throwing Bread

I remember hearing about a story in which an Italian King (hundreds of years ago) drew a circle and randomly threw bread behind his shoulder, and calculated the percent of bread that landed inside the ...
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is there any size of a circle where the area is an integer and the radius is an integer? [closed]

The formula for the area of a circle is pi times the radius squared. The radius is the diameter divided by 2. Imagine a line, like the axis, but instead, it doesn’t go past the edges. Now, the length ...
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is it $e^\pi<\pi^e$ or $\pi^e<e^\pi$? and how do we prove it? [duplicate]

is it $e^\pi<\pi^e$ or $\pi^e<e^\pi$? it's an interesting question that just came to my mind, well of course I used the calculator to get the answer, However I need to like prove it and I don't ...
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Approximating factorial using identity $\frac{1}{x}!\frac{2}{x}!\cdots\cdot\frac{x}{x}!=\frac{ {x}!\cdot(2\pi)^{\frac{x-1}{2}} }{ x^x\cdot\sqrt{x} }$

I created a function that describes the product of the inverse multiples of a factorial $$ m(x) = \frac{1}{x}!\cdot\frac{2}{x}!\cdot\frac{3}{x}!\cdots\frac{x-1}{x}!\cdot\frac{x}{x}!$$ for some reasons ...
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2 votes
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Area of a circle as sum of infinite squares

Take a quarter of a circle and divide it in squares as shown in the (very badly drawn) picture: Every time you draw a new square take as much space as possible from the circle. This way every square ...
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1 answer
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How do I put this in product notation?

I'm trying to create a chain rule formula for multiple composition of functions using the product notation. I want to explain to my students that: For instance, if we have 4 composition of functions, ...
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1 answer
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BBP formula question

I have to program something but I don't understand the task. I'm not sure if this is the right place to ask but I thought I can try. I have to approximate $\pi$ with the BBP formula. $\displaystyle\pi=...
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2 votes
2 answers
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Why is $2i\ln(-i)=\pi$

So I always wanted to make my own approximation for pi. I thought of using trigonometric functions to help me start, so I decided to use arccosine. I put $\cos^{-1}(x) = \pi $ in a calculator to solve ...
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4 votes
1 answer
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Formulas for $\pi$ of the form $2\sum_{k=0}^\infty\binom{2k}{k}\frac{a^{2k+1}+b^{2k+1}+c^{2k+1}}{4^k(2k+1)}$

Third edit: For those interested in the Sagemath-code to produce your own formula, given three natural numbers $x<y<z$, it can be found here. I am sharing those formulas in public domain, for ...
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-1 votes
1 answer
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Could there be an irrational number $x$ such that the product of $x$ and $\pi$ are rational?

I know that irrational numbers cannot be the quotients of any two rational numbers, and an irrational number times a rational number is thus also irrational. But, could there be an irrational number, ...
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1 answer
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Is there a way to calculate a specific digit of PI

Is there any mathematical I could find a specific digit of 𝛑 If I had f(x) = ... what would the function to return the x digit ...
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$ \left\lfloor 10^{\lfloor n \rfloor} \pi \bmod 10 \right\rfloor $ - does this function give the nth decimal place of pi?

Function to round the nth decimal place of pi to the nearest integer. For example, for pi, n = 0, y = 3. n = 1, y = 1. n = 2, y = 4. And so on and so forth. Gives me good results until n = 17, which ...
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How to prove $\pi^{e} + 1 > e^{\pi}$? [duplicate]

I have already know how to prove $\pi^{e} < e^{\pi}$(solution), but I cannot figure out how to prove $\pi^{e} + 1 > e^{\pi}$. You can use approximation to prove it but calculators are prohibited....
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1 vote
1 answer
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Show that $f^{(j)}(\pi)$ and $f^{(j)}(0)$ are integers.

In the article $\textit{A simple proof that } \pi \textit{ is irrational}$, see: https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-53/issue-6/A-simple-proof-that-...
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1 vote
2 answers
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$\text{Angle}=\dfrac{\text{Arc length}}{\text{Radius}}$, a result or a definition?

There is a commonly used formula to find an angle $$ \text{Angle}=\dfrac{\text{Arc length}}{\text{Radius}}. $$ My question is whether this is a deduced formula or it is the very definition of an angle?...
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What would the possible compression be with Pi

Imagine a data file with data (a movie, an image, etc.). A file is just a sequence of 1 and 0. Since Pi contains infinite many binary digits, that image should be found in Pi somewhere and instead of ...
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An easy way to calc PI with powers of 2 and square roots

I don't know if this algorithm is know, but I think it's an easy way to calc $\pi$ more quickly than others. The reason of this, it's because when I search for computing algorithms, I only find ...
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1 vote
0 answers
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Approximating $\pi$ with the help of a regular $k$-sided polygon

I'm reading "An Introduction to Computational Physics" by Tao Pang. In it, he writes the following. In general, if the side length of a regular inscribed $k$-sided polygon is denoted as $l_k$...
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8 votes
5 answers
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Showing $\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\left(\frac{e}{2}-\frac{1}{e}\right)<1$ without a calculator

Show that: $$\sqrt{\frac{e}{2}}\cdot\frac{e}{\pi}\left(\frac{e}{2}-\frac{1}{e}\right)<1$$ I have tried power series of exponential around $0$ wich is : $$e^x=1+x+\frac{x^2}{2}+O(x^3)$$ We can ...
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Why can $-n\cdot \pi $ be changed to $n\cdot \pi $?

This task was just to solve this equation: $\cos2x=\cos4x$. I solved it correctly apart from one step. My book somehow changes $-n\cdot \pi $ to $n\cdot \pi $. How is it possible?
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1 answer
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Integral of a circle to find pi

I was thinking of ways to calculate pi, and one idea I had was to integrate $x^2+y^2 = 1$ between 0 and 1 to get $\frac{1}{4}\pi$ Because the total area of such a circle would be $\pi$ I computed this ...
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Eulers number divided by Pi

Whether $\pi e$ and $\pi+e$ etc. are irrational or not are famously unsolved problems in math. Is the irrationality of $\frac{e}{\pi}$ or $\frac{\pi}{e}$ equally hard to prove or is it trivial? Haven'...
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1 answer
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Proving $\pi^2=18\sum_{n=0}^{\infty}\frac{n!n!}{(2n+2)!}$

How to prove $$\pi^2=18\sum_{n=0}^{\infty}\frac{n!n!}{(2n+2)!}$$ I saw this as an exercise in Hobson's Treatise on Plane Trigonometry, pg.297. The $\pi^2$ has me flustered, I assume that there is some ...
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2 votes
2 answers
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How is e to the hyperbola what pi is to the circle?

I'm trying to find a nice similarity between e and pi and I thought of conic sections. If you have a circle then the perpendicular conic section to that is a hyperbola. So this seems pretty similar. I ...
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1 vote
1 answer
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Is there a unit equal to 2pi radians?

We can cut up circles in whatever size chunks we choose -- we normally choose to cut them up so that the size of the angle of an entire circle is $2\pi$ or 360. Said differently, we choose units to be ...
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1 vote
2 answers
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What is the integral of the secant between $0$ and $\pi$?

The secant function has a discontinuity at $\pi/2$, so I separated the integral as: $\int_0^{\pi}{\sec(\theta) d\theta} = \int_0^{\pi/2}{\sec(\theta) d\theta} + \int_{\pi/2}^{\pi}{\sec(\theta) d\theta}...
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3 answers
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Why does $\int_{-\infty}^\infty \operatorname{sech} x \, \mathrm{d}x =\pi\;$? [closed]

According to Desmos, $$\int_{-\infty}^\infty \operatorname{sech} x \, \mathrm{d}x = 3.14159265359$$ Why is $\pi$ here in this definite integral?
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2 votes
1 answer
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Request for proof: Regularity of pi's continued fractions

Notice In this post, a continued fraction is represented as follows $$ a + \cfrac{1^2}{b+\cfrac{3^2}{b+\cfrac{5^2}{\ddots}}} = a +K^\infty_{k=1}\frac{(2k-1)^2}{b} $$ When I was checking the ...
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Seeking infinite product representations of $\pi$ (other than Wallis)

I am looking for a list of as many representations of $\pi$ using "$\prod$" product notation; i.e., an infinite product for $\pi$. An example would be the Wallis product $$\pi=4\prod_{n=1}^{\...
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PI calculation and error with approximations

$\Pi$ can be calculated with infinite product, series or polygonal methods. It's my understanding approximation using these methods give a bound on what the value of $\Pi$ could be. For instance, ...
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Is $\frac{0}{\pi}$ rational?

In math lesson, our teacher showed this formula. $Q = \{\frac{a}b\}\land (a\land b\in Q)\land (b\neq 0)$ According to this formula, $\frac{0}\pi$ is... strange. You know, $\frac{0}\pi$ is 0 and 0 is ...
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3 votes
0 answers
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Why Stirling formula at n = -1 and n= -2 is so close to 2*pi*i [closed]

We know asymptotic formula for gamma function as Stirling formula $$\Gamma(z+1) \approx F(z+1) = \sqrt{2\pi z}\left({\cfrac{z}{e}}\right)^z\cdot \left({1+\cfrac{1}{12z} +\cfrac{1}{288z^2}+...}\right)$$...
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An uncommon continued fraction of $\frac{\pi}{2}$

I'm currently stuck with the following infinite continued fraction: $$\frac{\pi}{2}=1+\dfrac{1}{1+\dfrac{1\cdot2}{1+\dfrac{2\cdot3}{1+\dfrac{3\cdot 4}{1+\cdots}}}}$$ There is an obscure clue on this: ...
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0 votes
2 answers
127 views

Why this is equal to $\pi$?

I just recently came across: $$\lim_{N\to \infty}\bigg[2\prod_{k=1}^{N}\biggl(\frac{(2k)^2}{(2k)^2-1}\biggr)\biggr]$$ which appears to be approaching $\pi$... Can anyone explain to me why this is, or ...
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5 votes
3 answers
142 views

Are $\pi$ and $\tan^{-1}\left(2\right)$ rational multiples of each other? [duplicate]

For a proof of quantum universality, I need to show that $\tan^{-1}\left(2\right)$ is not a rational multiple of $\pi$. How do I show this? I feel like showing algebraic independence over the ...
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Proving that the n-dimensional Wallis sieve has the same hypervolume as the n-ball

The Wallis sieve is a variation on Sierpinski's carpet, where you start with a square, and in the $i$th step you divide each square into $(2i+1)^2$ smaller squares and remove the middle one. The total ...
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