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.

$\pi$ is the ratio of a circle's circumference to its diameter. Its definition is modern analysis is (by Karl Weierstrass) $$ \pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}. $$ An alternative definition, popularised by Landau is: Define $\frac{\pi}{2}$ as the smallest positive root of the cosine function.

It can also be given by the Gregory-Leibniz series (exhibits sublinear convergence) $$ \pi = 4 \sum_{j=0}^\infty \frac{(-1)^j}{2j+1}. $$ $\pi$ has the approximate numerical value $3.14159265358979323846\dots$, can be approximated by fractions, for example, $\frac{22}{7}, \frac{333}{106}, \frac{355}{113},\dots$, and is both irrational and transcendental.

It is part of Euler's famous identity:

$$e^{i\pi}+1=0.$$

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The staircase paradox, or why $\pi\ne4$

What is wrong with this "proof"? Is $\pi=4?$
Pratik Deoghare
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Does $\pi$ contain all possible number combinations?

$\pi$ Pi Pi is an infinite, nonrepeating $($ sic $)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever…
Chani
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Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger.

Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than $1,$ and show that $n < x < n+1$ for some integer $n$ .…
MJD
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$\pi$ in arbitrary metric spaces

Whoever finds a norm for which $\pi=42$ is crowned nerd of the day! Can the principle of $\pi$ in euclidean space be generalized to 2-dimensional metric/normed spaces in a reasonable way? For Example, let $(X,||.||)$ be a 2-dimensional normed vector…
CBenni
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Is there an integral that proves $\pi > 333/106$?

The following integral, $$ \int_0^1 \frac{x^4(1-x)^4}{x^2 + 1} \mathrm{d}x = \frac{22}{7} - \pi $$ is clearly positive, which proves that $\pi < 22/7$. Is there a similar integral which proves $\pi > 333/106$?
anon
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Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or what the motivation for it is?
Nick Alger
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Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but…
users31526
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How to sum this series for $\pi/2$ directly?

The sum of the series $$ \frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1} $$ can be derived by accelerating the Gregory Series $$ \frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2} $$ using Euler's Series Transformation . Mathematica…
robjohn
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Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I compare (without calculator or similar device) the values of $\pi^e$ and $e^\pi$ ?
Mirzodaler
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How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?

How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$ I found the above interesting identity in the book $\bf \pi$ Unleashed . Does anyone knows how to prove it? Thanks.
Neves
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What do $\pi$ and $e$ stand for in the normal distribution formula?

I'm a beginner in mathematics and there is one thing that I've been wondering about recently. The formula for the normal distribution is: $$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\displaystyle{\frac{(x-\mu)^2}{2\sigma^2}}},$$ However, what are $e$…
pimvdb
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Interesting and unexpected applications of $\pi$

$\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$ Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$ and the generalization of $\zeta (2k)$, my…
MT_
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How to convince a layperson that the $\pi = 4$ proof is wrong?

The infamous "$\pi = 4$" proof was already discussed here: Is value of $\pi = 4$? And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this thing to a non mathematician. The main missing point, in…
Gadi A
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Proof that Pi is constant (the same for all circles), without using limits

Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof?
Chris Card
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How can one prove that $e<\pi$?

This question is inspired by another one , asking to prove that something approximately equal to $1.2$ is bigger than something approximately equal to $0.9$. The numerical answer to this question was (expectedly) downvoted, though in my opinion it…
Start wearing purple
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