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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Review of my T-shirt design

I'm a graphics guy and a wanna-be mathematician. Is the T-shirt design below okay? Or if there's a bone headed error, I'd appreciate a heads up. `
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How do we know the ratio between circumference and diameter is the same for all circles?

The number $\pi$ is defined as the ratio between the circumeference and diameter of a circle. How do we know the value $\pi$ is correct for every circle? How do we truly know the value is the same for every circle? How do we know that $\pi =…
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Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ $$u_2=v_2=1$$ $$\lim_{n \to \infty}…
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Calculating pi manually

Hypothetically you are put in math jail and the jailer says he will let you out only if you can give him 707 digits of pi. You can have a ream of paper and a couple pens, no computer, books, previous pi memorization or outside help. What is the best…
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Does $\zeta(3)$ have a connection with $\pi$?

The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Several $\zeta$ values are connected with $\pi$,…
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What is the formula for pi used in the Python decimal library?

(Don't be alarmed by the title; this is a question about mathematics, not programming.) In the documentation for the decimal module in the Python Standard Library, an example is given for computing the digits of $\pi$ to a given precision: def…
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False proof: $\pi = 4$, but why?

Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I know from Calculus, but I emphasize that I have…
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Simplest way to get the lower bound $\pi > 3.14$

Inspired from this answer and my comment to it, I seek alternative ways to establish $\pi>3.14$. The goal is to achieve simpler/easy to understand approaches as well as to minimize the calculations involved. The method in my comment is based on…
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Fibonacci infinite sum resulting in $\pi$

I found the following identity. While trying to prove it, I found some things that I don’t quite understand: $$\frac{\pi}{4}=\sqrt{5} \sum_{n=0}^{\infty} \frac{(-1)^n F_{2n+1}}{(2n+1) \phi^{4n+2}}$$ (where $\phi=\frac{\sqrt{5}+1}{2}$). What I…
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What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? My choice of $\pi$ for this question isn't…
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Why is $\pi$ irrational if it is represented as $C/D$?

$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction. Then why is $\pi$ an irrational number?
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Why is $e^\pi - \pi$ so close to $20$?

$e^\pi-\pi\approx 19.99909998$ Why is this so close to $20$?
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Why is this series of square root of twos equal $\pi$?

Wikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+\cdots+ \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, like Taylor series or something like that, so I…
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Intuitive reasoning behind $\pi$'s appearance in bouncing balls.

This video is about an interesting math/physics problem that when cranked out churns out digits of $\pi$. Is there an intuitive reason that $\pi$ is showing up instead of some other funky number like $e$ or $\phi$, and how can one predict without…
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How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet $$\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$$ which looks like unbelievable. (I forgot…
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