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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Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference?

Pi appears a LOT in trigonometry, but only because of its 'circle-significance'. Does pi ever matter in things not concerned with circles? Is its only claim to fame the fact that its irrational and an important ratio?
Gordon Gustafson
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Are my results new?

I'm eighteen and sometimes I like doing math on my own when I'm inspired. I would like to know if some of my "discoveries" are new (I don't think so :) ). These are some of the results I found in the last 3 years: Infinite radical converging to…
Francesco Scavella
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Simple numerical methods for calculating the digits of $\pi$

Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that can be computed by hand in order to compute the…
e.James
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A novelty integral for $\pi$

My lab friends always play a mentally challenging brain game every month to keep our mind running on all four cylinders and the last month challenge was to find a novelty expression for $\pi$. In order to stick to the rule, of course we must avoid…
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Geometric intuition for $\pi /4 = 1 - 1/3 + 1/5 - \cdots$?

Following reading this great post : Interesting and unexpected applications of $\pi$, Vadim's answer reminded me of something an analysis professor had told me when I was an undergrad - that no one had ever given him a satisfactory intuitive…
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Is there a proof that $\pi \times e$ is irrational?

A little reading suggests: It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is transcendental. If we just want irrationality rather than…
idmercer
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$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can approximate just about any number. In formal…
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There's something strange about $\sum \frac 1 {\sin(n)}$.

Clearly, $$\sum_{n=1}^\infty \frac 1{\sin(n)}$$ Does not converge (rational approximations for $\pi$ and whatnot.) For fun, I plotted $$P(x)=\sum_{n=1}^x \frac 1{\sin(n)}$$ For $x$ on various intervals. At first, I saw what you might…
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How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? How does it compare to other irrational numbers…
Sean
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How are first digits of $\pi$ found?

Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am talking about accurate digits by either…
Confuse
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Why is it that this gives a good approximation of $\pi$?

At the end of a Physics examination, I decided to play around with my calculator, as I always do when I have time left over, and found that: $$\frac{1}{100} \cdot 11^{\ln(11)} \approx 3.14159789211,$$ where $\ln(x)$ is the natural logarithm of $x$,…
Taylor
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Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$?

I've been struggling for a while with the following problem: Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$? Needless to say software aid is not allowed. All manual calculations should be possible to be performed in reasonable time (for…
VividD
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What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula was not immediately apparent to a student upon being…
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A question comparing $\pi^e$ to $e^\pi$

I was doing an algebra problem set following a chapter on logarithms and exponentiation, and it presented this "bonus question": Without using your calculator, determine which is larger: $e^\pi$ or $\pi^e$. I wasn't able to come up with anything,…
ivan
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How to find continued fraction of pi

I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are: $\pi=[3:7,15,1,292,.....]$ and many others given here. Similarly some nice continued fractions for $e$ and it's derivatives are given…
happymath
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