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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3
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2answers
64 views

Can e be expressed in pi's digits? (or vice-versa)

Given that both pi and e's decimal places are completely random and infinite... I was wondering about how people say that "Every phrase ever uttered can be expressed in the digits of pi" -- which, for ...
7
votes
1answer
87 views

Is $\pi = 3.14159…$ first-order definable in the reals?

Given first-order logic with equality and the real field $\mathbb{R} = (R, 0, 1, <, +, \cdot)$, is $\pi$ first-order definable? By first-order definable, I mean a sentence of the form $\exists x \;...
2
votes
0answers
31 views

How to use Bailey–Borwein–Plouffe formula(BBP formular) Step by step

From this link BBP I want to know how BBP formula exactly work. And i need simple example. Maybe find 3th or 5th digit of pi. I am not good at math,i think step by step solution good for me. Thank ...
0
votes
2answers
101 views

Proving that $\pi$ and $e$ are rational numbers [duplicate]

Maybe this question is too dumb to be asked, but it's really bugging me so I decide to ask it anyway. I hope you bear with me. Okay, it's known that both sides of the following series equal. $$\pi=...
7
votes
3answers
143 views

Proving $\pi \gt e+\frac{1}{e} \gt \pi-\frac{1}{\pi} \gt e$

I created this problem for myself as a fun exercise. I want to prove the following statement: $$\pi \gt e+\dfrac{1}{e} \gt \pi-\dfrac{1}{\pi} \gt e$$ I found that the following upper/lower bounds ...
2
votes
1answer
54 views

Pi Appoximation: Simpler Solution to Limit?

May be a ridiculous question, but I wanted to see if MSE had "simpler" proofs for Viete's approximation (specifically, using an equation derived from Viete's formula) of $\pi$: $$\lim_{x \to \infty}2^...
7
votes
4answers
103 views

Proof of Leibniz $\pi$ formula

I found the following proof online for Leibniz's formula for $\pi$: $$\frac{1}{1-y}=1+y+y^2+y^3+\ldots$$ Substitute $y=-x^2$: $$\frac{1}{1+x^2}=1-x^2+x^4-x^6+\ldots$$ Integrate both sides: $$\...
0
votes
0answers
65 views

Formula for $\pi$ using primes

In one of his videos (https://www.youtube.com/watch?v=HrRMnzANHHs), Matt Parker introduces the following formula for $\pi$ using primes: $$\left(1-\frac{1}{3}\right)\cdot\left(1+\frac{1}{5}\right)\...
-5
votes
1answer
118 views

An integral and $\pi$ [closed]

I want to give you this question as an enigma. Can you prove that the following integral is equal to $\pi$ ? $$\int_0^\infty \sqrt{\frac{256x^4}{x^{12}+6x^{10}+15x^8+35x^4+6x^2+1}}=\pi.$$ This ...
26
votes
2answers
576 views

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 ...
8
votes
5answers
233 views

Compute $\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$

How can I evaluate the following integral? $$I=\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$$ I tried it with Wolfram Alpha, it gave me a numerical solution: $0.785398$. Although I immediately ...
0
votes
0answers
35 views

digits of pi, the nature of its randomness, immunity to place selections (Von mises) reichenbach partial limits

The digits of $\pi$ uniform distribution non-standard analysis, and its partial limit I was wondering whether there has been any investigation into the distribution of the decimals of $\pi$ using non-...
1
vote
0answers
59 views

(Solved) Deriving Pi from Euler's Identity

I was tinkering with Euler's Identity and I come to wonder if it was possible to derive $\pi$ from it. I know $\pi$ can't be expressed as a fraction of two rational numbers but neither $i$ nor $e$ ...
0
votes
1answer
59 views

How we create our own $\pi$ finder formula/function?

1) Nilakantha Somayaji; $\pi=3+\dfrac{4}{3^3-3}-\dfrac{4}{5^3-5}+\dfrac{4}{7^3-7}-\dfrac{4}{9^3-9}+.....$ 2)Franciscus Vieta; $\pi=2.\dfrac{2}{\sqrt2}.\dfrac{2}{\sqrt{2+\sqrt2}}.\dfrac{2}{\sqrt{2+\...
1
vote
0answers
17 views

Ratio of a circle's circumference to its diameter in Non-Euclidean geometries

Preface: The Windows calculator has an "Inverse" button, which transforms $\ln$ into $\exp$, for example. Oddly enough, it also transforms $\pi$ into $2\pi$, which is nonsensical, since that would ...
0
votes
1answer
57 views

Formulas for $\pi$ [closed]

Given a formula for $\pi$ like: $$4⋅\sum^\infty_{k=1} \frac{(−1)^{k+1}}{2k−1} = 4⋅(1−1/3+1/5−1/7+1/9−1/11…).$$ or some of the several others; how can you know that it holds true to any $k$? ...
1
vote
2answers
51 views

Why does convention not recognize $2\pi$ as the fundamental quantity? [duplicate]

It looks as though my question might turn out to be a duplicate. If so, it does not need an answer, after all, thanks. ORIGINAL QUESTION Why does convention recognize $\pi\approx 3.14$, rather than $...
2
votes
2answers
86 views

Complicated series converges to $\pi$.

How do I get this result? $$\frac {426880 \sqrt {10005}}{\large \sum_{k = 0}^{\infty}\frac {(6k)!(545140134k + 13591409)}{(k!)^3 (3k)! (-640320)^{3k}}} = \pi$$ It seems formidable. Context: I came ...
14
votes
4answers
557 views

My teacher said that $2\pi$ radians is not exactly $360^{\circ}$?

A few days ago, my math teacher (I hold him in high faith) said that $2\pi$ radians is not exactly $360^{\circ}$. His reasoning is the following. $\pi$ is irrational (and transcendental). $360$ is a ...
1
vote
1answer
43 views

Confusion with Bellard's algorithm for Pi

I've found the following algorithm for calculating the nth digit of pi in base B: It all makes sense, until you reach the $b=$ inside the second for loop. The whole line is: $$b=\frac{k}{a^{v(n,k)}...
1
vote
0answers
49 views

Can the sum of a finite series equal $\pi$?

Can the sum of a finite series equal $\pi$? I'm assuming of course that no element in the series is some fraction of $\pi$. I'm wondering since all methods I've seen of calculating $\pi$ involve ...
26
votes
2answers
700 views

Show that $\sum_{n=0}^{\infty}\frac{2^n(5n^5+5n^4+5n^3+5n^2-9n+9)}{(2n+1)(2n+2)(2n+3){2n\choose n}}=\frac{9\pi^2}{8}$

I don't how prove this series and I have try look through maths world and Wikipedia on sum for help but no use at all, so please help me to prove this series. How to show that $$\sum_{n=0}^{\...
7
votes
3answers
803 views

How to find sequence of digits in pi?

I saw this project on github https://github.com/philipl/pifs, where they are trying to compress files in the pi number after the decimal. I guess this makes sense because apparently every finite ...
2
votes
3answers
54 views

Solving For Sin Using Pi

Solving for sin using pi I was messing around with calculating pi by finding the perimeter of a many sided polygon, and dividing it by the diameter (Like the thing Archimedes did). The equation I ...
3
votes
0answers
50 views

I discovered a sequence that should converge to $\pi$ , but how to prove that it really converges to $\pi$? [duplicate]

So yesterday I took paper and pencil and did some constructions of geometrical nature and I discovered a sequence that should converge to $\pi$. It goes like this: Define $a_1=\sqrt{2}$ and for ...
1
vote
0answers
22 views

Are there infinite self-locating strings in the decimal expansion of $\pi$?

I came across the following interesting objects. Self-locating strings within $\pi$: numbers $n$ such that the string $n$ is at position $n$ (after the decimal point) in decimal digits of $\pi$. The ...
2
votes
0answers
58 views

Proof of $\sum\limits_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$ using the Fourier series of $|x|$

I'm sure easier proofs exist, but I have to specifically use the method in the picture: This is what my attempt is: First, I did some manipulation to figure out that $$ \sum_{k=1}^\infty \frac 1{k^...
32
votes
5answers
664 views

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 ...
-2
votes
1answer
75 views

How to find the correct value of pi? [duplicate]

Pi is defined as the ratio of $\frac{c}{r}$. Many ancient scintist try to find the value of pi. Some of the values are $\frac{22}{7}$(good hold upto 10 decimal point), $\frac{355}{113}$ (good hold ...
-1
votes
3answers
82 views

Improve upon: $\sqrt[4]{3^4+2^4+\frac{1}{2+(\frac{2}{3})^2}} \approx \pi$ [closed]

So here we have an approximate value of $\pi$. $$\sqrt[4]{3^4+2^4+\frac{1}{2+(\frac{2}{3})^2}} \approx \pi$$ $$3.14159265262 \ldots \approx 3.14159265358\ldots$$ How could one get a better ...
1
vote
0answers
64 views

Pos properties.

Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as: $$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$ When given $n\in\mathbb{N}$, this function gives the 'position' of $...
2
votes
1answer
51 views

Why are these sums equal?

I've been looking at some pretty cool proofs of $\zeta(2)=\frac{\pi^2}{6}.$ recently, and the one proof that was the easiest to understand for me was how Euler originally presented it, by finding and ...
0
votes
1answer
31 views

Understanding this trigonometric identity $\frac{n}{2} (2 \cos \frac{\alpha_n}{2} \sin \frac{\alpha_n}{2}) = \frac{n}{2} \sin \alpha_n$

I was looking an example that motivates rounding errors using the quadrature of a circle (?). And at a certain point there's this identity: $$\frac{n}{2} \left( 2 \cos \frac{\alpha_n}{2} \sin \frac{\...
9
votes
1answer
1k views

Why are there so many primes in the convergents of Pi?

Recently, I was looking into fractional approximations of pi, such as $\frac{22}{7}$ or $\frac{355}{113}$. I found that there was a name for these approximations, 'convergents' of pi, and I found a ...
2
votes
0answers
41 views

Representation of e

I was on this wikipedia page https://en.wikipedia.org/wiki/List_of_representations_of_e which has a list of representations of the constant e. I came across this one representation that looked ...
8
votes
7answers
244 views

Area of a circle $\pi r^2$

So, today I learned that the area of a circle is $\pi r^2$. So, I thought that since $r$ is $1$ dimensional, $r^2$ will be $2$ dimensional. In this case, a square, as you only multiply $2$ dimensions (...
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votes
0answers
25 views

Alternative formulas for the volume of an n-ball

The volume of an n-ball is \begin{align}\dfrac{\pi^{n/2}r^n}{\Gamma\left(n/2+1\right)}\end{align} However if we define $\alpha = \sqrt{\pi}$, and use the $\Pi$-function, we get \begin{align}\...
6
votes
1answer
77 views

What is $\int_{0}^{\infty}\frac{\cos^{n}x \sin x \ln x}{x} dx$?

Letting $\gamma$ denote the Euler-Mascheroni constant, evaluating $\int_{0}^{\infty}\frac{\cos^{n}x \sin x \ln x}{x} dx$ for $n \in \{ 1, 2, 3, 4 \}$, we have that: $$\begin{align*} \int_{0}^{\...
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vote
0answers
93 views

Why does $63725\pi$ give four approximations to $\pi$?

The fraction $\frac{355}{113}$ was first used for approximating $\pi$ by the Chinese mathematician and astronomer Zu Chongzhi in the 5th century (see Milü). An isolated case? The almost-integer $113\...
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vote
0answers
64 views

Ways to determine $\pi$ [duplicate]

I have read that it is possible to determine the value of a single digit, say the 874th of $\pi$. I know that it is a trascentental number, how is that possible? How many ways are there to determine ...
1
vote
0answers
80 views

The Digits of Pi and e

Im a math learner so the questions may seem obvious. With it being pi day the 14th of this month the digits of pi have been in my thoughts. The BBP algorithm for Pi enables the computation of any ...
13
votes
1answer
110 views

Proving a curious formula of $\pi$?

I have recently come across this statement without proof. $$ \pi = 128 \arctan\frac{1}{40} -4\arctan\frac{1}{239} -16\arctan\frac{1}{515} -32\arctan\frac{1}{4030} -64\arctan\frac{1}{32060}$$ I'd put ...
1
vote
4answers
89 views

How do i mechanically generate pi?

Here's a question with very "real-world" implications... I want to produce pi in my basement woodworking shop. (And no, using pi on the calculator or computer is not allowed, nor are books.) The shop ...
6
votes
0answers
108 views

An integral for $2\pi+e-9$

Motivation Lucian asked about the almost-integer $2\pi+e\approx9$ in a comment to a partially answered why question about $e\approx H_8$. This is more involved than approximations to $\pi$ and ...
26
votes
5answers
2k views

Why are turns not used as the default angle measure?

Why is $2\pi$ radians not replaced by $1$ turn in formulas? The majority of them would be simpler. If such a replacement was proposed earlier, why was it declined?
3
votes
2answers
116 views

Why is the sin of n times pi always 0

My instructor says that $\sin(\pi \cdot n)$ is always equal to $0$. However, when playing with jsconsole.com, I find that this is not the case. ...
4
votes
3answers
115 views

Find sequences such that $\lim_{n \to \infty} (\sqrt{a_n}-\sqrt{b_n})=\pi$, with $a_n,b_n \in Q$, increasing and defined by recursion

For any real number $r$ we can find a pair of natural numbers $N$ and $M$, such that $\sqrt{N}-\sqrt{M}$ will approximate $r$ with any given precision (if we choose $N,M$ large enough). That's why I ...
0
votes
1answer
51 views

Limit of regular polygons approaching pi - earliest proofs

Archimedes used areas of regular polygons to approximate pi. He calculated both inner and outer polygons and realized that more sides yielded closer results to each other. There's surviving proof that ...
3
votes
2answers
99 views

Why is the Leibniz method for approximating pi so inefficient

I've been playing around with algorithms for computing pi. One that I noticed is the leibniz algorithm. It states that pi can be approximated like this $n = 1$ (...
2
votes
4answers
110 views

Is there a geometry in which $\pi$ is a natural number?

Is there a geometry in which $\pi$, the ratio of a circle's circumference to its diameter, is an integral number such as $3$ or $4$?