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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Show that $\pi$ is a primitive recursive number [on hold]

Can anyone provide a proof that $\pi$ is a primitive recursive number, or suggest how I might prove it? Thanks
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45 views

Can any number sequence be found in $\pi$? [duplicate]

I read somewhere that any finite number sequence must be found in $\pi$. For example, $0998975645455$ must be somewhere in the digits of $\pi$. The reason for this was that $\pi$ is irrational, ...
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1answer
65 views

Why would we use the radius of a circle instead of the diameter when calculating circumference?

Forgive me if this question is a little too strange or maybe even off. Mathematics has never been my strong point, but I definitely think it's the coolest... Anyway, I was looking into tau, pi's up-...
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90 views

Does the speed of light depend on $\pi$? [on hold]

The speed of light is given by $c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$; $\epsilon_0$ is the vacuum permittivity while $\mu_0$ is the vacuum permeability. From Wikipedia, the definition of $\mu_0$ is $\...
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1answer
37 views

Computing pi with maps on the rationals

There are numerous ways to compute approximations to $\pi$. Is it possible to find a mapping $f:A \to A$ where $A \subseteq \mathbb{Q}$ such that the iterates $f^{n}(x)$ tend towards $\pi$ for any ...
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2answers
150 views

$\pi$ and $e$ as coded trajectories

Question about the number $\pi$ and $e$ and their unpredictability. We know that $\pi=3.141592653589793238462643383279502884...$ Suppose that we are in the origin of the plane i.e. at the point $(0,0)...
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5answers
348 views

$e^{\left(\pi^{(e^\pi)}\right)}\;$ or $\;\pi^{\left(e^{(\pi^e)}\right)}$. Which one is greater than the other?

$\newcommand{\bigxl}[1]{\mathopen{\displaystyle#1}} \newcommand{\bigxr}[1]{\mathclose{\displaystyle#1}} $ $$\large e^{\bigxl(\pi^{(e^\pi)}\bigxr)}\quad\text{or}\quad\pi^{\bigxl(e^{(\pi^e)}\bigxr)}$$ ...
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2answers
94 views

Proof of $\pi^e$ and $e^\pi$ Being Irrational

By contradiction, if $\pi^e$ were rational, then we could write $\pi^e=\frac{a}{b}$ where $a,b\in\mathbb{I}^+$ and $b\neq0$. So: $$\begin{align} \\ \pi^e&=\frac{a}{b} \\ e\ln(\pi)&=\ln(a)-\...
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26 views

how can i now position my character set in Pi?

i have characters (10 numerals) how can i count position number of beginning my collection in PI like ...
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1answer
63 views

sequence of diophantine approximants of $\pi$

I define the sequence of optimal diophantine approximants of $\pi$ to be the sequence $u_m = \frac{n}{m}$ where $n$ is given by $\min_{\forall n \in \mathbb{N}} |\frac{n}{m}-\pi|$ and we define $\...
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3answers
49 views

How to calculate the sine cosine or tangent of an angle(Simply Explained)

I wanted to know how a calculator finds the sine Or any other trig function with only knowing the value of the angle. I have been looking on the internet for answers because i was really interested ...
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1answer
53 views

Why do we say pi is the ratio of the circumference to the diameter, and not diameter to the circumference?

Everywhere I see it written, $\pi$ is described as "the ratio of the circumference to the diameter". I know $\pi = C/d$, but the ratio $3.14...:1$ is $3.14$... diameters to $1$ circumference. So ...
3
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2answers
67 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 ...
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1answer
88 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 \;...
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0answers
32 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 ...
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2answers
104 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=...
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2answers
149 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
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1answer
57 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^...
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4answers
105 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: $$\...
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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)\...
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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 ...
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2answers
602 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 ...
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241 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 ...
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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-...
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62 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 $e$ ...
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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+\...
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18 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 ...
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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 $k$? ...
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2answers
53 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 $...
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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 ...
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4answers
571 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 ...
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1answer
47 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)}...
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51 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
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2answers
724 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}^{\...
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3answers
823 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 ...
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3answers
58 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 ...
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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 ...
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0answers
25 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 ...
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59 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^...
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4answers
710 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 ...
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1answer
77 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 ...
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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 ...
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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 $...
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1answer
52 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 ...
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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{\...
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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 ...
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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 ...
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251 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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26 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
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1answer
78 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}^{\...