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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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
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2answers
77 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
494 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
38 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: ...
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0answers
47 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 ...
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3answers
783 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
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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 ...
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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 ...
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0answers
19 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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0answers
57 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 ...
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5answers
620 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
73 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
81 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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63 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
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1answer
49 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 ...
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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 ...
2
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0answers
40 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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7answers
231 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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0answers
23 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 ...
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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*} ...
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0answers
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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 ...
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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 ...
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0answers
65 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 ...
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1answer
107 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 ...
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4answers
86 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
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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 ...
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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?
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2answers
107 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. ...
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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 ...
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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 ...
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2answers
98 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 $n = 1$ ...
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4answers
104 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 $4$?
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1answer
108 views

Extracting the nth digit of pi using Plouffe's formula?

I have come upon the following formula to extract the nth digit of pi in base 10: $$\pi + 3 = \sum_{n=1}^{\infty} \frac{n 2^n n!^2}{(2n)!} $$ But this just seems to be a formula for pi. How can I use ...
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5answers
232 views

Is $22/7$ an often used approximation for $\pi$?

It is $\pi$-day and the internet is full of stories about $\pi$. One story mentions that "an approximation -- $22/7$ -- is used in many calculations." I have never actually used $22/7$ as an ...
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6answers
210 views

How can never ending decimal numbers represent finite lengths? e.g. pi(π), $\sqrt{2}$

Recently, I was in a discussion with a colleague that, whether the πd really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came ...
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0answers
115 views

Infinitely nested radical expansions for real numbers

Conjecture. For any real number $x \in (0,1]$ there exists a unique expansion in the form $x=-2+\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}$ with $a_k$ being natural numbers from the set $(2,3,4,5,6)$. ...
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84 views

A double inequality for $\frac{\pi}{2}$

Approximating $\frac{\pi}{2}$ from above Since $$\left(\frac{\pi}{2}\right)^9\approx 58.220897$$ the root $$58^\frac{1}{9}\approx 1.5701$$ is not far from $$\frac{\pi}{2}\approx 1.570796$$ This ...
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1answer
69 views

An algorithm for creating a circle on a discrete plane and a limit for $\pi$

I know there is a well known algorithm which uses the circle equation to approximate it with pixels. However, I wanted to approach this problem from the most basic principles. So we start with a ...
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2answers
69 views

Interesting Ways to Find the value of Pi [closed]

Is there any interesting ways to find pi? Thanks.
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2answers
256 views

What's the formula for this series for $\pi$?

These continued fractions for $\pi$ were given here, $$\small \pi = \cfrac{4} {1+\cfrac{1^2} {2+\cfrac{3^2} {2+\cfrac{5^2} {2+\ddots}}}} = \sum_{n=0}^\infty \frac{4(-1)^n}{2n+1} = \frac{4}{1} - ...
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1answer
62 views

The irrationality of $\pi/e$ is listed as open yet the infinite product formula for it seems to suggest a way to prove it.

And the formula of all rational products seems to suggest that taking some n as n approaches infinity, the formula will have an always increasing amount of uncancelled primes(so provably non ...
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1answer
74 views

Value Of $\pi$ obtained using limits!

What i thought was simple, a circle can be formed by increasing the number of sides of regular polygon( like pentagons, hexagons, etc ) up to infinity by keeping the distance between the center and ...
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0answers
40 views

Is there a simple way to prove pi is irrational using math, which is known understandable for someone doing HL Maths (Advance High School Maths)?

In school I have learned from early on that pi is irrational, but is there a simply proof which I could understand to show that this is the case. I am doing HL Maths (IB, with discrete as the option ...
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1answer
42 views

What is the relation between the gaussian integral and the volume of the n-ball?

Even if I've red other threads treating this question, it's still obscure to me what deeply relates the multiple Guassian integral $\int e^{-x^2} = \sqrt \pi$ and the area of a $n$-ball. Someone ...
2
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2answers
396 views

Pi's Recursiveness [duplicate]

I don't know if this will make sense, but: If $\pi$ is infinite and contains all strings of numbers including those of infinite length, then it must contain $\sqrt2$, and if $\sqrt2$ is infinite and ...
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2answers
108 views

Why do ratios of these Fibonacci-type sequences approach $\pi$?

Define $A_n$ by $A_1=12$, $A_2=18$, and $A_n=A_{n-1}+A_{n-2}$ for $n\ge3$. Similarly define $B_n$ by $B_1=5$, $B_2=5$, and $B_n=B_{n-1}+B_{n-2}$ for $n\ge3$. Terms of $A_n$: $12, 18, 30, 48, ...
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140 views

Why isn't there a formula for $\zeta(k)=\sum_{n=1}^\infty\frac{1}{n^k}$ involving $\pi$ when $k$ is odd?

Now we know that $\sum \frac{1}{n}=\text{divergent}, \sum \frac{1}{n^2}=\frac{\pi^2}{6}$ but now this for $\sum \frac{1}{n^3}=1.20....$ and again $\sum \frac{1}{n^4}=\frac{\pi^2}{90}$ .Now somewhere ...
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0answers
78 views

Why is Gelfond's constant transcendental?

I have seen a proof of $\pi$ being transcendental by conclude that transcendental number powered by algebraic number must be transcendental and algebraic number powered by algebraic number must be ...
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1answer
24 views

Formula for cycloid?

Is there a formula for cycloid? My approximation is $((2\times(x\div(\pi\div2)))-(x\div(\pi\div2))^2)^.626$.