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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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 (...
0
votes
0answers
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
votes
1answer
79 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}^{\...
1
vote
0answers
95 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\...
1
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
82 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
112 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
96 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
109 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
121 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
117 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
101 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
111 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 $4$?
4
votes
1answer
118 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 ...
-1
votes
5answers
238 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 ...
4
votes
6answers
240 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 ...
9
votes
0answers
122 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 $(2,3,4,5,6)$. ...
1
vote
0answers
85 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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vote
1answer
72 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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votes
2answers
74 views

Interesting Ways to Find the value of Pi [closed]

Is there any interesting ways to find pi? Thanks.
15
votes
2answers
257 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} - \...
0
votes
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 ...
0
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1answer
75 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 ...
1
vote
0answers
46 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 ...
0
votes
1answer
45 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
votes
2answers
400 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 ...
8
votes
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 $A_n$: $12, 18, 30, 48, 78,\dots$...
6
votes
2answers
156 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 ...
0
votes
0answers
98 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 ...
1
vote
1answer
25 views

Formula for cycloid?

Is there a formula for cycloid? My approximation is $((2\times(x\div(\pi\div2)))-(x\div(\pi\div2))^2)^.626$.
0
votes
0answers
63 views

Show that $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$ [duplicate]

$(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$ How can we show this equivalent without using a calculator?
2
votes
0answers
43 views

What is wrong with my statement?

So with Euler identity I used $2 \pi(\tau)$ instead and got $e^{i2\pi} = 1$, and took natural logarithm $\ I 2\pi = 0$? But is see online the answer is $6.28....i$.
0
votes
1answer
36 views

Need an Ambiguous example

Suppose I have a floating-point number with $m$ where $(m > 0)$ digits after the decimal point. Now if I want to round it up to $d$ where $(0 ≤ d < m)$ digits after the decimal point, sometimes ...
37
votes
1answer
697 views

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=...
0
votes
2answers
90 views

Use two formulas to approximate $\pi$

Hi guys. Any hints on part c? How are we going to approximate $\pi$ by computing $P_k$ and $p_k$
1
vote
2answers
111 views

Finding sine of an angle in degrees without $\pi$

The following is found using a combination of: (a) a polygon with an infinite number of sides is a circle, (b) the perimeter of that polygon is the circumference of the circle that it becomes (of ...
9
votes
0answers
249 views

Interpretation of $\frac{22}{7}-\pi$

Integral and series proofs that $\frac{22}{7}>\pi$ We can prove that $\frac{22}{7}$ exceeds $\pi$ by using Dalzell integral $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ or its ...
8
votes
0answers
126 views

Limit approximation for $\pi$ in the four fours puzzle?

The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:...
5
votes
3answers
86 views

How to show $\lim\limits_{n\rightarrow \infty}\left(\raise{5pt}\frac{2^{4n}}{n\binom {2n} {n}^{2}}\right)=\pi $?

How to show this is true? $$\lim_{n\rightarrow \infty}\left(\raise{3pt}\frac{2^{4n}}{n\binom {2n} {n}^{2}}\right)=\pi $$
1
vote
1answer
59 views

Geometric interpretation of Leibniz formula for $\pi$

We know $\pi=4(\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}....)$. I'm wondering, is there a geometric interpretation of this identity. Can we prove this identity by finding a different way to ...
3
votes
2answers
117 views

Elementary proof that $\pi$ is irrational

I'm trying to understand the first proof in this page. So we have $$S=\frac{\pi }{4}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}=S_{n}+R_{n}$$ where $S_{n}=\sum_{k=1}^{n}\frac{(-1)^{k-1}}{2k-1}$ and $...
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votes
2answers
156 views

A series of positive terms to prove $\pi>\frac{333}{106}$

This is a consequence of the answer to that question. A proof that $\pi > \frac{333}{106}$ is given by the series of positive terms $$\pi-\frac{333}{106} \\ =\frac{48}{371} \sum_{k=0}^\infty \frac{...
1
vote
1answer
263 views

Series and integrals for inequalities and approximations to $\pi$

Fundamentals Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral $$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ (see Why do we need an integral to prove ...
9
votes
3answers
602 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
13
votes
2answers
430 views

Is $\pi^k$ any closer to its nearest integer than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to its nearest ...
6
votes
1answer
192 views

Is there a series to show $22\pi^4>2143\,$?

This extends this post. I. For $\pi^3$: $$\pi^6-31^2 =\sum_{k=0}^\infty\left(-\frac{63}{(2k+2)^6}+\frac{31^2}{(2k+3)^6}\right) =\sum_{k=0}^\infty P_1(k)\tag1$$ As pointed out by J. Lafont, when ...
4
votes
1answer
133 views

Why is “$\pi^2= g $” where $g$ is the gravitational constant?

Some months ago a professor of mine showed us a 'proof' of why $g\approx 9.8 ~\text{m}/\text{s}^2$ (the gravitational acceleration at the surface of the Earth) is 'equal' to $\pi^2\approx9.86\dots$ ...
6
votes
0answers
232 views

A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$

This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$. I. $\pi$ and $\pi^2$ There are series with all terms positive for $\...