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.

learn more… | top users | synonyms

3
votes
2answers
108 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 ...
-3
votes
2answers
148 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 ...
1
vote
1answer
237 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
572 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
420 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
176 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
130 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
228 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 ...
4
votes
1answer
53 views

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could then the $\pi$ also be an integer?

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?
0
votes
1answer
60 views

What do people mean by “finding the end of $\pi$” [closed]

So I have been wondering. I have heard many times statements like "if we find the end of $\pi$ then we might be in a virtual reality" or "new computer can calculate $X$ digits of $\pi$". My ...
1
vote
1answer
20 views

How does the speed of convergence of these formulae for calculating PI compare with the best algorithms?

I came across some series many years ago for calculating PI. I found that the first member of that series has been known for a long time in the math world. It is the set of series defined by: $$ ...
-1
votes
2answers
308 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
4
votes
1answer
239 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
1
vote
1answer
27 views

How to express an angle in terms of pi

I have the complex number $z = 5 + 6i$ in polar form $$z = \sqrt{61} (\cos \theta + i\sin \theta)$$ and $$\theta = \tan^{-1}\left(\frac{6}{5}\right) = 0.87605805059 \text{ rad}$$ But I need that ...
2
votes
1answer
190 views

A series related to $\pi\approx 2\sqrt{1+\sqrt{2}}$

This question follows a suggestion by Tito Piezas in Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$? Q: Is there a series by Ramanujan that justifies the approximation ...
1
vote
1answer
78 views

Simpler derivation to $\pi$ [closed]

I'm an amateur in mathematics, being in 9th grade. I have been trying to derive $\pi$. During this I reached a limit to find the value of $\pi$. $$\lim_{x \to 0} \frac{180\sin x}{x}$$ Where $x$ is in ...
0
votes
1answer
267 views

Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$?

The approximation $$\pi\approx\frac{22}{7}=3+\frac{1}{7}$$ suggests that the closest integer to $\frac{1}{\left(\pi-3\right)}$ is $7$. However, $$ \frac{1}{\left(\pi-3\right)^2}\approx49.879 $$ is ...
1
vote
1answer
85 views

Infinite tetration of $i$

Proof Euler's identity; $$e^{i\pi} + 1 = 0$$ can be manipulated in order to obtain the result: $$e^{i\pi} = -1$$ Raising both sides of the equality to the power of $i$ gives, after ...
0
votes
1answer
47 views

Getting theta of Line Equation

Please forgive my lack of knowledge, which i think it's one of those basic formula related to Trigonometry. Let's look at visual example: ...
3
votes
0answers
311 views

generalized 2-terms Machin's formula (an efficient way to compute $\pi$)

Looking at Machin's formulas in this post Machin's formulas and cousins, and digging a bit, I've finally computed the next formulas, allowing to generate an infinite number of 2-terms Machins's ...
6
votes
1answer
123 views

Why does this sequence converge to $\pi$?

Over at our friends at codegolf.SE, I asked a question about programs that seemed to converge to $\pi$, but didn't actually do that. One of the answers (by ...
0
votes
1answer
179 views

Does Pi contain itself? [duplicate]

Alright, recently there was a question on 9gag whether the digits of $\pi$ may contain $\pi$ itself here's the original. One user had - in my opinion - a really plausible answer: Here's his answer. ...
0
votes
2answers
139 views

Prove that $\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots =\frac{\pi^2}{8}$ [closed]

Prove that $$ \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+ \cdots =\frac{\pi^2}{8}$$ and $$\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2} +\cdots =\frac{\pi^2}{24}.$$ I do ...
6
votes
0answers
250 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...
0
votes
1answer
51 views

Finding suqares in the rectangle $(0,0,1,1)$ which are “divided with” $\frac{\pi}{4}$ by the unit circle.

I want to write an algorithm which calculates the following: Find all suqares $(x_0,y_0,x_1,y_1)$ which are "in" the suqare $(0,0,1,1)$ and are divided by the unit circle so that their inner area ...
14
votes
1answer
494 views

Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
1
vote
0answers
112 views

Riemann sum formulas for $\text{acos}(x)$, $\text{asin}(x)$ and $\text{atan}(x)$

In this post just another $\pi$ formula, I gave a kind of Riemann sum to compute the area of a quarter of circle based on a very simple geometric trick, and same reasoning can be used to compute any ...
5
votes
4answers
176 views

How does $-[-\pi]$ equal 4?

For Christmas I got a math watch and for 4 it was $-[-\pi]$. I know that $\pi$ does not equal 4 so how does $-[-\pi]$ equal 4? Thank you.
-4
votes
5answers
200 views

How can you determine rigorously if $e$ or $\pi$ are points on the real line?

This question was a part of a discussion at an interview. QUESTION: How can you determine rigorously if $e$ or $\pi$ are points on the real line? MY OPINION: They should be, since they are defined ...
6
votes
3answers
102 views

Digits of $\pi$ using Integer Arithmetic

How can I compute the first few decimal digits of $\pi$ using only integer arithmetic? By 'integer arithmetic' I mean the operations of addition, subtraction, and multiplication with both operands as ...
2
votes
2answers
209 views

Approximation of $\pi$ using Brahmagupta's Identity

Brahmagupta, an ancient Indian Mathematician, gave an pretty efficient algorithm for finding integer solutions to the famous Pell's Equation, far before Fermat propounded this before the European ...
1
vote
1answer
134 views

Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$ using Fourier series

Consider the function $f(x) = \frac{x}{2}$, defined over the interval $[0, 2\pi]$. Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$.
4
votes
5answers
293 views

Infinite sums of reciprocal power: $\sum\frac1{n^{2}}$ over odd integers [duplicate]

The infinite series I need to solve is $$\sum_{n=1,3,5...}^{\infty}\frac{1}{n^{2}}$$ and because the point of interest lies in the value of odd n, the infinite series can be expressed as ...
5
votes
4answers
244 views

How fundamental is Euler's identity, really?

Euler's identity, obviously, states that $e^{i \pi} = -1$, deriving from the fact that $e^{ix} = \cos(x) + i \sin(x)$. The trouble I'm having is that that second equation seems to be more of a ...
0
votes
0answers
71 views

Decimal digits in $\pi$

Around ten years ago I had read somewhere that there was a question in an exam for application for software engineer position in a big company which states: "What is the one billionth digit of $\pi$?" ...
0
votes
1answer
24 views

Identify the class of language?

Given a set $$S=\{x∣ \text{there is an x-block of 5's in the decimal expansion of π}\}$$ (Note: x-block is a maximal block of x successive 5's). Identify class of language? Somewhere it ...
1
vote
1answer
81 views

How can I prove without using a calculator that $\frac{1}{e} > \frac{\ln \pi}{\pi}$? [duplicate]

Without using a calculator. I can see that $\ln \pi$ is close to $1$ but a little bit greater... Since $e$ is less than $\pi$, $\frac{1}{e}$ has to be a larger number. I don't understand how someone ...
6
votes
1answer
492 views

just another $\pi$ formula

I've found this $\pi$ formula: $$ \pi =\lim_{n\to \infty }4\sum_{k=1}^{n} \frac{2 n^3 (1-2 k)^2 \left((k-1) k+n^2\right)}{\left(k^2+n^2\right)^2\left((k-1)^2+n^2\right)^2} $$ What is interesting is ...
2
votes
0answers
37 views

Using $\pi$ to number by X-mas presents

Context: For this X-max, I will make $50$ presents by myself for my family. I would like to label them so that the labels are totally ordered and all different. However, I think that the traditional ...
0
votes
3answers
97 views

How are irrational numbers, fixed points on the number line?

Please, while answering/reading this question, only keep in mind my point of view only. The question is, that how come an irrational number on a number line is a fixed point. To make things more ...
2
votes
1answer
70 views

Ceiling and Floor function

I believe that I have found a trigonometric expression for both the ceiling and floor function, and I seek confirmation that it is, indeed, correct. Update
4
votes
0answers
108 views

Two formulae for $\pi$, probably known?

I stumbled upon (in the literature) two identities for $\pi$, but they were not referenced as they are probably well-known. Hoping someone could point out who found them first. Basically, the ...
10
votes
7answers
769 views

What are your favorite relations between e and pi? [closed]

This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for ...
0
votes
1answer
23 views

Is it possible to construct an incrementally accurate rectification of a circle?

Exact rectification of a circle (construction of a segment exactly the length of the circumference of a given circle) has been proven impossible. There is a number of rectification constructions that ...
-1
votes
1answer
53 views

If π is irrational, does that means no one may ever draw a perfect circle? [closed]

Just had a thought today regarding PI. I'm not very good at geometry: If π is irrational, does that means no one may ever draw a perfect circle? This is just my assumption... .
2
votes
0answers
35 views

A variation of Viète's formula — an infinite product of nested radicals [duplicate]

Viète's formula expresses an infinite product of nested radicals in terms of $\pi$. Let $$a_1=\sqrt2,\quad a_n=\sqrt{2+a_{n-1}}.$$ Note that $\lim\limits_{n\to\infty}a_n=2.$ Then ...
0
votes
1answer
86 views

Longest chain of digits in $\pi$.

What is the longest chain of same digits in $\pi$? This question comes into my mind while reading about the Feynman Point in a book. So is there any longest known chain? Like $999999$ of Feynman ...
-2
votes
1answer
40 views

Find a physical quantity equal to $1/\sqrt\pi$ [closed]

What physical quantity gives a good sense of $1/\sqrt{\pi}$? An example of a "physical quantity that gives a good sense" of $\pi$ would be the area of a disk with unit radius.
6
votes
0answers
71 views

The irrationality of Pi [duplicate]

Pi is defined as circumference/diameter, but it is an irrational number. And by definition an irrational number can't be defined by a fraction. So how is it that pi is circumference/diameter and on a ...
0
votes
1answer
98 views

Why does this expression equal pi?

I was fiddling with numbers when I noticed that $$50 \times 1.05^{168} \times \frac{12600}{727767941} \approx \pi$$ I understand it's an approximation. Does anyone know why?