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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1answer
41 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
16 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
136 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
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1answer
72 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
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1answer
224 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 ...
0
votes
1answer
70 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
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1answer
37 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
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0answers
229 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
88 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
169 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
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2answers
131 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
220 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
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1answer
49 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
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1answer
450 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, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
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0answers
100 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
165 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.
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5answers
188 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 ...
5
votes
3answers
81 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
170 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 ...
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1answer
89 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
278 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
227 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
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0answers
58 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
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1answer
23 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 ...
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1answer
79 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
466 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
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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
87 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 ...
1
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1answer
61 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
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0answers
106 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
698 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
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1answer
17 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
49 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
33 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
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1answer
82 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 ...
0
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0answers
31 views

convering speed of machin's formula vs leibniz

i've been writing a program calculating pi with machin's formula π/4=4arctan15−arctan1239. However, because I only learned up to calc 1 (My college hasn't started), i don't understand why machin's ...
-2
votes
1answer
35 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.
5
votes
0answers
66 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
90 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?
0
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1answer
28 views

Is this number a Liouville number?

Suppose I have a binary constant $q = 0.1010000000000000000000000000000000001001..._2$. In base 10 this number is $q $~$ .6250000000077325..$ and is defined as $$q = \sum_{\rho}^{\infty} ...
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0answers
69 views

Chudnovsky binary splitting and factoring

In this article, a fast recursive formulation of the Chudnovsky pi formula using binary splitting is given. For $S(a,b)$: $$ m = (a + b) / 2 $$ $$P(a,b) = P(a,m) P(m,b)$$ $$Q(a,b) = Q(a,m) Q(m,b)$$ ...
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1answer
26 views

How to Find the Differential of $y=2\sin^2(x)$ when $x = \pi/4$ and $dx = 0.49$

I am wondering how to find the differential of $y=2\sin^2(x)$ when $x = \pi/4$ and $dx = 0.49$. I realize that I should be finding the derivative of $y=2\sin^2(x)$, which is $4\sin(x)\cos(x)$. And, I ...
1
vote
1answer
47 views

Is this a valid proof of the Area of a Circle, assuming we do not already know $\pi$?

In my AP Calculus class my friend and I decided we wanted to prove the area of a circle, without knowing $\pi$ to begin with. My friend and I each did it a different way. My method was to split the ...
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0answers
49 views

A set of integers related to $\pi$

Let $\Psi$ be the maps \begin{array}{lrcl} \Psi : & {\mathbb N}^{(\mathbb N)} & \longrightarrow & \mathbb R^+\cup\{+\infty\} \\ & A=(a_i)_{i\in \{0,\ldots,n-1\}} & \longmapsto ...
6
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0answers
58 views

Divisibility sequence resulting in limit with pi

Consider the following sequence of operations : Start with a natural number $n$ and then round it up to the closest multiple of $n-1$ .Then round up this new number to the closest multiple of ...
0
votes
1answer
40 views

“Error, (in Student:-calculus1:-roots) Cannot determine if this expression is true or false” [closed]

So I am trying to do the calculation which states: intervalsolve(sin(t) = .7, t = 0 .. 4*PI) but whenever I do it, I get the error which says: ...
5
votes
4answers
140 views

What is the relation between $\Gamma(a)$ and circles?

Looking through my calc textbook, it states that $$\int_0^\infty x^{a-1} e^{-x} \text{d}x =\Gamma(a)$$ As I have read ahead, I can understand most of the fairly basic concepts behind this function, ...
5
votes
2answers
122 views

Where, if ever, does the decimal representation of $\pi$ repeat its initial segment?

I was wondering at which decimal place $\pi$ first repeats itself exactly once. So if $\pi$ went $3.143141592...$, it would be the thousandth place, where the second $3$ is. To clarify, this ...
13
votes
1answer
241 views

Prove without using a calculator $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$

Prove without using a calculator $$(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$$ I want to know if there is an easy way to prove this inequality without using a calculator.
2
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0answers
34 views

Using Euler for estimating $\pi$

From Euler we know that $\pi ^{2} / 6 = \sum 1/ n^{2}$. Is this a good approximation for estimating $\pi$ ?