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

$(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$ How can we show this equivalent without using a calculator?
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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 $6.28....i$.
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1answer
35 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 ...
36
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0answers
628 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: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ ...
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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 $p_k$
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2answers
100 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 ...
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226 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 ...
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0answers
120 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 ...
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3answers
83 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? $$\lim_{n\rightarrow \infty}\left(\raise{3pt}\frac{2^{4n}}{n\binom {2n} {n}^{2}}\right)=\pi $$
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1answer
53 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 ...
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2answers
111 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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2answers
149 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 ...
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1answer
240 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 ...
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3answers
584 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 ...
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2answers
423 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 ...
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1answer
180 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 ...
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1answer
131 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
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0answers
229 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 ...
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1answer
55 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?
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1answer
70 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 ...
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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: $$ ...
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2answers
313 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 ...
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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 ...
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1answer
50 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 ...
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1answer
195 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 ...
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1answer
79 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 ...
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1answer
270 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 ...
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1answer
89 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 ...
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1answer
53 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: ...
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313 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 ...
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1answer
124 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 ...
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1answer
189 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. ...
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2answers
143 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
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0answers
252 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 ...
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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 ...
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1answer
497 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
116 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 ...
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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.
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5answers
202 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 ...
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3answers
108 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
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2answers
215 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
158 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 $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$.
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5answers
300 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 ...
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4answers
247 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 ...
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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 $\pi$?" ...
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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 ...
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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
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1answer
500 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 ...
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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
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3answers
100 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 ...