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Questions tagged [pi]

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 this 2Pi-Periodic function looks like this?

I am given this function 1 on an interval $[0,\pi]$ and am asked to make a sketch of it on an interval $[-\pi,\pi]$. When plugging it into Maple, I am given a sketch graph like this 2, however, that ...
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1answer
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Confusion over the word “ratio” in the definition of $\pi$

According to Wikipedia, pi is "a mathematical constant that was originally defined as the ratio of a circle's circumference to its diameter." However, when I think of the word "ratio", something like ...
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What is the formula for pi used in the Python decimal library?

(Don't be alarmed by the title; this is a question about mathematics, not programming.) In the documentation for the decimal module in the Python Standard Library, ...
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Nice result that I can't prove: $\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x²\text{erf}(x)) \bigg) \;dx=\pi$

I'm always trying to find the integral representation of $\pi$ using some interesting special function, at this time I have got the below representation $$I=\int_{-2}^{2} \tan^{-1} \bigg( \exp(-x^2\...
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4answers
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A nice relationship between $\zeta$, $\pi$ and $e$

I just happened to see this equation today, any suggestions on how to prove it? $$\sum_{n=1}^\infty{\frac{\zeta(2n)}{n(2n+1)4^n}}=\log{\frac{\pi}{e}}$$
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41 views

Does this limit $lim_{n\to\infty}\sum_{i=0}^n 1/n \sqrt{1 - i^2/n^2}$ converge to $\pi/4$?

While trying to find an approximate area of a quarter of a circle by splicing it into small rectangles and summing their areas I've reached a point where I have this formula: $$\sum_{i=0}^n 1/n \sqrt{...
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3answers
69 views

Silly Question about $π$ [closed]

In our junior classes, we learnt that $π$ is an irrational number. Now, also we know about rational numbers. So, if I say that I have a thread of 44cm long and we may convert it into a circle. Then ...
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1answer
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Help understanding the cause of this pattern when writing π as an infinite series with double factorials

I made a post about a year and a half ago: $\pi$ as an Infinite Series using Taylor Expansion on Equation of a Circle where essentially I used the Taylor series expansion on $\ y = \sqrt{r^2-x^2}$ (...
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10answers
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How are the known digits of $\pi$ guaranteed?

When discussing with my son a few of the many methods to calculate the digits of $\pi$ (15 yo school level), I realized that the methods I know more or less (geometric approximation, Monte Carlo and ...
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1answer
62 views

How is Euler's Formula Wrong?

I figured out that if $x^{y} = z$, then $z^y = x^{y^2}$. Then we know Euler's Formula: $$e^{πi} = -1,\quad (e^π)^i = -1,\quad (e^{2π})^i = 1$$ Now, using the formula above, let $e^{2π}$ act as x, ...
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Has the arithmetic mean of the Leibniz series been used for computing $\pi$?

I'm sure this was asked before, but I wasn't able to find the answer. If we have an alternating series with slow convergence, can't we just use the arithmetic mean to get a much better approximation ...
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2answers
58 views

Is $2\pi\sum_{n=0}^\infty\frac1{n!(n+1)!}$ equal to $5/\pi$?

I was working through some contour integration questions, and when finding the residues of the integral $\int e^{z+1/z}\,dx$, I found that it was equal to the infinite sum $$2\pi\sum_{n=0}^\infty\...
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2answers
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Adding $2\pi$ inches to a string about the Earth's equator makes a new circle reaching how far above the ground? [closed]

Sofia wraps a long string around the equator of the earth, pulling it snug. If the earth were a perfect sphere, the string would be touching the ground all along its length. Suddenly, Sofia stretches ...
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1answer
189 views

Can The Existence Of $\pi$ Be Proved Without Formal Analysis?

I hope this question is not too long, but I have included some extra information to clarify the context of the question and hopefully avoid the 'circular' arguments which inevitably occur on this ...
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106 views

Is there an intuitive explanation for the occurrence of e and pi in Stirling's approximation? - $n!\approx \sqrt{2\pi n} (n/e)^n$

Is there an intuitive explanation for the occurrence of e and pi in Stirling's approximation? $$n!\approx \frac{n^n}{e^n}\sqrt{2\pi n}$$ Any help would be much appreciated.
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1answer
112 views

Is there a closed form for $\zeta(\pi)$?

What is $\zeta(\pi)$? I know that $\operatorname{Re}(\pi)>1$, thus $$\zeta(\pi)=\sum_{n\geq1}\frac{1}{n^\pi}$$ Yet I have no idea how to even begin evaluating this series. It's probably ...
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2answers
32 views

EXACT measurements of a circle. Possible? [closed]

I understand simple circle equations like c=pi×d and a=pi×(rr) (sorry keyboard doesn't have pi or exponents) if pi is irrational with an infinitely long decimal, doesn't that make it impossible to ...
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34 views

How many terms it takes for the Leibniz series to converge to three decimal places of accuracy?

I need to find out how many terms it takes for the this series to converge to three decimal places of accuracy of Pi. e.i how many it terms it takes to obtain the value 3.141 from, the series: Leibniz ...
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1answer
99 views

Proof of Bellard's formula

I'm reading Bellard's proof for his eponymous formula computing pi digits, and I can't get past the first line. Given that: $\displaystyle-\ln(1-x) = \sum_{n=1}^\infty \frac{x^n}{n}$ for $|x| < 1$...
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What are some statistical distributions with the irrational numbers e and pi in their functions? (apart from the most common - Normal, Poisson)

I've been researching on the application and origin of irrational numbers in probability theory and statistical distributions, so far having derived a unique proof of Stirling's approximation, and ...
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If two real numbers are immesurable can an integer sum between the two get as close to any real number as we like?

Say for example we have $\pi$ and $1$. Can the sum $m\pi + n$ for $ m, n\in \mathbb{Z}^+$ get as close to a real number as we like? At first I tried using the fact that you could have $n = -floor(m \...
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Construction of the quadratrix by one motion only

Ancient geometers geometrically constructed the number $\pi$ using a special curve, called the quadratrix of Hippias (or Dinostratus). One way to construct the quadratrix is by tracing the path of the ...
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112 views

Which is greater, $\left(\frac{e}{2}\right)^\sqrt{3}$ or $(\sqrt{2})^{\pi/2}$? (no calculators)

From a math contest in 1985: Determine which of the following is greater: (no calculators) $$\left(\frac{e}{2}\right)^\sqrt{3} \, \hspace{3mm} \text{or} \hspace{3mm} \, (\sqrt{2})^{\pi/2}$$ Hints ...
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1answer
75 views

How can I proof that a circle's circumference is well defined? And How do I find it? [closed]

I searched here and I got the definition that the circumference of a curve would be the smallest upper bound of the sequence of the sum of lenghts of polygonal paths along the curve, but how can I ...
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0answers
42 views

Compute Machin's function by hand

I use Machin's function \begin{equation} \pi = 16 \cdot \left(\frac{1}{5} - \frac{1}{3\cdot 5^{3}}+ \dotsm\right)-4 \cdot\left(\frac{1}{239} - \frac{1}{3\cdot 239^{3}} +\dotsm\right) \end{equation} ...
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1answer
41 views

let define the “waved” factorial as $\prod_{i=1}^n (\text{if } \bmod(i,2)=={0\text{ or } 1}\text{ then } [2/i] \text{ else } [i])$

Consider these two products: EvenWavedFactorial = $$\prod_{i=1}^n \text{if } (i\bmod 2 ==0)\text{ then } \left(\frac{2}{i}\right) \text{ else }(i)$$ OddWavedFactorial = $$\prod_{i=1}^n \text{ if } ...
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1answer
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Partial sums of $\frac{\pi}{2}=\sum_{n=0}^\infty \frac{(2n-1)!!}{2^n\cdot n!\cdot (2n+1) }$.

Recently, I have found a formula for $\pi$. That is $$\frac{\pi}{2}=\sum_{n=0}^\infty \frac{(2n-1)!!}{2^n\cdot n!\cdot (2n+1) }$$ However, the problem arises when you take the partial sums. For ...
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Is Pi over Pi rational?

A student asked this question in class today, and I wasn't sure of the answer. On the one hand, since Pi is irrational itself, Pi/Pi doesn't fit the definition of a rational number (namely a number ...
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Can A be proved using B when B was proved using A?

Many questions have been posted on this site about the irrationality of $\pi$; I'll be referring to one such question here. The accepted answer mentions that $\pi$ is irrational because it is the ...
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Prove that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancendental over $\Bbb{Q}$.

We want to prove that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancedental over $\Bbb{Q}$. Attempt. We use proof by contradiction and so assume that $\alpha \in \Bbb{C}$ is algebraic over $\...
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Is there an equivalent to trigonometry for solid angles?

Intuitively I would say that it would make no sense, but this question crossed my mind : Is there an equivalent to trigonometry for solid angles?? I haven't found anything yet. Thanks for answering!...
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What is the probability that a normal number will look periodic?

Let's take for example $\pi$. The numbe $\pi$ is irrational, hence the decimal pattern is non-periodic. However, could it happen, that by just observing a huge but finite amount of digits of $\pi$, it ...
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Is there a sufficiently reachable plausibility argument that $\pi$ is irrational?

I was teaching someone earlier today (precisely, a twelve-year-old) and we came upon a problem on circles. Little did I know in what direction it would lead. I was able to give a quick plausibility ...
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How to find the $\pi$? [duplicate]

We know as we increase the number of sides in regular polygon, after infinite repetition it will give us a circle. So, is there any way to find a function which approaches to value of $\pi$ when we ...
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Cardinality of number of digits [closed]

What is the cardinality of the number of digits (in decimal form) of an irrational number like $ \pi $?
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67 views

What is the exact value of this : $5\int_{0}^{\infty}\exp(-x^2 \text{erf}(x))x^{\sin x+\frac12}dx$?

My curiousity is to get more integrals about the constant $\pi$ using special functions. I have used some special functions as shown below in the integral: $$5\int_{0}^{\infty}\exp\left(-x^2 \text{...
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1answer
34 views

How to find intersection of two and three sine waves on x axis intercept? (Biorhythms)

I'm currently studying the trigonometry behind biorhythms. I was reading through the Wikipedia article on the topic (https://en.wikipedia.org/wiki/Biorhythm) which states that: Basic arithmetic ...
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1answer
59 views

Is there any way to finitely represent all the information in pi? [closed]

Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any ...
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195 views

Showing $\pi$ is irrational using taylor's theorem

To prove the irrationality of $$e = \sum ^\infty _ {n=0} \frac{1}{n!}$$ we can show that $e \lt 3$ by using a suitable geometric series. By Taylor's theorem (applied to $a=0$ and $b=1$) we know that, ...
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1answer
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Proving a formula for $\pi$

I found a formula for $\pi$ in this paper. However, I could not find any proof of this formula, and I don't know how to approach to it. Is there good explanation for it? $$ \pi + 3 = \sum_{n=1}^\...
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1answer
65 views

Approximating pi rate of convergence

I have been reading about a method for approximating $\pi$ using two uniform distributions and the ratio of points that lie within the circle compared to the square formed by the two uniform ...
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1answer
67 views

Is this :$\int_{-1}^{1}\exp (-\tanh^{-1} (x))dx=\pi$ a well known integral representation for $\pi $?

I have tried to get interesting integral for interesting transcendental number , I have got the following :$\int_{-1}^{1}\exp (-\tanh^{-1} (x))dx=\pi$ , really I didn't accross that integral in the ...
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2answers
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General P-series rule

This is a p-series: $$\sum_{n=1}^\infty \frac{1}{n^p}$$ There are 2 p-series (to my knowledge) that somehow reach pi: $$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$$ $$\sum_{n=1}^\infty \frac{...
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1answer
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Representation of $\pi$ using algebra and exp/log.

Can $\pi$ be represented exactly using a mixture of algebraic as well as exp/log functions, all real valued? I know it can't be done using only algebra since its transcendental, but what if we ...
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1answer
160 views

How Gelfond find his limit for $\exp(\pi) $? [duplicate]

$$ a_0 = \frac{1}{\sqrt 2} $$ $$ a_{n+1} = \frac{( \sqrt {1 - a_n^2} -1)^2}{a_n^2} $$ $$ \lim_{n \to \infty} \frac{4^{\frac{1}{2^n}}}{a_{n+1}^{\frac{1}{2^n}}} = \exp(\pi) $$ How did Gelfond find ...
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Proving $\pi = 48\tan^{-1}\frac{1}{18} + 32 \tan^{-1}\frac{1}{57} - 20\tan^{-1}\frac{1}{239}$

The below equation represents $\pi$ to some decimals using tangent inverse. I need to prove that the left hand side of the equation equals the right hand side. $$ \pi = 48\tan^{-1}\frac{1}{18} + 32 \...
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2answers
91 views

Evaluation of infinite series of ratios (with denominators given as finite products) in terms of rational numbers and $\pi\sqrt3$

How can we prove whether $$\sum_{k=0}^\infty \frac{1}{\prod_{i=1}^{6n+2}(3k+i)}=q_1+q_2\sqrt{3}\pi$$ for all natural $n$ with rational $q_1$ and $q_2$? Some series related to rational ...
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1answer
112 views

Can we find every finite sequence of $\pi$ within $\pi$?

Inspired by one of the top posts I was wondering if $\pi = 3.14159\dots$ were normal--as in you could find every finite string of numbers within $\pi$'s digits--would that mean we could find every ...
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1answer
24 views

Decimal representations containing every possible sequence

I know that pi contains every finite number sequence, but i am not sure why. Since you can take any irrational number and change every $9$ to an $8$, then clearly not all irrational numbers are the ...
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2answers
87 views

Why does pi has so many digits? [closed]

Pi is used in calculating circles, spheres and round things. In calculate, pi is written down as 3.14. But the full pi is incredibly long (over a TRILLION digits). So, why there are so many digits in ...