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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Solving a problem of Ramanujan's interest

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
6
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2answers
92 views

Calculating value of $\pi$ independently using integrals.

Recently I noticed this integral: $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}7-\pi\approx0$$ Which is a very interesting result which gives us the value of ...
26
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6answers
3k views

What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula ...
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2answers
94 views

Why Doesn't this equation work?

$$\frac{n2^nn!^2}{(2n)!}$$ Is supposed to output the nth decimal of pi, this works fine with $$n = 1$$, but why not with $$n = 2$$? (I ommited the sigma, equation was found here)
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3answers
42 views

How Does This Evaluate?

I just relized, I have no idea how to evaluate: $$n2^2n!^2$$ Google didn't help me find a breakdown could anybody just space it out?
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8answers
225 views

Calculate $\pi$ By Hand?

All over the internet the only hand equation i found was $$\frac\pi4 = 1 - \frac13 + \frac15 - \frac17+\cdots.$$ But this takes something like a thousand iterations to get to four digits, is there a ...
2
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1answer
89 views

Plouffe's formula for $\pi$

Plouffe established the following formula for $\pi$ in $2006$ $$\pi = 72\sum_{n = 1}^{\infty}\frac{1}{n(e^{n\pi} - 1)} - 96\sum_{n = 1}^{\infty}\frac{1}{n(e^{2n\pi} - 1)} + 24\sum_{n = ...
3
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1answer
76 views

Proving Archimedes Sequences Equal $\pi$.

I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
6
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1answer
99 views

Show that $\lim_{n \to \infty}\prod_{k = n}^{2n}\dfrac{\pi}{2\tan^{-1}k} = 4^{1/\pi}$

MathWorld states that (see equation $(130)$) $$\lim_{n \to \infty}\prod_{k = n}^{2n}\dfrac{\pi}{2\tan^{-1}k} = 4^{1/\pi}$$ and attributes it to Gosper. I believe an approach to establish the formula ...
1
vote
1answer
64 views

Can $\pi$ and the $\pi$ in radians simplify?

I saw in a proof for the limit $$\lim_{x\rightarrow 0}\frac{\sin(x)}{x}=1$$ that, in one of the steps, you had to take the area of a section of a circle, in which you had to do $\frac{\pi r^2 ...
2
votes
3answers
61 views

Why does $y = x\sin(\frac{180}{x})$ approach $\pi$?

A few days ago I was playing on my scientific calculator and I ran over an interesting little equation: $180\sin(1)$ is extremely close to $\pi$. At first I thought it was a coincidence, but then I ...
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0answers
38 views

Calculate digits of pi without needing to reuse them

I am looking for some algorithms that can calculate digits of pi. without needing to reuse previous digits. I would like to find the most simple and fast algorithms possible. Thanks!
2
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1answer
50 views

Find Pi using integral

I am just started learning calculus and wonder why: $$\int_0^1 \frac{4}{1+x^2}$$ Allows to find $\pi$? It would be great if someone could provide very detailed explanation.
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2answers
84 views

Finding the exact area of a circle?

Background: I recently began taking calculus and it has come to alter the way I look at circles, and curves. The equation of a circle is $\pi r^2$, traditionally in school we have always left the ...
7
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1answer
109 views

Showing that $ \int_{0}^{\pi / 4} \arctan \! \left( \sqrt{\frac{\cos 2x}{2 \cos^{2} x}} \right) \mathrm{d}{x} = \frac{\pi^{2}}{24} $.

I was wondering if an expert in integration could kindly solve the following problem, which was posed in a mathematics competition (I can’t remember which one) and was unsolved by any participant. ...
2
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2answers
68 views

How can I visually imagine the area of a circle divided by $\pi$?

If I have a circle with an area of 100 units^2, and I divide it by $\pi$, how can I imagine that visually in my mind? Since 100 / $\pi$ =~ 31.83, and the square of that is =~ 5.64, I currently ...
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2answers
73 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.
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2answers
412 views

Ramanujan's approximation for $\pi$

In 1910, Srinivasa Ramanujan found several rapidly converging infinite series of $\pi$, such as $$ \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}. ...
2
votes
1answer
63 views

Continued fraction expansion of Pi (oeis A001203). [duplicate]

I would like to understand how you get the numbers $$3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+...}}}}$$ i.e. $\{3,7,15,1,292,...\}$ (A001203). In the comments of A046965 is explained a method ...
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0answers
40 views

What is the limit of $k^2|\pi-n(k)/k |$, where $k$ minimizes $|k\pi -n|$?

Let $k\in \mathbb N$ and for any such n, let $k=k(n)$ minimizes the distance $|k\pi-n|\leq 2 \pi$. It is clear that, by fixing the value of $n$, it is possible to choose $k$ (and vice versa). ...
4
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1answer
92 views

Intuitive reason for why the Gaussian integral converges to the square root of pi?

This is a very famous problem, which is commonly taught when students begin learning about multivariable integration in polar coordinates. However, it has always bothered me that we recieved such an ...
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2answers
136 views

Lambert's Original Proof that $\pi$ is irrational.

I am trying to find Lambert's original proof that $\pi$ is irrational. Wikipedia has a little description but it is quite lacking. Can someone direct me to Lambert's original proof or post his proof ...
4
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1answer
132 views

How do you use the BBP Formula to calculate the nth digit of π?

I know what the Bailey-Borweim-Plouffe Formula (BBP Formula) is—it's $\pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} ...
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2answers
179 views

What is the importance of $\pi$ in mathematics? [closed]

In what specific fields is $\pi$ relevant in mathematics and how is its accuracy important? Is there any field in which its precision leads to some results despite others?
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3answers
491 views

Another integral for $\pi$

Here is a new integral for $\pi$. $$\int_{0}^{1}\sqrt{\frac{\left\{1/x\right\}}{1-\left\{1/x\right\}}}\, \frac{\mathrm{d}x}{1-x} = \pi $$ where $\left\{x\right\}$ denotes the fractional part of ...
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3answers
63 views

Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ (corrected inequation)

Prove that Prove that $n\sin\frac{2\pi}{n}-\frac{1}{4}n\sin\frac{4\pi}{n}>\pi$ algebraically or geometrically. $n\sin\frac{2\pi}{n}-n\sin\frac{\pi}{n}$ means the area of a regular n-gon + the area ...
5
votes
1answer
156 views

Finding a specific sequence of digits in pi

Looking at the pifs project on GitHub and this question on SO has made me curious as to how feasible it is to find a specific sequence of digits within Pi. Essentially, on average, how many digits of ...
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1answer
84 views

How is PI used to predict weather patterns?

I've heard that using PI to predict weather patterns is possible. I would like verification on this, and how this is possible. I can't seem to find any other sources explaining this concept. My ...
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2answers
256 views

Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?

I have heard that it is unknown whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer. How can this be? $\pi$ is known to many digits and it seems like only a matter of time on a computer to find ...
26
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4answers
536 views

Geometric intuition for $\pi /4 = 1 - 1/3 + 1/5 - \cdots$?

Following reading this great post : What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?, Vadim's answer reminded me of something an analysis ...
3
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2answers
112 views

What is the probability of a specific sequence of 11 digits occurring in a random sequence of one billion digits?

This isn't homework, I'm actually (please don't ask me why) wondering how likely it is that any particular 11-digit telephone number will occur in the first billion digits of pi. My probability course ...
6
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1answer
83 views

A formula for $\pi$ and an inequality

For any $n\in \mathbb{N}$ prove the identity : $$\pi =\sum_{k=1}^{n}\frac{2^{k+1}}{k\dbinom{2k}{k}}+\frac{4^{n+1}}{\dbinom{2n}{n}}\int_{1}^{\infty}\frac{\mathrm{d}x}{(1+x^2)^{n+1}}\tag{1}$$ and thus ...
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0answers
91 views

Curious set of $n\sin^2(n)$

I consider this set $S_{0}=\{ n \in \mathbb{N}: n\sin^2(n) < 1 \}$. And I have some questions. See the elements of $S_{0}$= $\{ 1,3,6,19,22,25,44,47,66,69,88,110,132,154,157,176,179,$ (common ...
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vote
2answers
207 views

Which is greater, $e^{\pi}$ or $\pi^e$? [duplicate]

I'm familiar with a simple method of demonstrating that $e^\pi$ is greater: $f(x) = \ln|x|/x$ $f'(x) = (1 - (\ln|x|))/(x^2)$ so f's max is at $(e, 1/e)$ so $1/e > \ln(\pi)/\pi$ and $e^{\pi} > ...
10
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6answers
209 views

When the approximation $\pi\simeq 3.14$ is NOT sufficent

It's common at schools to use $3.14$ as an appropriate approximation of $\pi$. However, here it's stated that for some purposes, $\pi$ should be approximated to $32$ decimal places. I need an example ...
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1answer
75 views

Pi for non mathematician

I've been long gone from math (shamefully) and have trouble using some quite familiar concepts... Consider the following picture in which I render two circles with radius 32 (...
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4answers
66 views

Why is it safe to approximate $2\pi r$ with regular polygons?

Considering this question: Is value of $\pi = 4$? I can intuitively see that when the number of sides of a regular polygon inscribed in a circle increases, its perimeter gets closer to the perimeter ...
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1answer
206 views

Help me ID this weird $\pi$ formula

I remembered, and managed to find, still gathering dust in a forgotten corner of the Internet, an old QuickBASIC program which, with a trick, can rapidly sum up a HUGE amount of terms of the famous ...
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votes
4answers
199 views

Why does $e^{i\pi}=-1$? [duplicate]

I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are. As far as I know ...
8
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0answers
121 views

The “trick” functions in the “$\pi$ is transcendental” proofs

I was reading this paper and I wondered how did Hermite decide to define a function $$f(x)=\frac{x^{p-1}(x-1)^p\cdots (x-m)^p}{(p-1)!}$$ Are these functions only tricks or there is a deeper meaning?
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1answer
86 views

In the hyperbolic geometry, is there a range of $\pi$?

In Euclidean space, $\pi$ is the constant value $3.14159\dots$ But I tried to measure the value of $\pi$ and found that $\pi$ is not constant! So I wonder if there is a range of $\pi$. If so, is ...
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1answer
216 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
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0answers
61 views

Comparison of Pi with rational numbers

Is there a way to compare (i.e. to know if it is greater or less) Pi with any given rational number (given by a/b, where a and b are integers), and to know beforehand the maximum of steps (development ...
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3answers
134 views

Is it possible that $\pi$ is finite in other numerical bases?

In base $\pi$, the number $\pi$ is $1\cdot \pi^1 + 0\cdot \pi ^ 0 $, which is equal to $10$. So, is $\pi$ an irrational number in all bases or not?
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3answers
288 views

Using loop to approximate pi (Monte Carlo, MATLAB)

I've written the following code, based on a for loop to approximate the number pi using the Monte-Carlo-method for 100, 1000, 10000 and 100000 random points. ...
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1answer
92 views

Why do sine and cosine functions intersect the multiples of pi at x-axis?

Why do sine and cosine functions intersect the multiples of pi at x-axis? Maybe a dumb question, but I can't figure out why is that... And since $\pi$ is a transcendental number, we can't find the ...
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1answer
32 views

Is there any combination of numbers which upon division gives the exact number of P?

In other words, there are (probably) infinite combination of numbers/operations which leads to irrational numbers. So I wonder, if there is one which gives exact number representation of P(π)? Do we ...
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4answers
79 views

there is any relation between $\pi$, $\sqrt{2}$ or a generic polygon?

I'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make use ...
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2answers
71 views

Is it true that the formulas of the volume and surface area of an n-dimensional sphere are best expressed in terms of $\eta = \frac{\pi}{2}$?

Someone told me that the formulas of the volume and surface area of an n-dimensional sphere get simplified a lot if we express them in terms of $\eta = \frac{\pi}{2}$ instead of $\pi$. . In terms of ...
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3answers
85 views

If $\pi $ is normal, can it be used as a random number generator?

If one day we finally prove the normality of $\pi $, would we be able to say that we have ourselves a sure-fire truly random number generator?