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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4
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1answer
68 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 ...
1
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0answers
76 views

Comment on this Pi Formula? [on hold]

I compute Pi in my own way as follows: $$\frac{\pi}{2}=1+\frac{1}{1+\frac{1}{\frac{1}{2}+\frac{1}{\frac{1}{3}+\frac{1}{\frac{1}{4}+\frac{1}{\vdots}}}}}$$ and use ...
0
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2answers
68 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 ...
0
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2answers
75 views

If pi (or tau) is a constant for a circle, what is the equivalent of a sphere called? [closed]

I never learned this in math class, but was just thinking that a sphere should have some equivalent to pi, and that it too should be a constant... what would that be? Thinking more about it, if the ...
3
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0answers
79 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} ...
4
votes
2answers
121 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?
8
votes
3answers
268 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
55 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 ...
4
votes
1answer
104 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 ...
1
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1answer
68 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 ...
8
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2answers
189 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
votes
4answers
501 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
votes
2answers
87 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
votes
1answer
76 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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2answers
161 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} > ...
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6answers
182 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
67 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
64 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 ...
8
votes
1answer
194 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 ...
2
votes
4answers
180 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
115 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?
0
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1answer
66 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 ...
6
votes
1answer
177 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
57 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 ...
3
votes
3answers
127 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
55 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. ...
-2
votes
1answer
57 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 ...
0
votes
1answer
29 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
74 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 ...
0
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2answers
54 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 ...
3
votes
3answers
73 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?
3
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2answers
97 views

Understanding proof that $\pi$ is irrational

Reading this: Simple proof that $\pi$ is irrational, I fail to understand the following part: Since $n!f(x)$ has integral coefficients and terms in $x$ of degree not less than $n$, $f(x)$ and ...
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votes
1answer
59 views

Why there are two different values of θ for same quadrant?

Let Sin θ = 1/2 is function. Let us find its solution set. sine is +ve in I and II quadrant with reference angle π/6 θ = π/6 (I quadrant) Now here is my problem. We can use π-θ and (π/2)+θ to find ...
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vote
1answer
31 views

2 pi term in sinusoidal signal

My intuition is that the $2\pi$ term in the sinusoidal signal equation: $$x(t) = \sin(2\pi\,f\,t)$$ Is indicative of the fact that this signal can be described as movement around a circle, is that ...
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3answers
117 views

Is there any geometry where ratio of circle's circumference to its diameter is rational?

In Euclidean geometry, the ratio of the circumference of a circle to its diameter is an irrational number, 3.14159 and so on. But if you change to non-Euclidean geometries, you get other values for ...
2
votes
1answer
103 views

Irrationality of $\pi$ and circumference to diameter ratio.

How is $\pi$ actually defined? If it is defined as the ratio of the circumference of a circle to its diameter then from this definition itself either of the circumference and diameter has to be ...
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4answers
110 views

Is 1/113 a rational number?

Before two days , one of my physics si told me if one wants to use value of pi more accurately 355/113 can be considered as value of pi to get more accurate result. I want to know is 1/113 a ...
0
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2answers
49 views

Is an anomaly in base-n arithmetic discoverable in base-m arithmetic?

I have always been fascinated by the book "Contact" by Carl Sagan. The final chapter of the book (not the film!) reports about an anomaly in the n-millionth decimal of pi, optimally visible when pi is ...
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2answers
49 views

Precalculus converting radians to degrees.

I'm studying for a precalc test and I've kind of hit a brick wall with conversion. I've searched for help but only found things on converting from pi radians. I found some questions that consist of ...
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votes
5answers
261 views

How do i prove that $3<\pi<4$?

Let's not invoke the polynomial expansion of $\arctan$ function. I remember i saw somewhere here a very simple proof showing that $3<\pi<4$ but i don't remember where i saw it.. (I remember ...
7
votes
1answer
128 views

Calculating $\pi$ via the $\zeta$ function?

I was fooling around, trying to come up with a rapid way to compute $\pi$. Then I remembered that we always have: \begin{equation} \zeta(2n)=c\pi^{2n}, \end{equation} where $n$ is a positive integer ...
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3answers
310 views

Why is $\pi$ so close to $3$? [closed]

$\pi\approx 3.141592654$ Why is it so close to $3$? I find this intriguing, this cannot be a coincidence.
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3answers
225 views

is 22/7 is real value of pi? [duplicate]

I am just new to this so pardon me if my question is some silly. I was googleing about value of pi. in wikipedia it has 3.141592653589793238462643383279502884197169399 as I have studied its 22/7. my ...
3
votes
1answer
200 views

Characterization of $\pi$

Let $x$ be the ratio of a circle's circumference to its diameter. Let $y$ be twice the smallest positive number $t$ for which $\cos(t)$ equals $0$. How to prove $x=y$? Thanks.
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4answers
94 views

why the occurrence of 4,5,6 and 9 in pi differs?

i´m playing around with pi, i have this document with the first 5million decimal numbers after comma. http://www.aip.de/~wasi/PI/Pibel/pibel_5mio.pdf and i build a script that i put in for example ...
0
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1answer
30 views

Critical Numbers Problems

Okay so I found the critical number no problem, it being cos x=-1/2, but on my answer sheet it says that the critical numbers are ...
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3answers
857 views

How to find continued fraction of pi

I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are: $\pi=[3:7,15,1,292,.....]$ and many others given here. Similarly some nice continued ...
2
votes
1answer
103 views

Doubt in Ivan Niven's proof of irrationality of pi.

In the proof, how do we get the upper limit for $f(x) \sin{x}$ as $\pi^n \cdot \frac{a^n}{n!}$ ? I thought $f(x) \sin{x}$ would be maximum at $x=\pi/2$ when its value would be: $$\pi^n \cdot ...
3
votes
3answers
114 views

Proof without words for $\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$

$$\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$$ $$1-\frac13+\frac15-\frac17+\frac19-\cdots=\frac\pi4$$ Does anyone know of a proof without words for this? I am not looking a for a just any proof, since I ...