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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0
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1answer
32 views

The probability of a number appearing in an approximation of an irrational number?

I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 ...
4
votes
0answers
34 views

Looking for reference on the appearance of $\pi$ in the central limit theorem

There are many books on $\pi$ out there, but I have not yet been able to find one that makes a serious attempt to make sense of the fact that $\pi$ (traditionally a geometric constant) appears in the ...
16
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2answers
267 views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...
0
votes
1answer
59 views

Is there an O(1) operation to find the Nth digit of Pi?

I'm afraid that 10 minutes of googling isn't finding references to a paper that I thought existed. I remember seeing some years ago a reference to a paper that claims to have proven an algorithm that ...
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0answers
56 views

Is $\pi$ wrong? [closed]

Which is more fundamental $\tau$ or $\pi$? Where $\tau=2\pi$ In particular, a circle is defined as the set of points a fixed distance the $\textbf{radius}$ from a given point, the center. While ...
1
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1answer
42 views

Randomness in pi and other irrational numbers [duplicate]

This is a post I read about pi while looking for stuff about tau -which is two times as much as pi. This makes me wonder, why does only pi contain such randomness? Don't other non-repeating and ...
-4
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1answer
95 views

How to approximate $0.714286$ as a fraction of $\pi$?

I'm doing an exercise that tells me that the answer must be a multiple of pi, like $12\pi$ or $\dfrac23\pi$. I need to approximate $0.714286$ as a fraction of $\pi$. How do I achieve this?
3
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4answers
701 views

Alternate method to calculate an infinite string of numbers that's not $\pi$, and contains any string

So, rather than using $\pi$, is there any way that isn't overly complicated, (and can be calculated on a computer without taking a year) in which I could generate an infinite string of numbers that ...
0
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1answer
63 views

Gosper Formula for inv $\pi$, properties.

I need to understand very good how the properties of this formula $\frac{4}{\pi} = \frac{5}{4} + \sum_{N \geq 1} \left[ 2^{-12N + 1} \times(42N + 5)\times {\binom {2N-1} {N}}^3 \right] $ Taken from ...
3
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0answers
38 views

$\tau$-ists and the History of Radian Measure?

Recently, I have been reading about the $\tau$ vs $\pi$ debate. One of the arguments for $\tau$ was that $1\tau$ radian is the whole circle, thus fractions of $\tau$ correspond to the fractions of the ...
8
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0answers
72 views

Generalizing Bellard's “exotic” formula for $\pi$ to $m=11$

Bellard's "exotic" pi formula has the form, $$a\pi+b = \sum_{n=1}^\infty \dfrac{P(n)}{{\displaystyle \binom{mn}{2n}2^{n-1}}}$$ where $a,b,m$ are integers and he uses $m=7$. However, it seems there ...
1
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1answer
25 views

Limit(s) of a Sequence from the decimal expansion of $\pi$

I found a statement in a book concerning the decimal expansion of $\pi$ that I do not really understand. The statement is my problem number 2, where problem number 1 really looks like a reference ...
22
votes
5answers
507 views

Why is it that this gives a good approximation of $\pi$?

At the end of a Physics examination, I decided to play around with my calculator, as I always do when I have time left over, and found that: $$\frac{1}{100} \cdot 11^{\ln(11)} \approx ...
14
votes
2answers
131 views

Bellard's exotic formula for $\pi$

In his webpage, Fabrice Bellard mentions an exotic formula for $\pi$ as follows $$\pi = \frac{1}{740025}\left(\sum_{n = 1}^{\infty}\dfrac{3P(n)}{{\displaystyle \binom{7n}{2n}2^{n - 1}}} - ...
-4
votes
4answers
349 views

Is $\pi \cdot 7$ actually $22$? [closed]

The value of $\pi$ is $\frac {22}{7}$. Now if I multiply, $\pi$ by $7$, it gives 22 (as $\frac{22}{7} \cdot 7 = 22$). But, a when we multiply a rational number with a irrational number, we are ...
2
votes
3answers
135 views

How do mathematicians invented and introduced $\pi$ term in the case of circle?

This is basic question. Since childhood I am mugging the mathematical formulae areas of square, rectangle and circle etc. Now,it is possible for me to understand formula of area of square I.e. ...
1
vote
1answer
70 views

What would a base $\pi$ number system look like?

Imagine if we used a base $\pi$ number system, what would it look like? Wouldn't it make certain problems more intuitive (eg: area and volume calculations simpler in some way)? This may seem like a ...
9
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0answers
63 views

First order definition of $\pi$

$e$ has a very short first-order definition. It's the only constant that makes this true: $$\forall x,e^x\ge x+1$$ What about $\pi$? What's the shortest first-order definition we could give it? I'm ...
7
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0answers
127 views

Looking for an effective irrationality measure of $\pi$ [migrated]

Most standard summaries of the literature on irrationality measure simply say, e.g., that $$ \left| \pi - \frac{p}{q}\right| > \frac{1}{q^{7.6063}} $$ for all sufficiently large $q$, without giving ...
3
votes
1answer
50 views

Can we show that the decimal expansion of $\pi$ doesn't occur in the decimal expansion of the Champernowne constant?

This question was inspired by an answer and some comments to this question. Recall that the Champernowne constant is obtained by concatenating all natural numbers written in base 10 and then put ...
11
votes
2answers
744 views

Decimal Expansion of Pi

Sorry if this has been asked before, but I have a query about the notion that the decimal expansion of pi contains every possible string of numbers (please note that I am only a "casual" maths ...
1
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2answers
46 views

Number system and PI

Ok, we all use the decimal system with numbers from 0 to 9. And we have PI with an infinite number of decimals. We also have a boolean system or hexadecimal. Is there any decimal system where PI has ...
0
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0answers
41 views

Extensions of the algebraic numbers

I have two extensions of the algebraic numbers, and I'd like to know whether they are equivalent. Definition 1. $x \in \mathbb E_1$ iff $x$ is a root of $e^{P(x)} + Q(x)$ with $P, Q$ some ...
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0answers
35 views

Euler's Equation [duplicate]

I'm new around here, I'm fourteen, and I am in Ninth Grade. Can somebody tell me what Euler's equation exactly is, and why it's important, and what we can use it for? The whole $e^{i\pi} = -1 $ thing. ...
2
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1answer
43 views

What's the fastest known running time for a spigot algorithm for computing an arbitrary digit of $\pi$?

That is, for the fastest known algorithm for doing so, how many steps will it compute the $n^{\text{th}}$ digit of $\pi$ in? I know some people define running time as the number of steps it will take ...
0
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0answers
24 views

Calculating the speed of a rotating object on its own axis and its speed around a point

I'm seeking help calculating the speed of rotation of the blue object $s_2$ in relation to $s_1$ with a distance 'd' from point $A$. In other words, the goal is to keep the object always facing $A$ if ...
6
votes
1answer
148 views

Does this curiousity have any meaning?

If $\pi$ is calculated to the $360$-th digit after the decimal point, the last digits are $360$ : ...
1
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0answers
54 views

Integral tending to an integral for $\pi$

I am examining: $$\int_0^1 (1-ax)^{1/2} dx$$ If we differentiate: $$\dfrac{d}{dx} \left[\dfrac{-2(1-ax)^{3/2}}{3a}\right]$$ we get to the function in the integral. The idea now is consider various ...
0
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0answers
33 views

Need proof of integration of sine parametrized functions [duplicate]

Yesterday, i encountered an integral formula (actually it's a generalization, i think). This : $$\int_0^\pi x f(\sin x)\,dx = \frac{\pi}{2} \int_0^\pi f(\sin x)\,dx$$ For simple functions like ...
14
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0answers
189 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$: (Mathematica code) ...
28
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5answers
957 views

Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$?

I've been struggling for a while with the following problem: Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$? Needless to say software aid is not allowed. All manual calculations should be ...
5
votes
4answers
656 views

Number raised to power of irrational number

What is the consequence of raising a number to the power of irrational number? Ex: $2^\pi , 5^\sqrt2$ Does this mathematically makes sense? (Are there any problems in physics world where we ...
6
votes
0answers
48 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of ...
16
votes
8answers
436 views

Explain why $e^{i\pi} = -1$ to an $8^{th}$ grader?

I am an $8^{th}$ grader that is taking Algebra I. But nearly everyday I try to learn things outside of what I am learning in class. Quite a while ago I discovered that $e^{i\pi} = -1$. This ...
15
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3answers
2k views

How to convert $\pi$ to base 16?

According to this Wikipedia article $\pi$ is approximately 3.243F in base 16 (i.e. hexadecimal). Can someone explain this? (Note: I understand how to convert an integer to base 16) Thanks
4
votes
2answers
62 views

Deriving the value of $\pi$ from a dart board

I saw this on a website and it was pretty interesting: The circle inscribed in the square has a radius of $1$ and the square has a side length of $2$. This means that the area of the circle is: ...
0
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1answer
69 views

How To Calculate a Tangent In Degrees Without a Calculator

So the other day in my Geometry class, I was bored so I decided to try and calculate pi (which is one of many things I do when I am bored). During that class, I finally developed an equation to ...
2
votes
3answers
97 views

Is it possible to intuitively explain, how the three irrational numbers $e$, $i$ and $\pi$ are related?

I read a bit about this equation: $e^{i\pi}=-1$ For someone knowing high school maths this perplexes me. How are these three irrational numbers so seemingly smoothly related to one another? Can this ...
9
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2answers
177 views

Does π start with two identical decimal sequences?

Let d$(x,y)$ be the sequence of decimals of π from the x:th one to the y:th one. My question: is there a number $n$ such that d$(1,n)$ = d$(n+1, 2n)$? I.e., does π start with two identical decimal ...
0
votes
3answers
109 views

How does atan(1) * 4 equal PI?

I needed the PI constant in C++, and I was lead to the answer that: const PI = atan(1) * 4 Note that despite involving code, I'm asking this from a mathematics ...
0
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1answer
29 views

ratio of “diameter” of a n-gon to perimeter

So say I have a regular polygon with n sides, and I bisect the an angle E such and the line (EF). Assume line segment EF has length b, while the polygons side length is s. What is $b/(n*s)$, and as n ...
8
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3answers
234 views

Show that $\pi =4-\sum_{n=1}^{\infty }\frac{(n)!(n-1)!}{(2n+1)!}2^{n+1}$

Show that $$\pi =4-\sum_{n=1}^{\infty }\frac{(n)!(n-1)!}{(2n+1)!}2^{n+1}$$ I found the formula of $\pi$ by using the numerical calculation but I dont have the proving. any Help
0
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3answers
93 views

Repetition in pi

If there are infinite digits in $\pi$ and any group of digits occurs in $\pi$. Then does all the digits of pi occur in itself infinite times over? Therefore $\pi$ repeats. What is wrong with my ...
1
vote
2answers
77 views

Where do the numbers come from, to calculate pi?

As we all know, pi is the ratio of a circle's circumference to its diameter. When you divide the circumference by the diameter, the result is pi. But, here's my question: When you enter the ...
3
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2answers
97 views

$ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $

If $ \sum _{n=1}^{\infty} \frac 1 {n^2} =\frac {\pi ^2}{6} $ then $ \sum _{n=1}^{\infty} \frac 1 {(2n -1)^2} $ Dont know what kind of series is this. Please educate. How to do such problems?
6
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2answers
103 views

If $\dfrac{\mathrm {circumference}}{\mathrm {diameter}}$ is the same for all circles, does the surface have to be flat?

Given a two dimensional Riemannian manifold with the property that the ratio of the circumference and the diameter is the same for all circles. What can be said about it? Does it have to be the ...
41
votes
7answers
3k views

Calculating pi manually

Hypothetically you are put in math jail and the jailer says he will let you out only if you can give him 707 digits of pi. You can have a ream of paper and a couple pens, no computer, books, previous ...
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1answer
69 views

Pi Day question

Everyone today is talking about Pi Day and the match to 3/14/15 at 9:26:53 AM. As I've become old, my brain doesn't work so well, so I could be way off on this, but if we include decimal fractions of ...
1
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1answer
87 views

Buffons needle crossing both lines?

Buffon's Needle Problem : Given a needle of length $l$ dropped on a plane ruled with parallel lines $t$ units apart, what is the probability that the needle will cross a line? I am working out ...
4
votes
1answer
68 views

History Behind Integral Error Between $\pi$ and $22/7$

Looking at an expression for $\pi$ $$\pi = \frac{22}7-\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \ dx$$ it seems to me that the integral expression is the error between the approximation $\frac{22}7$ and ...