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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6
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0answers
53 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
452 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 ...
16
votes
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
65 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
votes
1answer
102 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
99 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
votes
2answers
219 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
219 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
votes
1answer
33 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
votes
3answers
249 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 would be appreciated.
0
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3answers
106 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
80 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
votes
2answers
101 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
votes
2answers
105 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
4k 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 ...
0
votes
1answer
71 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
vote
1answer
94 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
73 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 ...
4
votes
5answers
2k views

why is PI considered irrational if it can be expressed as ratio of circumference to diameter? [duplicate]

Pi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where both ...
2
votes
1answer
98 views

Is $\pi$ approximately algebraic?

As we know, $\pi$ is transcendental, meaning that there is no rational numbers $a_0,\ldots,a_n\in\mathbb{Q}$ such that $$a_0+a_1\pi+\cdots+a_n\pi^n=0.$$ But I was wondering if we can get this as a ...
1
vote
2answers
48 views

How to prove that this is equal to $\pi$?

I was trying to prove the formula for the area of a circle (without using integrals), so I started with the $\frac{Pa}{2}$ formula and started to manipulate it until I got, for an infinite number of ...
2
votes
1answer
109 views

Why does Archimedes Method to calculate Pi decrease in precision after a certain time?

i`m using the following recursive formula to calculate Pi based on Archimedes ideas. $$ S' = \sqrt{2-\sqrt{4-S^2}} $$ The formula gives back the edge length of a Polygon B based on the edge length of ...
0
votes
1answer
65 views

what are some of the oldest and most accurate approximations of pi?

I read that there is a tradition in the Jewish literature of an approximation of pi given in the prophets was very accurate ($\frac{3\times111}{106} \approx 3.14150\ldots$ - difference of about ...
0
votes
2answers
137 views

Is there a difference between the calculated value of Pi and the measured value?

The mathematical value of Pi has been calculated to a ridiculous degree of precision using mathematical methods, but to what degree of precision has anyone actually measured the value of Pi (or at ...
3
votes
2answers
135 views

Am I cheating in this case to evaluate $\pi$?

Since $\lim_{x \to 0}$$\sin x \over x$$=1$,here let $x=$$\pi\over n$ , then we have $\lim_{{\pi\over n} \to 0}$$\sin {\pi\over n} \over {\pi\over n}$$=1$ , which implies $\pi=$$\lim_{n \to\infty}$$\ ...
-2
votes
1answer
37 views

Adding 90° to atan2 result

I have a question since im using Atan2 that correctly results in -pi/pi problem is the object that im using the rotation on has its source rotation at -90 so for it to work coorecly i wanna ...
2
votes
0answers
40 views

Find Pi number using Turing Machine

What is the most convenient and fast way to find first $n$ binary digits of $\pi$ using Turing Machine?
3
votes
1answer
55 views

Let $x$ be a positive real number. Inequality problem with $\pi$ and $x$ terms.

Let $x$ be a positive real number. Then (A) $ x^{2} + \pi^{2} + x^{2\pi}> x\pi+ (\pi+x)x^{\pi} $ (B) $ x^{\pi} + \pi^{x}> x^{2\pi}+ \pi^{2x} $ (C) $ \pi x +(\pi+x)x^{\pi}>x^2 + \pi^2 + ...
0
votes
2answers
285 views

find the point (x,y) on the unit circle that corresponds to the real number t

t=π/4 I tried to solve this problem but i dont even know where to start! i thought you had to divide the pie into 4 then put it on a number line, but when i checked my answer it was like in quadratic ...
6
votes
2answers
141 views

Strange approximation of $\pi$?

I was playing with my calculator (Casio fx-991MS) the other day. I input $$\arcsin(\sin(2))$$ The result came out as $$1.141592653\ldots$$ I immediately noticed that the digits seem to resemble $\pi$. ...
0
votes
1answer
54 views

Prove a limit using the formal definition of the limit

So I have a sequence {a_n} = π/2^n where n=1,2,3,4.... And I need to prove that its limit is 0. Here is what have done, can someone check and tell me if this is correct.? Definition: A sequence ...
1
vote
1answer
48 views

An algorithm for computing $\pi^{-1}$

Wikipedia: Borwein's algorithm claims Start out by setting $$\begin{align} a_0 & = \frac{1}{3} \\ s_0 & = \frac{\sqrt{3} - 1}{2} \end{align} $$ ...
29
votes
8answers
6k views

How are first digits of $\pi$ found?

Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am ...
0
votes
1answer
71 views

Find Number of Iterations of Euler's Method in order to approximate $\pi$

I am given the function $x(t)=4 \arctan t$ and told that routine computations will show that $x(0)=0$ and $x(1)=\pi$. I must determine a differential equation for $x(t)$ of the form $x'(t)=f(t,x)$. ...
3
votes
1answer
83 views

Is there a number $x\neq0$ whose products with $\pi$ and with $e$ are both rational?

Does there exist a number $x\neq0$, such that $[x\cdot\pi\in\mathbb{Q}]\wedge[x\cdot{e}\in\mathbb{Q}]$? I thought this question would be easy to answer, but it turns out otherwise. Obviously ...
3
votes
2answers
117 views

It's possible to calculate the frequency of distribution of digits of $\pi$?

It's possible using mathematical formula to calculate frequency of distribution of digits of $\pi$ or other constant? I know that there are already plenty of data available with statistics and you ...
11
votes
1answer
208 views

Gosper's unusual formula connecting $e$ and $\pi$

Wolfram MathWorld quotes (see equation $(26)$) Gosper gives the unusual equation connecting $\pi$ and $e$ $$\sum_{n = 1}^{\infty}\frac{1}{n^{2}}\cos\left(\frac{9}{n\pi + \sqrt{n^{2}\pi^{2} - ...
2
votes
0answers
56 views

Find the Langitude and Longitude of the centre point of a circle given a point on the circumference.

I couldn't find a similar question! Given I have the latitude and longitude (x,y) of a point on the circumference of a circle, and I want the circumference to be 1000m. An example of a lat lang I ...
-1
votes
4answers
43 views

Prove the inequalities without calculating the integrals

$$ \int_{0}^{\frac{\pi}{2}} \sin^4x dx \le \int_{0}^{\frac{\pi}{2}} \sin^3xdx$$ I have tried to define 2 functions $ f, g:[0, \frac{\pi}{2}] \rightarrow \mathbb{R}$ and say that $ f(x) = \sin^4x$ ...
1
vote
4answers
190 views

What does Pi equal to [duplicate]

What is the approximation of pi in a fraction form. I am very curious to know what it is. I have been seeing pi almost everywhere.
4
votes
2answers
145 views

Value of $\pi$ by Aryabhata

Aryabhata gave accurate approximate value of $\pi$. He wrote in Aryabhatiya following: add 4 to 100, multiply by 8 and then add 62,000. The result is approximately the circumference of circle of ...
0
votes
1answer
98 views

How to prove that $\cos(\pi÷11)+\cos(3\pi÷11)+\cos(5\pi÷11)+\cos(7\pi÷11)+\cos(9\pi÷11)=0.5$? [duplicate]

I need to prove that $$\cos\dfrac{\pi}{11}+\cos\dfrac{3\pi}{11}+\cos\dfrac{5\pi}{11}+\cos\dfrac{7\pi}{11}+\cos\dfrac{9\pi}{11}=\dfrac{1}{2}$$ How to do it?
0
votes
3answers
305 views

Patterns in pi in “Contact”

In Carl Sagan's novel Contact, the main character (Ellie Arroway) is told by an alien that certain megastructures in the universe were created by an unknown advanced intelligence that left messages ...
2
votes
4answers
145 views

Proof of the rationality of $e + \pi$ via their series representation?

I found just one question similar to this, but it had been edited, so hopefully this isn't asked too often. Given the formulas via infinite sums for expressing $e$ and $\pi$... $$ e = ...
-1
votes
3answers
117 views

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

Recently I asked a question on Maths SE Proof that at most one of $e\pi$ and $e+\pi$ can be rational after solving this one one I was thinking whether $e^\pi$ is greater or $\pi^e$ ? On calculating ...
1
vote
2answers
173 views

How can I improve my explanation of why the ratio $\pi=\frac{C}{d}$ holds for all circles?

I'm trying to informally explain why $\pi$ holds for all circles. I would like to know if there is anything pertinent that I can add, or that is wrong with this explanation. It's an explanation, not ...
6
votes
1answer
80 views

Rational approximation of pi

I found this problem intriguing: $355 / 113 = 3.14159292035398\ldots$ gives the approximation of $\pi$ in $7$ correct numbers, say $C(355/113)=7$, but it number of digits in numerator + number of ...
1
vote
0answers
87 views

How to show$\frac{\pi}{4} = \frac{2\cdot4\cdot4\cdot6\cdot6\cdot8 \dotsm}{3\cdot3\cdot5\cdot5\cdot7\cdot7 \dotsm}$?

I am doing the exercises of Structure and Interpretation of Computer Programs. In exercise 1.31 the following equation is casually shown as an approximation of $\pi$: $$\frac{\pi}{4} = ...
10
votes
2answers
224 views

Proof that at most one of $e\pi$ and $e+\pi$ can be rational

$e$ and $\pi$ are rather peculiar numbers. It turns out that, in addition to being irrational numbers, they are also transcendental numbers. Basically, a number is transcendental if there are no ...
9
votes
2answers
307 views

$e^{\pi\sqrt N}$ is very close to an integer for some smallish $N$s. What about $\pi^{e\sqrt N}$?

Heegner numbers (1, 2, 3, 7, 11, 19, 43, 67, 163 - let's use symbol $H_n$) are know for peculiar property that $e^{\pi\sqrt{H_n}}$ are almost integers: $$e^{\pi \sqrt{19}} \approx 96^3+744-0.22$$ ...