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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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 ...
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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 ...
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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?
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2answers
104 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 ...
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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 ...
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1answer
90 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 ...
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1answer
71 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 ...
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5answers
1k 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 ...
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1answer
97 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 ...
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2answers
46 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 ...
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1answer
105 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 ...
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1answer
63 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 ...
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2answers
121 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 ...
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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}$$\ ...
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1answer
36 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 ...
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0answers
39 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?
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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 + ...
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2answers
184 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 ...
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2answers
139 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 $\pi$. ...
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1answer
51 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 ...
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1answer
48 views

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

Wikipedia: Borwein's algorithm claims Start out by setting $$\begin{align} a_0 & = \frac{1}{3} \\ s_0 & = \frac{\sqrt{3} - 1}{2} \end{align} $$ ...
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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 ...
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1answer
69 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 $x'(t)=f(t,x)$. ...
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1answer
78 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 ...
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2answers
97 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 ...
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1answer
204 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} - ...
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0answers
50 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 ...
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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$ ...
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4answers
188 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.
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2answers
138 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 ...
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1answer
97 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?
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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 ...
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3answers
115 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 ...
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2answers
171 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
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1answer
78 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 ...
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0answers
85 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} = ...
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2answers
185 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 ...
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2answers
304 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: $$e^{\pi \sqrt{19}} \approx 96^3+744-0.22$$ ...
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1answer
262 views

How was this approximation of $\pi$ involving $\sqrt{5}$ arrived at?

The Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. ...
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2answers
232 views

“Bizarre” continued fraction of Ramanujan! But where's the proof?

$$\frac{e^\pi-1}{e^\pi+1}=\cfrac\pi{2+\cfrac{\pi^2}{6+\cfrac{\pi^2}{10+\cfrac{\pi^2}{14+...}}}}$$ "Bizarre" continued fraction of Ramanujan! But where's the proof? i have no training in continued ...
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2answers
138 views

Geometrical interpretation of $\pi=\int_0^1\frac{4}{1+x^2}dx$.

How to show that $$\pi=\int_0^1\frac{4}{1+x^2}dx?$$ I know how to do it symbolically by using that $\frac{d}{dx}\arctan x=\frac{1}{1+x^2}$. But is there a geometrical interpretation of this result?
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4answers
296 views

A calculation that goes awfully wrong if we let $\pi=22/7$

Me and one of my friends had an argument and he said that using $22/7$ as value of $\pi$ is sufficient for any calculation. Can we always take it $22/7$, or is there some example of some calculation ...
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2answers
349 views

Simple proof that $\pi$ is irrational - using prime factors of denominator

Simple proof that $\pi$ is irrational Consider the Gregory - Leibniz series for $\pi/4$: $$\frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 + \cdots $$ Let $A_n/B_n$ be the irreducible fraction given by ...
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1answer
79 views

How do I correctly measure the circumference of a circle

I found How exactly do you measure circumference or diameter? but it was more related to how people measured circumference and diameter in old days. BUT now we have a formula, but the value of PI ...
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2answers
485 views

$\pi$ in terms of $4$?

I'm trying to define $\pi$ in terms of $4$ by placing a unit circle inside a square, and subtracting the corners of the square. I'm attempting to use summation to define the area of a corner, then ...
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2answers
69 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
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1answer
29 views

Proof for $-4\pi^2+48\ne A+B\pi+C\pi^2$ when $(A,B,C)\ne (48,0,-4)$

I have to prove $-4\pi^2+48\ne A+B\pi+C\pi^2$ when $(A,B,C)\ne (48,0,-4)$ $A,B,C \in \mathbb{Q}$ As a part of question I try to solve.
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593 views

Finding Reference Angles in Precalculus?

I'm reviewing for an exam, and having some trouble with reference angles depending on the quadrant they lie in. For example, my book shows the following: I get the part about subtracting 12pi/6, ...
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193 views

$\pi + e$ is rational or $\pi-e$ is rational

I was asked to find the truth value of the statement: $$ \pi + e \; \text{ is rational or } \pi - e\; \text{ is rational } $$ I am only allowed to use the fact that $\pi, e $ are irrational ...