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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44 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
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1answer
97 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
61 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
115 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
35 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
54 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
134 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
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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
50 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
47 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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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 ...
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1answer
64 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
75 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
89 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
191 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
47 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
187 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
131 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
94 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
110 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
168 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
74 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
84 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
174 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
302 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
228 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?
6
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4answers
268 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
329 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
70 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
484 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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1answer
50 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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2answers
505 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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2answers
189 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 ...
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1answer
127 views

What's the value of tau?

I've seen $\tau$ on a title of a YouTube video and I need help knowing what the value is. I'm serious. I've never heard of the value. So, what is it? Also, is it rational or irrational (this part ...
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1answer
134 views

Value of Pi derivation

Derive the value of Pi. I want with explanation. Is there any possible way? How do scientists calculate it?
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1answer
151 views

Does $\pi$ contain any zeroes?

Let's say we have two functions, $f$ and $g$. $f:\mathbb{R}\mapsto [0,1]$ where $0,1$ denote true, false respectively. $f(x)=1$ when $x$ contains any zeroes as a digit; $f(x)=0$ otherwise. Now let's ...
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1answer
177 views

How did Pi originate?

What methods/calculations were used to calculate the value of pi (3.14....). Was it simply determined by calculating the circumference of a circle then dividing by the diameter, or some other method?
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236 views

How come $\pi$ is usually approximated as 3.14 or 22/7?

I've heard that $\pi$ is usually approximated as 3.14, but it can also be approximated as 22/7, which is equal to 3.142857142857142857.... Guess what? $\pi$ can also be approximated as 355/113, ...
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1answer
79 views

Prove this inequality $ \sqrt{5} > \frac {13 + 4\pi}{24 - 4\pi} $ [closed]

$$ \sqrt{5} > \frac {13 + 4\pi}{24 - 4\pi} $$
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1answer
27 views

Does the perimeter of a 2-D object “count” toward its area?

I'm writing a quick Monte-Carlo simulation for a class in Matlab in order to estimate the value of pi as demonstrated in this gif: http://en.wikipedia.org/wiki/File:Pi_30K.gif However, I'm not sure ...
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3answers
150 views

Can $\pi$ be a ratio of angles?

I know that $\pi$ is the ratio between various measurements in 2 and 3 dimensional shapes. (For example, $V=\pi r^2h$ for a right circular cylinder can be written as $\pi=\frac{V}{r^2h}$) The golden ...
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1answer
102 views

Will the Declaration of Independence ever show up in pi? [duplicate]

If pi goes on forever and is completely random, if ascii would be mapped onto pi would you eventually find the Declaration of Independence in it? If so, by what digit of pi can we reasonably expect ...
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2answers
83 views

An Inequality involving integration

Show that $$\int_{0}^{\pi} \left|\frac{\sin nx}{x}\right| dx \ge \frac{2}{\pi}\left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right)$$ How do I go about proving this inequality ?