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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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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0answers
49 views

Most Famous Formula in Mathematics? [closed]

$1 + e^{i \pi} = 0$ . This is the most famous formula in mathematics. Why is it called so???
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2answers
287 views
+50

$\pi$ 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
20 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
26 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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27 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
168 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
84 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
57 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
103 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
40 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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Why is it a good idea to approximate $\pi$ to 3.14 or 22/7 rather than 3? [closed]

I think 3.14 or 22/7 are better approximations for $\pi$ than 3 because they're closer to $\pi$ than 3. See that? More accuracy! What do you think?
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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
63 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
23 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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2answers
63 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
61 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
64 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 ?
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Evaluate the integrals in $\int_{0}^{1} \frac {x^4(1-x)^4}2\, dx \le \int_{0}^{1} \frac {x^4(1-x)^4}{1+x^2}\, dx \leq \int_{0}^{1} {x^4(1-x)^4}\, dx$

Note that when $0\le x \le 1$ we have $$\frac 12 \le \frac 1 {1+x^2} \le 1.$$ Hence, $$\int_{0}^{1} \frac {x^4(1-x)^4}2\, dx \le \int_{0}^{1} \frac {x^4(1-x)^4}{1+x^2}\, dx \leq \int_{0}^{1} ...
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1answer
32 views

Could we calculate pi using an iterative series

I know that, as a hobbyist mathematician, this is generally a term we can use to express pi \begin{equation*} \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} ...
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2answers
163 views

Is it known if $\pi + e$ is transcendental over the rational numbers?

I recall reading a comment on reddit that had stated that it is not known if $\pi + e$, (nor $\pi e$) is transcendental over $\mathbb{Q}$, nor even if it is irrational. Is this true? It strikes me as ...
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3answers
107 views

What is the probability of getting the exact number of expected digits ($0-9$) in $10^6$ digits of $\pi$?

I noticed that at $1$ million digits of $\pi$, none of the digits has the "perfect" expected $100{,}000$ occurrences. My question is what is the probability (if the digits are truly random) of at ...
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1answer
185 views

Random digits of $\pi$ seem somewhat predictable! Why is this not generally believed? [closed]

I decided to test my theory that I can guess the next digit of $\pi$ with greater than $10$% ($1$ out of $10$) confidence based alone on knowledge of previous digits. I just used a simple spreadsheet ...
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1answer
93 views

Relationship between Pi and Phi using the Great Pyramid of Giza?

In a documentation about the Great Pyramid of Giza, I heared following three theses about its measurements and the numbers $\pi$ and $\phi$ (the golden ratio). Measurement The Great Pyramid of ...
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2answers
71 views

How to make π degree angle?

Can we make π degree angle? π is a decimal and angles are divided into minutes and seconds, but, I think (I'm not sure), we can still divide 1 degree into decimal parts (we can divide 1 degree into ...
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2answers
409 views

(UPDATED) Why didn't Archimedes further approximate $\pi$ this way (or did he)?

Update is at bottom of my post. I saw on YouTube (https://www.youtube.com/watch?v=_rJdkhlWZVQ) a way to approximate $\pi$ starting with a hexagon inscribed inside a circle of unit radius. It uses ...
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1answer
85 views

$\pi$ normal to the base $10$ [closed]

If $\pi$ is normal to base $10$, why would we expect to find a string of ten $0$'s in its decimal expansion?
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1answer
70 views

Show that $\int_0^r \frac{\mathrm{d}t}{\sqrt{r^2 - t^2}} $ is independent of $r$

I'm trying to show that the integral $$\int_0^r \frac{1}{\sqrt{r^2 - t^2}}\mathrm{d}t$$ is independent of $r$, without using trigonometric functions (namely, $t=\cos s$ and such). Can it be done? ...
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1answer
74 views

Is this a true statement? [duplicate]

This is a 9GAG picture I saw tonight. The way it's put, it is evidently false, since 0.10100100010000… (the powers of 10 all in a row) is definitely decimal, infinite and nonrepeating (or in one ...
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1answer
41 views

Do the $2^n$ hyper-octants of a $n$-sphere always have a $n$-dimensional right angle? Is $\pi/2$ only fundamental in $2$ and $3$ dimensions?

In $2$ dimensions, a $2$-sphere can be divided into $2^2 = 4$ congruent pieces, the $4$ quadrants, each of angle $\pi/2$ radians. In $3$ dimensions, a $3$-sphere can be divided into $2^3 = 8$ ...
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2answers
156 views

A Mathematical Coincidence, or more?

According to the paper "Ten Problems in Experimental Mathematics", $$\int_0^\infty \cos(2x)\prod_{n=1}^\infty \cos\left(\frac{x}{n}\right)dx \quad = \quad \frac{\pi}{8}\color{blue}{-7.407 \times ...
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1answer
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Proving that $\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = {1\over\pi}$

While trying to prove that $$(1)\qquad x\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = 1 \implies x=\pi$$ I got to a point, using W|A, where I have to prove that $$\color{red}{(2)\qquad ...
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2answers
71 views

Methods for calculating $\pi$ that use the sphere?

The area of the unit circle is $\pi$ and its circumference is $2\pi$. Consequently, many elementary methods for calculating and approximating $\pi$ use a geometric approach on the circle, such as ...
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1answer
96 views

Prove this formula for $\pi$

I have to use a certain approximation for $\pi$ for my computer science class, but I don't really understand what's going on, other than that this is related to the Taylor polynomial for arctangent. ...
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167 views

How prove $\pi^2>2^\pi$

show that $$\pi^2>2^\pi$$ I use computer found $$\pi^2-2^\pi\approx 1.044\cdots,$$ can see this I know $$\Longleftrightarrow \dfrac{\ln{\pi}}{\pi}>\dfrac{\ln{2}}{2}$$ so let ...
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2answers
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Looking for a closed form for $\sum_{k=1}^{\infty}\left( \zeta(2k)-\beta(2k)\right)$

For some time I've been playing with this kind of sums, for example I was able to find that $$ \frac{\pi}{2}=1+2\sum_{k=1}^{\infty}\left( \zeta(2k+1)-\beta(2k+1)\right) $$ where $$ ...
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1answer
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Approximating Pirrational Numbers

A while back I wrote this question on PPCG.SE about the numbers I termed Pirrational numbers. They are defined as follows: Let $P_i$ be the $i$th Pirrational number for some $i \in \mathbb{N}_0$ ...
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1answer
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A question about infinitie series and pi

This is the sequence that can be used to find an exact value of pi 4/1−4/3+4/5−4/7+4/9−4/11…..(to infinity) = 𝜋 Or (1/1−1/3+1/5−1/7+1/9−1/11….. (to infinity) )= 𝜋/4 Given that we have this ...
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6answers
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$\lim_{n\to\infty}\sqrt{6}^{\ n}\underbrace{\sqrt{3-\sqrt{6+\sqrt{6+\dotsb+\sqrt{6}}}}}_{n\text{ square root signs}}$

We have the following representation of pi: $$\pi=\lim_{n\to\infty}2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\dotsb+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}}}}}_{n\text{ square root ...
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How do we know that the first few digits of an approximation for $\pi$ are correct?

For Gregory–Leibniz series, wikipedia has - "after 500,000 terms, it produces only five correct decimal digits of π.". But how do you know that those five decimal values are correct when you reach ...
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Different ways of approximation of $\pi$

i am studying trig n knows that pi can be approxated using gregory series ruthrford sries etc . Aso its strange n mysterious that pi is just ratio of cirumference n diameter . this profoundly shows ...
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Correcting Error in the Leibniz $\pi$ formula… why does it work?

You are probably familiar with the Leibniz $\pi$ formula: $$ 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \frac{\pi}{4} $$ For a CS homework assignment I had to write a ...
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1answer
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evaluating Pi with imaginairy unit i leads to contradiction!

I was reading about evaluating $i^i$ so I tried that with Mathematica and got a real (R) result and Mathematica suggested an alternative form being $e^{-pi/2}$ so I solved for $\pi$: $i^i = ...
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2answers
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Prove that $\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$

How to prove the following identity $$\sum_{k=0}^\infty \frac{1}{16^k} \left(\frac{120k^2 + 151k + 47}{512k^4 + 1024k^3 + 712k^2 + 194k + 15}\right) = \pi$$ I am totally clueless in this one. Would ...
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238 views

Which number its greater $\pi^3$ or $3^\pi$?

$ \pi^3$ or $3^\pi$ using algebra please, I arrive the solution whith $a^x > 1 + x$ but I am interested in more solutions.
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2answers
25 views

What is the meaning of the PI function.

I am solving for a configuration problem and i have seen a function π This is a function not 3.14 which is the value of pi. While accessing some lectures i found out that they also call this symbol ...
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1answer
150 views

Why does summation of infinite series end up in powers of pi?

For instance, we have $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+...=\frac{\pi}{4}$$ $$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...=\frac{\pi^2}{6}$$ ...
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3answers
63 views

Why does this loop yield $\pi/8$?

Was doing some benchmarks for the mpmath python library on my pc with randomly generated tests. I found that one of those tests was returning a multiple of $\pi$ consistently. I report it here: ...
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1answer
54 views

Bailey–Borwein–Plouffe formula

$\displaystyle \pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right]$ How does the BBP formula for ...
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1answer
77 views

What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? [duplicate]

"homework" What can be said about $\pi+e$ and $\pi e$? Are these numbers rational or irrational? I know that both $\pi$ and $e$ are irrational. What can be said about $\pi+e$, and $\pi e$?