# Questions tagged [pi]

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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### What is the fastest way to calculate $\pi$ without use of square root? [closed]

What is the fastest way to calculate $\pi$ without use of square root? From all the research I did, the literatore point to Ramanujam/Chudnovsky's formula and others have suggested Arithmetic/...
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### Proving $\pi=(27S-36)/(8\sqrt{3})$, where $S=\sum_{n=0}^\infty\frac{\left(\left\lfloor\frac{n}{2}\right\rfloor!\right)^2}{n!}$ [closed]

I have to prove that: $$\pi=\frac{27S-36}{8\sqrt{3}}$$ where I know that $$S=\sum_{n=0}^\infty\frac{\left(\left\lfloor\frac{n}{2}\right\rfloor!\right)^2}{n!}$$ Where do I get started?
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### Prove that if $m= \sin x+ \sin\frac{x}{2}$ has $2$ real roots $x, y$ on $(0, 2\pi]$ then $\frac{1}{x}+ \frac{1}{y}> \frac{3}{2m}$ .

Prove that if $m= \sin x+ \sin\dfrac{x}{2}$ has $2$ real roots $x, y$ on $(0, 2\pi]$ then $$\frac{1}{x}+ \frac{1}{y}> \frac{3}{2m}$$ I want to see other solution(s) that is not as same as one ...
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### Elementary proof that $\pi$ is transcendental

A popular (and maybe the only) approach to showing that $\pi$ is transcendental is to first prove that for every non-zero algebraic number $a$, the number $e^a$ is transcendental. That requires ...
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### Help me with this interesting problem please. You have some A, B. And you can get (A - B) / B. [closed]

The problem is following you have some apparatus that read some $A$ and $B$, then for the output, it will give $(A - B) \over B$. So you have numbers N, M and $\pi$. You should get $N - M$, $N + M$ ...
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### Rational or Irrational?

Here is my question: can an Irrational number; like $e$, be equal to a second irrational number, $\pi$, times an integer, then divided by a second rational number? Such as: $e = \frac{\pi a }{b}$ , ...
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### MCQ: comparing two quantities; $\tan(1)$ and $\frac{\pi}{2}$ [duplicate]

Without using a calculator, given that $\pi \approx 3.14159$, compare: FIRST VALUE: $\tan(1)$ [the angle is in radians]. SECOND VALUE: $\frac{\pi}{2}$ Options: (A) FIRST VALUE is greater than the ...
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### Why is 4PI = 0 mod 2PI? [closed]

question is basically in the title. I just do not know what is meant by mod2 in this case.
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### Should pi, the constant, be italicized or not?

I know of a certian ISO standard that says "don't italicize constants" but it isn't in widespread use. So should $\pi$ be in italics or not? What's more common in actual usage? Feel free to tell me ...
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### what is the longest number that pi can reach

Where does π (pi) end? this is what I know: 3.14159265359. This is from the Google search calculator. I know it never ends but I want to know how far we got π to.
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### $\pi$ is defined as the ratio of circumference to diameter, so why is it called “irrational”? [closed]

By definition, $\pi$ is the ratio of circumference and diameter of a circle. So, by definition itself it is a rational number. Why it is called irrational?
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### $\sqrt{2} \ln \pi \approx 1.618033…$, the golden ratio. Why?

$\sqrt{2} \ln \pi = 1.618892$… is approximately equal to the golden ratio $\phi = 1.618033$… . Is this just a coincidence? Could it be some kind of first-order approximation?
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### Proving $\pi$ is irrational without arguing by contradiction

I know few proofs to prove that $\pi$ is irrational, but in all proofs one thing is common: it is done by the method of contradiction. So, I am looking for a proof without using contradiction. Is ...
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### Solving trigonometric equations without using calculator

How to solve the following trigonmetric equations without using calculator? 1)$\pi +\arccos{(x)}=\cot{(\arcsin{(x)})}=\frac{\sqrt{1-x^2}}{x}$ 2)$\theta-\tan{(\theta)}=2\pi$ Answers: Using hp 50g ...
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### What is the simple continued fraction of $τ$ ($2π$)?

I cannot find any information on Google or Wolfram Mathworld to answer this question. I also don't have the skills to calculate it myself so I thought it would be good if someone with this knowledge ...
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### Can the radius of convergence be equal to 1 (arctangent function)?

I wanted to find a proof of the Leibniz $\pi$ formula, knowing that $\arctan(1) = \pi / 4.$ As such, I simply needed to find the Maclaurin Series of $\arctan$ and its rate of convergence. I want to ...
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### Would this be a proof of the Pythagorean Theorem. [closed]

Does the expression (1+1) contain within it the Pythagorean Theorem? When applying the operation of addition does it prove the Pythagorean Theorem? Here are some needed questions with answers that ...
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### Is it possible to put pi into terms of rationals [duplicate]

I don't know of that is the correct terminology but by this I mean in the way that $\phi$ can be put into the equation $(1+\sqrt 5)/2$. I have heard that it has never been achieved but I was wondering ...
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### What is the mathematical technique used to discover new decimal expansion of $\pi$? [closed]

I asked my self many times how mathematicians discovered new decimal expansion of $\pi$ each year or each 4 years... by means What is the technical basic used to find such new decimal expansion ? ...
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### Identify this derivation of PI

The book "Build supercomputers with Raspberry PI" (C. Morrison) calculates a numerical approximation of PI using the following integral: $${\pi} = \int_{0}^1 \frac{4}{1+x^2}\ dx$$ Who is credited ...
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### Why this rational approximation $\pi\sim\frac{80249}{25544}$ is not mentioned in OEIS?

I have checked sequence of Denominator of best approximation to $\pi$ with denominator $\le10^n$ in OEIS but I didn't find this rational $\frac{80249}{25544}$ however it is better than $\frac{22}7$, ...
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### Why are mathematicians still trying to calculate digits of pi? [closed]

Why do mathematicians still try to calculate digits $\pi$? In 2019 Haruka Iwao calculated the world's most accurate value of $\pi$; which included $31.4$ trillion digits, far past the previous ...
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In this post, the symbol $\sim$ means asymptotically equivalent. The relationship between $\pi$ and factorials hinges on Stirling's formula: $$n!\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n\implies n!... 2answers 44 views ### A trigonometric problem$$2 (\cos{⁡2x} )(\sin ⁡2x-1)=\sqrt{3} \cos{⁡4}x$$I have tried to reduce this equation and I got this: \cos(4x+\pi/6)=-\cos2x but I don't know how to do next 1answer 54 views ### Very interesting problem with integral,number theory and irrationality It's a problem that I found interesting because it gives an approximation of \frac{\pi}{2} and a sequence of term .I can add to my problem a bit of elementary number theory furthermore.Finally and ... 1answer 65 views ### Is there p,q\in \mathbb{Q} such that p = \pi /\sin{q} holds? I feel like such kind of things does not exist, but the only thing I could find is this question: Is sin(x) necessarily irrational where x is rational? According to this if x is nonzero rational, ... 1answer 35 views ### Did Archimedes squared the circle? What i can't understand is that I'm reading book " a History of Mathematics by boyer" and it says Archimedes made possible to construct a triangle equal in area to that of a circle by help of spirals.... 2answers 118 views ### Integral involving elliptic integral functiions Recently I came across this identity:$$\int _0^1\:\frac{K\left(x\right)}{1+x}dx$$Where:$$K\left(x\right)=\int _0^{\frac{\pi }{2}}\:\frac{1}{\sqrt{1-\left(x\sin \left(\theta \right)\right)^2}}d\...
I was recently interested in the formula equ 1: $$\sum_{n=0}^∞ 1/2^n$$ This made me curious about: equ 2: $$\sum_{n=0}^∞ \frac{1}{\pi^n}$$ I found that the second formula converges to 0....
### As $n$ increases, the value of $\sqrt[ne^{\pi}]{\pi}$ approaches $1$. Is there a name for this result?
I was messing around with $\pi$ and $e$ on the Desmos calculator, and came up with the observation that this value: $$\sqrt[ne^{\pi}]{\pi}$$ approaches $1$ as $n$ increases. (To be clear, ...