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.

Filter by
Sorted by
Tagged with
-1
votes
1answer
42 views

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/...
5
votes
1answer
82 views

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?
-2
votes
0answers
40 views

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 ...
3
votes
1answer
97 views

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 ...
-2
votes
0answers
67 views

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 $N - M$, $N + M$ ...
-1
votes
2answers
31 views

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} $ , ...
0
votes
2answers
36 views

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 ...
-1
votes
1answer
35 views

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.
1
vote
1answer
36 views

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 ...
0
votes
1answer
43 views

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.
-5
votes
2answers
66 views

$\pi$ 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?
1
vote
1answer
108 views

$\sqrt{2} \ln \pi \approx 1.618033…$, 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?
0
votes
0answers
74 views

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 ...
0
votes
0answers
29 views

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 ...
0
votes
1answer
44 views

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 ...
0
votes
1answer
13 views

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 ...
-2
votes
1answer
91 views

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 ...
0
votes
0answers
38 views

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 ...
0
votes
0answers
67 views

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 ? ...
10
votes
0answers
178 views

Why Is $\ln 23+\cfrac{1}{\color{red}{163}+\cfrac{1}{1+\cfrac{1}{\color{red}{41}}}}\approx\pi$

I know from reading that the Heegner number 163 yields the prime generating or Euler Lucky Number 41. Now apparently $\ln23<\pi$ and this can be shown without calculators. I noticed that $$ \pi-\...
2
votes
1answer
51 views

Proof of the surface area of a cone doesn't make sense (to me at least)

Even though the surface area of a cone is $(\pi R G + \pi R^2)$, it makes sense to think that it actually is $\pi R\times \text{Height} + \pi R^2$ as you could think that the side area $=$ average ...
-2
votes
0answers
21 views

How I prove or disprove this $|\pi-\frac{\sigma(n)+n}{\phi(n)}|\leq \frac{e^{20}}{\phi(n)^{1+e^{-20}}}$ for positive integer $n$?

I have tried to measure irrationality of $\pi$ using the following inequality $$\left|\pi-\frac{\sigma(n)+n}{\phi(n)}\right|\leq \frac{e^{20}}{\phi(n)^{1+e^{-20}}}$$ such that $\phi$ is the Euler ...
15
votes
1answer
154 views

On the formula, $\pi = \frac 5\varphi\cdot\frac 2{\sqrt{2+\sqrt{2+\varphi}}}\cdot\frac 2{\sqrt{2+\sqrt{2+\sqrt{2+\varphi}}}}\cdots$

I found a formula on google images when I was looking at some formulas for $\pi$ just for the fun of it, and I came across one that really startled me, and was quite reminiscent of Viète's product. ...
2
votes
2answers
90 views

Why we shouldn't believe that $\pi(x)$ has connection to the constant $\pi$?

It is well known that $\pi(x)$ is the prime counting function , This notation was introduced by number theorist Edmund Landau in 1909 if we define this function as: $\pi:\mathbb{Z}\to \mathbb{Z} $ ...
2
votes
1answer
54 views

Prove that $(\frac{\pi}{3})^{\frac{3}{\pi}}+(\frac{3}{\pi})<2$

Prove that : $$\left(\frac{\pi}{3}\right)^{\frac{3}{\pi}}+\frac{3}{\pi}<2$$ Straightforward proof : Since the function $f(x)=(x)^{\frac{1}{x}}+\frac{1}{x}$ is decreasing on$\left[1,\frac{\pi}{3}\...
1
vote
0answers
37 views

Value of π*φ and π/φ

I found the expression of $\pi\varphi$ and $\pi/\varphi$ on Twitter via @AnecdotesMaths (https://twitter.com/AnecdotesMaths/status/1241032301783461890) in terms of an integral, and I wondered if ...
3
votes
1answer
57 views

Does this infinite nested radical show that $\pi$ is irrational?

A $\sqrt{2}$ is at the end (?!) of this infinite nested radical expression. Is that enough to show that $\pi$ is irrational? Also, I'm curious if there is a better way to write this? Thanks! $$\lim_{n\...
-1
votes
2answers
47 views

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 ...
2
votes
1answer
64 views

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 $\frac{22}7$, ...
0
votes
2answers
112 views

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 ...
-2
votes
1answer
42 views

Is it possible :$\pi \sim{\frac{-\gamma}{(\sqrt{2}e^2-\gamma)^4}}-e^{-13}+2(17^{\frac13}-1)-2\phi^{-16}-\sum_{n=1}^{\infty}(\frac{1}{17})^{2n+3}$?

I have done many attempts to give another approximation for $\pi$ I have got this $$\pi \sim{\frac{-\gamma}{(\sqrt{2}e^2-\gamma)^4}}-e^{-13}+2(17^{\frac13}-1)-2\phi^{-16}-\sum_{n=1}^{\infty}(\frac{...
5
votes
1answer
121 views

prove that $\int_{0}^{1}\Big(\frac{\operatorname{li}(x)}{x}\Big)^2dx= \frac{\pi^2}{6}$

prove that : $$\int_{0}^{1}\Big(\frac{\operatorname{li}(x)}{x}\Big)^2dx< \frac{\pi^2}{6}$$ Where $\operatorname{li}(x)$ denotes the logarithm integral . The inequality is very very sharp .I have ...
2
votes
2answers
40 views

Is it true that $[\pi^n] $ is a prime number for only finitely many integers $n$?

let $[\pi^n] $ be the integer part of $(\pi^n)$, I did mathematica code up to $10^4$ to test primality of $[\pi^n] $ , I have got it could be prime for $n=1,3,4,12$ , Now are there other ? and Is ...
-1
votes
2answers
33 views

Floor, ceil problem

Edit-For what values of $k$, -$2$+ $\left\lceil\frac{(6k+1)\pi }{6}\right\rceil=\left\lfloor\frac{(6k-1) \pi}{6}\right\rfloor$ where $k$ is positive integer greater than equal to one?
6
votes
1answer
81 views

Succinct proof that $\frac\pi4+\frac\pi6+\log2\gt2$

In answering Average angle between two randomly chosen vectors in a unit square, I noticed that the average angle formed by two vectors uniformly picked in the unit square, $\frac\pi4+\log2-1\approx0....
-2
votes
1answer
55 views

What is the geometric interpretation of ${\pi^{\pi^{\pi^{…\pi}}}}$ $n$ times? [closed]

$\pi$ is the area of a circle of diameter $1$ then ${\pi^{\pi^{\pi^{......\pi}}}}$ is the power of finit area $n$ times of circle of diameter $1$, The question I should ask What is this finit ...
1
vote
5answers
55 views

why is $\sin(x) = \cos(x-\frac{\pi}{2})$? [closed]

I was playing around in desoms and then found out that $\sin(x) = \cos(x-\frac{\pi}{2})$, why? What is the relation between sin, cos functions and $\pi$?
1
vote
1answer
30 views

A class of generalized Integrals involving polygamma functions

I recently came across some nice integrals and I have a few questions about them: You may have heard of the Euler-Mascheroni constant $\gamma$ and if you did so, you may know the following integral: ...
2
votes
2answers
60 views

Nice integral $\int_{0}^{\infty}\ln\Big(\frac{x^3-x^2-x+1}{x^3+x^2+x+1}\Big)\frac{1}{x}dx=-\frac{3\pi^2}{4}$

Last integral of the day : $$\int_{0}^{\infty}\ln\Big(\frac{x^3-x^2-x+1}{x^3+x^2+x+1}\Big)\frac{1}{x}dx=-\frac{3\pi^2}{4}$$ I have tried integration by parts and some obvious substitution but I ...
1
vote
1answer
86 views

Continued fraction of $π$ using sums of cubes

Recently I came across this identity: $$\pi=3+\cfrac1{6+\cfrac{1^3+2^3}{6+\cfrac{1^3+2^3+3^3+4^3}{6+\cfrac{1^3+2^3+3^3+4^3+5^3+6^3}{6+\ddots}}}},$$ thus $$\pi=3+\cfrac{1}{6+\cfrac{(1\cdot3)^2}{6+\...
3
votes
0answers
42 views

Conjecture about a famous trigonometric integral

I don't know if this conjecture is well know but let me try it : Let $f(x)$ be a continuous , differentiable function on $[0;+\infty[ $ with $f(x)\geq 0 \quad\forall x\geq 0$ and such that :$...
-2
votes
1answer
56 views

Can hypotenuse = PI() be rewritten as 2 legs in a right triangle? [closed]

In a right triangle, where the larger acute angle $\measuredangle\alpha$ is calculated in radians as: $\alpha = \frac{x\pi()}{2(x+1)}$ where $x$ is the ratio $\alpha$ and the smaller acute angle, $x\...
7
votes
3answers
143 views

Proof that $\pi =\lim_{n\to\infty}\frac{2^{4n}n!^4}{n(2n)!^2}$

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!...
1
vote
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
1
vote
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 ...
1
vote
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, ...
2
votes
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....
6
votes
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\...
0
votes
1answer
77 views

What's the significance of the Σ(1/pi^n)?

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....
0
votes
2answers
61 views

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, ...

1
2 3 4 5
26