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.

learn more… | top users | synonyms

0
votes
0answers
70 views

Chudnovsky binary splitting and factoring

In this article, a fast recursive formulation of the Chudnovsky pi formula using binary splitting is given. For $S(a,b)$: $$ m = (a + b) / 2 $$ $$P(a,b) = P(a,m) P(m,b)$$ $$Q(a,b) = Q(a,m) Q(m,b)$$ ...
0
votes
1answer
26 views

How to Find the Differential of $y=2\sin^2(x)$ when $x = \pi/4$ and $dx = 0.49$

I am wondering how to find the differential of $y=2\sin^2(x)$ when $x = \pi/4$ and $dx = 0.49$. I realize that I should be finding the derivative of $y=2\sin^2(x)$, which is $4\sin(x)\cos(x)$. And, I ...
1
vote
1answer
47 views

Is this a valid proof of the Area of a Circle, assuming we do not already know $\pi$?

In my AP Calculus class my friend and I decided we wanted to prove the area of a circle, without knowing $\pi$ to begin with. My friend and I each did it a different way. My method was to split the ...
1
vote
0answers
49 views

A set of integers related to $\pi$

Let $\Psi$ be the maps \begin{array}{lrcl} \Psi : & {\mathbb N}^{(\mathbb N)} & \longrightarrow & \mathbb R^+\cup\{+\infty\} \\ & A=(a_i)_{i\in \{0,\ldots,n-1\}} & \longmapsto ...
6
votes
0answers
59 views

Divisibility sequence resulting in limit with pi

Consider the following sequence of operations : Start with a natural number $n$ and then round it up to the closest multiple of $n-1$ .Then round up this new number to the closest multiple of ...
0
votes
1answer
40 views

“Error, (in Student:-calculus1:-roots) Cannot determine if this expression is true or false” [closed]

So I am trying to do the calculation which states: intervalsolve(sin(t) = .7, t = 0 .. 4*PI) but whenever I do it, I get the error which says: ...
5
votes
4answers
141 views

What is the relation between $\Gamma(a)$ and circles?

Looking through my calc textbook, it states that $$\int_0^\infty x^{a-1} e^{-x} \text{d}x =\Gamma(a)$$ As I have read ahead, I can understand most of the fairly basic concepts behind this function, ...
6
votes
2answers
125 views

Where, if ever, does the decimal representation of $\pi$ repeat its initial segment?

I was wondering at which decimal place $\pi$ first repeats itself exactly once. So if $\pi$ went $3.143141592...$, it would be the thousandth place, where the second $3$ is. To clarify, this ...
13
votes
1answer
242 views

Prove without using a calculator $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$

Prove without using a calculator $$(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$$ I want to know if there is an easy way to prove this inequality without using a calculator.
2
votes
0answers
35 views

Using Euler for estimating $\pi$

From Euler we know that $\pi ^{2} / 6 = \sum 1/ n^{2}$. Is this a good approximation for estimating $\pi$ ?
0
votes
4answers
49 views

How do you determine how many digits of pi are necessary?

It is said that you only need to calculate pi to 62 decimal places, in order to calculate the circumference of the observable universe, from its diameter, to within one Planck length. Most of us are ...
0
votes
2answers
152 views

Is there a function whose limit approaches Pi?

I don't think my knowledge of Pi, irrationality, and transcendental numbers in general is complete. I've Googled for a day before posting this question. Intuitively, I understand why the ratio of ...
4
votes
3answers
93 views

Analyzing Euler's Identity

From Wikipedia: "Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x, $$ e^{ix} = cos(x) + isin(x) $$ where the inputs ...
0
votes
1answer
65 views

What is the closest fraction (that isn't something like 31415…/1000…) that gets you pretty close to pi?

I'm just wondering what is the closest fraction (question for math nerds and geniuses) that isn't like pi/length-of-pi that gets you relatively close (like accurate to the 20th place) to pi? For ...
0
votes
1answer
26 views

How to Find the Inverse of the Function 13cos(12x)+1

If the domain is between [0,pi/12], how would I get this answer? So far, I have tried the following: 1) Switch 'x' with 'y', so, x=13cos(12y)+1. 2) Try to get 'y' by itself. Therefore, ...
2
votes
0answers
35 views

How come pi is in this question? [duplicate]

I was doing my homework assignment and I did this question correctly. However, I'm interested in knowing the reason behind the logic that how $1/1^4 + 1/2^4 + 1/3^4$ .... to infinity evaluates to ...
2
votes
4answers
108 views

Find an increasing sequence of rationals that converges to $\pi$

I am not sure how to construct a sequence that would convey convergence to $\pi$. Except maybe $a_n=\{\pi + 1/n\}$ but the terms would not be rational. Looking for an adequate way to show to satisfy ...
1
vote
1answer
142 views

Area under quarter circle by integration

How would one go about finding out the area under a quarter circle by integrating. The quarter circle's radius is r and the whole circle's center is positioned at the origin of the coordinates. (The ...
5
votes
0answers
200 views

How can we prove $e^{\pi}-\pi\simeq 20$ geometrically? [closed]

Using a calculator we can easily check that $$\color{Green}{e^{\pi}-\pi}=19.999\cdots\color{Green}{\approx 20}$$ This article and this one provides some details about this almost near identity, but no ...
2
votes
1answer
96 views

Can $\pi$ be rational in some base radix

I am from a physics background and my mathematics is not very good, so pardon my insolence with the question. Editing based on the comments : We know that $\pi$ in decimal (i.e. base 10) is ...
0
votes
0answers
18 views

Does a listing for A002485 beyond n=26 exist?

I have been able to find a listing, A002486, for the denominator of the convergents of PI that runs to n=201. However, for the numerator, A002485, I can only find a listing that runs to n=26. I ...
13
votes
4answers
259 views

Showing $\pi/(2\sqrt3)=1-1/5+1/7-1/11+1/13-1/17+1/19-\cdots$

I am struggling to show that $$\dfrac \pi{2\sqrt3}=1-\dfrac 15+\dfrac 17-\dfrac 1{11}+\dfrac 1{13}-\dfrac 1{17}+\dfrac 1{19}-\cdots$$ by using the Fourier series $$\frac \pi2-\frac x2=\sum_1^\infty ...
1
vote
2answers
43 views

$\pi$ as $180^{\circ}$ or $3.14$ in formula of areas.

I know that may be it is a very simple question. I came across through a question where they canceled $2π$ by $360^{\circ}$. Case was of the area of sector of a circle. So I am not completely ...
1
vote
2answers
68 views

Is there a formula to approximate $\pi$ in the form of $\dfrac{p}{q}$?

Is there a formula which helps in approximation of $\pi$ as $\dfrac{p}{q}$ where $p,q \in \mathbb{Z}$? I got this site though : [http://qin.laya.com/tech_projects_approxpi.html ] which shows the ...
1
vote
1answer
72 views

How can we say that the area of any circle or circular based shape is finite?

I am saying this because the area of the circle is pi * radius * radius. We know that pi is a never ending value. So, if some one says I need a circle of 3 metre square area to make a rim of the ...
1
vote
4answers
93 views

How can pi have infinite number of digits and never repeat them?

I am very confused about this matter, even if I searched google about this already. Please show me how this is determined and/or at least explain to me. First, I saw this "Infinite Monkey Theorem" ...
4
votes
4answers
159 views

PI as an infinite set of integers

I just had an interesting conversation with my kid who asked an innocent question about the $\pi$: If $\pi$ is infinite - does that mean that somewhere in it there's another $\pi$? I looked ...
0
votes
0answers
66 views

How do we know $\pi$ cannot be expressed a root [duplicate]

In other words, is there a proof that $\pi^a\neq b$ where $a,b\in \mathbb{Z}$?
0
votes
0answers
30 views

How do I make sense of this PI calculation

I was reading some code when I stumbled upon the following PI = (3373259426.0 + 273688.0 / (1 << 21)) / (1 << 30); Ignoring the fact that all ...
4
votes
3answers
127 views

Why do we need an integral to prove that $\frac{22}{7} > \pi$?

We know this famous (and beautiful) integral which shows that $\dfrac{22}{7} > \pi$ as : $$0 < \int_0^1 \frac{x^4(1-x)^4}{1+x^2} \, dx = \frac{22}{7} - \pi$$ Now since the integrand is ...
2
votes
4answers
184 views

how can we show $\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + …$?

Let $f(x) = \frac4\pi \cdot (\sin x + \frac13 \sin (3x) + \frac15 \sin (5x) + \dots)$. If for $x=\frac\pi2$, we have $$f(x) = \frac{4}{\pi} ( 1 - \frac13 +\frac15 - \frac17 + \dots) = 1$$ then ...
1
vote
1answer
60 views

Is the Champernowne constant actually useful

Has the aforementioned constant ever been used in any major proofs? Can it be expressed in terms of $e$, $\pi$, or both? Does it appear in any sort of geometric sense like $\pi$ does? Or is it just a ...
-5
votes
1answer
82 views

Is π really infinite? [closed]

I have had trouble wrapping my head around this for a long time. I don't get how π can be infinite because that would make the diameter/radius/circumference of a circle infinite. So, the question I ...
6
votes
1answer
115 views

What are some of the implications of $\pi + e$ being rational?

Whether or not $\pi + e$ is rational is an open question. If it were rational, what would some of the implications be?
4
votes
1answer
70 views

What is the math behind this art project?

This is a fascinating piece of art that makes me wonder how the cut out was created. Can anyone explain to me, in layman mathematical terms, how the position and angle of the black stick, when ...
10
votes
8answers
554 views

What are better approximations to $\pi$ as algebraic though irrational number?

I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to ...
1
vote
1answer
123 views

Why is $\pi$ irrational?

I've a few questions in mind: Why is $\pi$ irrational? If it is, then how can 2 rational quantities (circumference, diameter) can produce irrational number? How are we able to determine digits of ...
5
votes
2answers
149 views

PI aproximation with the $ {x\over y^2 } \approx \pi $ pattern

there are commands in mathematica and maple for finding Rational integer forms that approximate a floating point or a decimal number ... Mathematica function Rationalize Maple function ...
1
vote
0answers
30 views

Determine which of the two numbers $e^{\pi} $ and $\pi ^e$ is greater. [duplicate]

Prove the inequality $e^x > (1 + x)$ using LMVT for all $ x \in R$ and use it to determine which of the two numbers $e^{\pi} $ and $\pi ^e$ is greater. I proved the inequality. Now how do I ...
3
votes
1answer
87 views

Algorithm for computing an approximate value of $\sqrt{\pi}$

I seek a simple algorithm or function that will produce an approximate value for $\sqrt{\pi}$, but none that begin with the actual value of $\pi$, then derive the root via approximation. This is very ...
2
votes
1answer
108 views

Geometrical representation of square of pi?

a) Let's say I have a circle with a diameter $1$. Then a perimeter is $π$ (pi) and ratio between a perimeter and a diameter is $π$. b) I want to use π as a diameter of another circle, which gives a ...
34
votes
3answers
2k views

Fibonacci infinite sum resulting in $\pi$

I found the following identity. While trying to prove it, I found some things that I don’t quite understand: $$\frac{\pi}{4}=\sqrt{5} \sum_{n=0}^{\infty} \frac{(-1)^n F_{2n+1}}{(2n+1) ...
1
vote
0answers
98 views

three fractions between π and 22/7

three fractions between π and 22/7 π=355/113 =3.14159 22/7=3.1428 using a/b 355/113<22/7 then a+b/c+d 1) 355+22/113+7 =377/120~ 3.14166 2) 377+22/12+7 =399/127~3.14173 3) 399+22/127+7 ...
-5
votes
1answer
58 views

In what base would $\pi$ have the fewest decimals? [closed]

What is the numeral system in which $\pi$ would have the lowest number of decimals possible?
9
votes
1answer
111 views

What is $\lim_{n\to\infty}2^n\sqrt{2-\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}$ for $negative$ and other $p$?

This was inspired by similar posts like this one. Define the function, $$F(p) = \lim_{n\to\infty}2^n\sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{p}}}}_{n \textrm{ square roots}}}$$ We know that, ...
3
votes
1answer
60 views

$\lim_{n \rightarrow \infty } 2^{n} \sqrt{2-\sqrt{2+\sqrt{2+…+\sqrt{2}}}}$ [duplicate]

$\lim_{n \rightarrow \infty } 2^{n} \sqrt{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}$ where the 2 inside the roots appear n times. For example if n = 2 : $2^{2} \sqrt{2-\sqrt{2}}$ I discovered this. Has this ...
1
vote
7answers
131 views

Is there any way to arrive at $\pi$ without mentioning the circle's radius or diameter? [closed]

Given a circle of arbitrary size, is there any way to arrive at $\pi$ or $\tau$ (if you will) without any reference to the circle's radius or diameter?
35
votes
2answers
1k views

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
7
votes
1answer
2k views

Is 4 the second or third digit of pi

If someone says that they know 10 digits of pi, does that mean that they know ten digits starting with the 3 in 3.14 or with the 1 in 3.14?
1
vote
1answer
98 views

Show that the integral can not exceed $\frac{\pi^2}{96}$

Show that $$ \int_{-\infty}^{\infty}\int_{-x}^{x}\int_{-y}^{y}\int_{-z}^{z}e^{-(x^2+y^2+z^2+w^2)}\dfrac{|zw|}{(1+x^2)(1+y^2)} \,dw \,dz\,dy\,dx\le\frac{\pi^2}{96} $$ I am not understanding how $\pi$ ...