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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10
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2answers
293 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: $$e^{\pi \sqrt{19}} \approx 96^3+744-0.22$$ ...
10
votes
1answer
253 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 $\mathbb{Q[\sqrt{5}]}$. ...
5
votes
2answers
189 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 ...
5
votes
2answers
135 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
votes
4answers
248 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 ...
5
votes
2answers
316 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 ...
0
votes
1answer
54 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 ...
22
votes
2answers
476 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 ...
0
votes
1answer
43 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?
0
votes
1answer
28 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.
0
votes
2answers
188 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, ...
8
votes
2answers
185 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 ...
0
votes
1answer
120 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 ...
0
votes
1answer
102 views

Value of Pi derivation

Derive the value of Pi. I want with explanation. Is there any possible way? How do scientists calculate it?
1
vote
1answer
136 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 ...
0
votes
1answer
153 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?
1
vote
2answers
184 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, ...
1
vote
1answer
74 views

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

$$ \sqrt{5} > \frac {13 + 4\pi}{24 - 4\pi} $$
2
votes
1answer
26 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 ...
1
vote
3answers
125 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 ...
1
vote
1answer
94 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 ...
3
votes
2answers
76 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 ?
1
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2answers
55 views

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} ...
1
vote
2answers
54 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} ...
3
votes
2answers
199 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 ...
3
votes
3answers
111 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 ...
0
votes
1answer
223 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 ...
1
vote
2answers
577 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 ...
1
vote
2answers
86 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 ...
18
votes
2answers
514 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 ...
0
votes
1answer
94 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?
3
votes
1answer
77 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? ...
-2
votes
1answer
84 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 ...
0
votes
1answer
49 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 $2^3 = 8$ ...
14
votes
2answers
168 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 ...
6
votes
1answer
100 views

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 ...
2
votes
2answers
115 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 ...
2
votes
1answer
102 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. ...
5
votes
4answers
184 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 ...
5
votes
2answers
111 views

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 $$ ...
0
votes
1answer
38 views

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 $i \in \mathbb{N}_0$ ...
0
votes
1answer
31 views

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 ...
20
votes
6answers
475 views

$\lim_{n\to\infty}\sqrt{6}^{\ n}\underbrace{\sqrt{3-\sqrt{6+\sqrt{6+\dotsb+\sqrt{6}}}}}_{n\text{ square root signs}}$

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 ...
2
votes
0answers
51 views

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 ...
0
votes
2answers
82 views

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 ...
3
votes
2answers
99 views

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 ...
0
votes
1answer
38 views

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 = ...
7
votes
2answers
138 views

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 ...
8
votes
5answers
254 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.
0
votes
2answers
29 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 ...