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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1answer
131 views

What is $\tau$ in base $12$?

I'm a big fan of both $\tau$ and the duodecimal system. And while I can find information for $\pi$ on both, I can't seem to find the number of $\tau$ in base $12$. $\tau$ is given as $\tau = 2\pi$. ...
3
votes
0answers
81 views

How to explain the significance of $\pi$ to a child? [closed]

In honor of $\pi$ Day, I thought I would pose this question. How would you explain the significance of $\pi$ to a child of, say, 9 years of age? While that's certainly an age that is old enough to ...
0
votes
2answers
162 views

Is my intuition wrong?

So we know that $\pi$ is irrational, that's fact! So we can't write it as $\frac{p}{q}$ where $p$ and $q$ are integers. We also know that the square root of a prime number is irrational/ But what ...
1
vote
1answer
56 views

Prove that : $\lvert s_n - \frac \pi 4\rvert \le \frac 1 {2n+1}$, where $s_n = \sum^{n-1}_{j=0} \frac {(-1)^j} {2j+1}$

Prove (Leibniz' series): $|s_n - \frac \pi 4| \le \frac 1 {2n+1}, \forall n \in \mathbb N$ where $s_n = \sum^{n-1}_{j=0} \frac {(-1)^j} {2j+1} = 1 - \frac 1 3 + \frac 1 5$ ... To prove the result ...
0
votes
1answer
76 views

Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
3
votes
5answers
133 views

Not pi - What if I used 3? Teaching pi discovery to K-6th grade

So, in ancient Mesopotamia they knew that they didn't really have the correct number (pi) to determine attributes of a circle. They rounded to 3. If you acted as though pi = 3, what shape would you ...
0
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3answers
38 views

Find the F(x) based on given points

Find an equation that satisfies the given sequence x | f(x) 1 | 2 2 | 4 3 | 6 4 | $π$ Normally, I would solve this myself but the f(4) = $π$ has really got me stumped
2
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0answers
75 views

What are the principle behind calculation of pi

It is possible that this is a duplicate, but I cannot find anything. I have always been wondering how to calculate pi. However, just using a given formula like the infinite series formulas does not ...
2
votes
2answers
88 views

Is it possible to calculate inverse sine without using pi?

I'm asking this in a programming context (because I'm a programmer) but I'm looking for general answers as well. In programming, all of the implementations of asin ...
30
votes
17answers
1k views

What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?

Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$ and the generalization of $\zeta (2k)$, my perception of $\pi$ has changed. I used to think of it ...
0
votes
2answers
99 views

how do i prove that $\sin(\pi/4)=\cos(\pi/4)$?

It's weird that i have not defined the tangent function yet. how do i prove that $\sin(\pi/4)=\cos(\pi/4)$? I have prove that $\tan:(-\pi/2,\pi/2)\rightarrow\mathbb{R}$ is a strictly increasing ...
1
vote
2answers
131 views

What would be an elementary way to prove that $3<\pi<4$

My definition for $\pi$ is twice the first positive real number such that $\cos(x)=0$. I think it's not even feasible to evaluate that $3.14 <\pi < 3.15$ in elementary level. Well, the only ...
1
vote
0answers
41 views

Ratios of right triangle integer multiples to PI

It is known that in a right triangle with angles 30 and 60 degrees the cathetus at the 60 angle is equal to the 0.5 of hypotenuse. In other words an angle with cosine 0.5 is equal to PI/3. Is there ...
2
votes
3answers
221 views

Why is the integral of sec^2(x) from 0 to pi infinity?

Why is it, if you take the integral of sec^2(x) from 0 to pi, my calculator returns "infinity" as the answer, but according to the second fundamental theorem of calculus, I got 0 with my own work. I ...
4
votes
3answers
214 views

Visual explanation of $\pi$ series definition

Can you visually explain why the following is true: $$ \frac{\pi}{4} = \sum\limits_{k=0}^\infty \frac{(-1)^k}{2k + 1} = \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\ldots\approx 78.5\% ...
1
vote
2answers
92 views

Proving that $\left(\frac{\pi}{3} \right)^{2}=1+2\sum_{k=1}^{\infty}\frac{(2k+1)\zeta(2k+2)}{3^{2k+2}}$.

I have asked a question or two like this one before and I've tryed to use similar methods to prove this identity(?), but I failed. By using WA it seems that numerically the LHS=RHS $$ ...
3
votes
2answers
157 views

Rational and irrational numbers under base pi

I am wondering, what would happen to the representation of a number like 2 in base pi? I know that things like $π^2$ would simply be 100 (right?), but what about numbers that are not of the form ...
6
votes
3answers
98 views

Recommendation for book on $\pi$?

I'm looking for a book that goes over the history of $\pi$, the mathematics of $\pi$ (like a discussion about the possible proof that all ten digits 0-9 occur with equal probability, viz., $\pi$'s ...
2
votes
2answers
141 views

Beautiful proof for $e^{i \pi} = -1$ [closed]

To celebrate the recent neuroscientific study that shows the beauty of math is in the mind, what is your most beautiful proof that $e^{i \pi} = -1$?
0
votes
0answers
22 views

Get 16*x[n-1] of a sequential BBP formula

So, I happened to find a slightly different version of the BBP algorithm (for calculating pi): Is there any way, perhaps by using the original BBP algorithm, to get that 16*x[n-1] term (without ...
2
votes
1answer
98 views

Is there a method to memorizing $\pi$? [closed]

The confirmed world record for memorizing the digits of $\pi$ goes to a Chinese graduate student named Lu Chao, who claims he has memorized up to 100,000 digits (although for the record breaking ...
4
votes
2answers
148 views

Why do many calculators evaluate $(-0.5)!$ to $\sqrt\pi$?

According to Wikipedia, factorial only is defined for non-negative integers. How come Spotlight, the Windows calculator and the Google search engine come up with $\sqrt\pi$ if you try to solve ...
0
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2answers
73 views

How $\pi$, $3.1415…$ and $180^o$ are adaptive together?!

I planed following to compute the circle's circumference. The circle's circumference finally can computable from: $$\lim_{\alpha\to0}{\frac{360^o}\alpha d} = 2\pi r$$ I don't want to follow above ...
4
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0answers
49 views

$\pi$ Monte-Carlo - Probability that O-Lock hit a Spoke?

(Edit: can someone please help me migrate this to physics stack? I think they would be more interested in helping me out with this problem. Thanks.) I have a bicycle with one of those O-locks on it ...
1
vote
2answers
76 views

Unquantifiable integral?

$$ \pi = \int_{-\sqrt{2}}^{\sqrt{2}} \sqrt{2-x^2} dx $$ So, thinking I was going to discover something amazing I reasoned that this is equal to pi, and that all I had to do to get the exact value of ...
6
votes
1answer
167 views

Predicting digits in $\pi$

Is it possible to predict next digit in $\pi$ using $N$ previous digits, so on and so forth? Or is this impossible because it's irrational? Basic assumption is that the person doesn't know a ...
4
votes
1answer
107 views

How do I make pi = 3?

This question emerges from a discussion on quora which concluded that if a circle was drawn on the surface of a sphere, the ratio of radius (from the circle's centre as projected to the sphere's ...
4
votes
3answers
141 views

Proving that $\frac{3}{2} \sum_{k=1}^{\infty} \frac{4}{k^3+k^2} = \pi^2-6$

I'm trying to prove that: $$\frac{3}{2} \sum_{k=1}^{\infty} \frac{4}{k^3+k^2} = \pi^2-6$$ I've tried looking at the partial sums, but no luck there. I just have no idea where to begin. Knowing that ...
2
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0answers
77 views

What consequence would there be if $\pi$ was not normal?

It is suspected that $\pi$ is normal, that is the distribution of its digits is uniform for any base. Would any results or algorithms, especially those that rely on probabilistic methods, be different ...
1
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0answers
57 views

Calculating custom bits of PI in hex or binary without calculating previous bits

I tried some spigot formulas to calculate custom hexadecimal PI digits. But any formula I tried definitely needed iterating and calculating sum from i=0 to N to get N-th digit. How to get N-th hex ...
1
vote
2answers
61 views

Solving infinite sums with primes.

Let $p_n$ denote the $n$'th prime number. How would one go about proving that infinite products like: $$\prod_{k=1}^\infty1 - \frac{1}{(p_k)^2} = \frac{6}{\pi^2}$$ or ...
0
votes
3answers
63 views

Are there any identities linking arithmetic functions and $\pi$?

The question is self-explainatory. For example are there any known identities involving Euler Totient function and $\pi$ ?
3
votes
7answers
534 views

Is there an identity that links $\pi$ and $\phi$ (the golden ratio)? [duplicate]

Is there some identity that shows a connection between $\pi$ and the golden ratio, $\phi$?
8
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4answers
355 views

Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$

After numerical analysis it seems that $$ \frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}} $$ Could someone prove the validity of such identity?
1
vote
1answer
84 views

For what values of $b\in \mathbb R$ is $\pi-b\in \mathbb Q$ true?

Just a simple short question. I'm looking for values $b$ such that $\pi-b$ is a rational number. Obviously $\pi$ is such a number, but are there more? Edit: $b$ is in $\mathbb R$
8
votes
3answers
134 views

How to demonstrate the equality of these integral representations of $\pi$?

Each of the following definite integrals are well known to have the value $\pi$: $$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}dx=2\int_{-1}^1\sqrt{1-x^2}dx=\int_{-\infty}^{\infty}\frac{1}{1+x^2}dx=\pi.$$ I ...
12
votes
3answers
281 views

Proving that $\left(\frac{\pi}{2}\right)^{2}=1+\sum_{k=1}^{\infty}\frac{(2k-1)\zeta(2k)}{2^{2k-1}}$.

Wolfram$\alpha$ says that we have the following identity $$ \left(\frac{\pi}{2}\right)^{2}=1+2\sum_{k=1}^{\infty}\frac{(2k-1)\zeta(2k)}{2^{2k}} $$ but, how does one prove such identity?
0
votes
1answer
74 views

Double Integration Problem for Buffon's Needle experiment

Numberphile has a video about the Buffon's needle experiment (Video). I am writing an essay on determining $\pi$ using probability and I need to show my understanding of the topic. I kind of already ...
1
vote
1answer
82 views

question about an inequality in calculus [duplicate]

Please, carefully show that $$ e^{\pi} > \pi^e $$ You are not allowed to use a calculator! thanks
0
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0answers
52 views

Demostrating that a trascendental function can be expressed as a product of its roots [duplicate]

I'm doing a work about the product of Wallis, a formula to calculate pi that is π/2= ∏ (2n/2n-1)· (2n/2n+1) from n=1 to n=∞. I have to prove the formula, and I have been searching in books and all ...
2
votes
1answer
125 views

Can integration get the real value of $\pi$?

If you take the equation of a cicle $x^2+y^2=1$ and re write it as $y=\sqrt{1-x^2}$, why can't you use $\int_{-1}^{1}(\sqrt{1-x^2})dx$ to get the real value of $\pi$ even if you end up with more ...
1
vote
1answer
62 views

Derivative of Trig. Function

if $f(x)=\tan(3x)$, then $f'(\pi/9)=$? I thought the answer was $4$ but my teacher marked it wrong. Work: $f'(x) = \sec^2(3x)\cdot 3 = \frac 3{\cos^2(3x)} = \frac{3}{\cos^2(\pi/3)} = 3/(3/4) = 4$.
4
votes
3answers
85 views

Why does it help to use $640320^3 = 8\cdot 100100025\cdot 327843840$ when you calculate $\pi$?

A Ramanujan-type formula due to the Chudnovsky brothers used to break a world record for computing the most digits of $\pi$: $$ \frac{1}{\pi} = \frac{1}{53360 \sqrt{640320}} \sum_{n=0}^\infty (-1)^n ...
10
votes
2answers
157 views

How to prove $\sum_{n=0}^\infty \left(\frac{(2n)!}{(n!)^2}\right)^3\cdot \frac{42n+5}{2^{12n+4}}=\frac1\pi$?

In an article about $\pi$ in a popular science magazine I found this equation printed in light grey in the background of the main body of the article: $$ \color{black}{ \sum_{n=0}^\infty ...
7
votes
5answers
691 views

Is π unusually close to 7920/2521?

EDIT: One can look at a particular type of approximation to $\pi$ based on comparing radians to degrees. If you try to approximate $\pi$ by fractions of the form $180n/(360k+1)$, you can find that ...
2
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1answer
64 views

Generalization of some infinite series containing binomial coefficients

On a page here. There are some infinite series in the form $$\sum_{k=1}^{\infty}\frac{k^n}{\binom{2k}{k}}=\frac{a}{b}+\frac{c \pi}{d}$$ Where $n \in {[0,1,2,3,4...]}$ and for some natural numbers ...
12
votes
10answers
476 views

How is the value of $\pi$ ( Pi ) actually calculated?

When I was a child I was taught $\pi$ (Circumference/Diameter) is an irrational number and can be approximated to $22/7$ but $= 3.(142857)(\ldots)$. But where does this value comes from? In ...
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1answer
83 views

A question about $\pi$

Is it correct? $$\pi=3+\cfrac{\log(\frac{1}{\Delta})}{\log(\frac{1}{\Delta^{\sqrt{\Omega}}})},$$ with $$\Omega=\cfrac{1}{(\pi-3)^{2}}\thickapprox49.8790939?$$
5
votes
2answers
259 views

Proofs without words of some well-known historical values of $\pi$?

Two of the earliest known documented approximations of the value of $\pi$ are $\pi_B=\frac{25}{8}=3.125$ and $\pi_E=\left(\frac{16}{9}\right)^2$, from Babylonian and Egyptian sources respectively. ...
6
votes
3answers
200 views

What is the fastest way to $\pi$?

There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$: (a) ...