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

2
votes
0answers
32 views

Irrationality of $\pi$ and circumference to diameter ratio.

How is $\pi$ actually defined? If it is defined as the ratio of the circumference of a circle to its diameter then from this definition itself either of the circumference and diameter has to be ...
1
vote
4answers
77 views

Is 1/113 a rational number?

Before two days , one of my physics si told me if one wants to use value of pi more accurately 355/113 can be considered as value of pi to get more accurate result. I want to know is 1/113 a ...
0
votes
2answers
41 views

Is an anomaly in base-n arithmetic discoverable in base-m arithmetic?

I have always been fascinated by the book "Contact" by Carl Sagan. The final chapter of the book (not the film!) reports about an anomaly in the n-millionth decimal of pi, optimally visible when pi is ...
1
vote
2answers
25 views

Precalculus converting radians to degrees.

I'm studying for a precalc test and I've kind of hit a brick wall with conversion. I've searched for help but only found things on converting from pi radians. I found some questions that consist of ...
-6
votes
0answers
41 views

Possible hidden messages in Pi [duplicate]

Assume that a proof that pi is normal existed. Is it then possible that starting at some finite position x in pi, from there >on every p(n)'th digit is 0, where p(n) is the n'th prime? I know ...
8
votes
5answers
235 views

How do i prove that $3<\pi<4$?

Let's not invoke the polynomial expansion of $\arctan$ function. I remember i saw somewhere here a very simple proof showing that $3<\pi<4$ but i don't remember where i saw it.. (I remember ...
7
votes
1answer
106 views

Calculating $\pi$ via the $\zeta$ function?

I was fooling around, trying to come up with a rapid way to compute $\pi$. Then I remembered that we always have: \begin{equation} \zeta(2n)=c\pi^{2n}, \end{equation} where $n$ is a positive integer ...
2
votes
3answers
212 views

Why is $\pi$ so close to $3$? [closed]

$\pi\approx 3.141592654$ Why is it so close to $3$? I find this intriguing, this cannot be a coincidence.
-5
votes
3answers
106 views

is 22/7 is real value of pi? [duplicate]

I am just new to this so pardon me if my question is some silly. I was googleing about value of pi. in wikipedia it has 3.141592653589793238462643383279502884197169399 as I have studied its 22/7. my ...
3
votes
1answer
195 views

Characterization of $\pi$

Let $x$ be the ratio of a circle's circumference to its diameter. Let $y$ be twice the smallest positive number $t$ for which $\cos(t)$ equals $0$. How to prove $x=y$? Thanks.
0
votes
4answers
86 views

why the occurrence of 4,5,6 and 9 in pi differs?

i´m playing around with pi, i have this document with the first 5million decimal numbers after comma. http://www.aip.de/~wasi/PI/Pibel/pibel_5mio.pdf and i build a script that i put in for example ...
0
votes
1answer
28 views

Critical Numbers Problems

Okay so I found the critical number no problem, it being cos x=-1/2, but on my answer sheet it says that the critical numbers are ...
3
votes
3answers
431 views

How to find continued fraction of pi

I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are: $\pi=[3:7,15,1,292,.....]$ and many others given here. Similarly some nice continued ...
2
votes
1answer
74 views

Doubt in Ivan Niven's proof of irrationality of pi.

In the proof, how do we get the upper limit for $f(x) \sin{x}$ as $\pi^n \cdot \frac{a^n}{n!}$ ? I thought $f(x) \sin{x}$ would be maximum at $x=\pi/2$ when its value would be: $$\pi^n \cdot ...
2
votes
3answers
105 views

Proof without words for $\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$

$$\sum_{i=0}^\infty(-1)^i\frac{1}{2i+1}$$ $$1-\frac13+\frac15-\frac17+\frac19-\cdots=\frac\pi4$$ Does anyone know of a proof without words for this? I am not looking a for a just any proof, since I ...
1
vote
0answers
47 views

How does making a line curvy save brick?

Recently, I visited a college. The University of Virginia, to be exact. Just more of a sightseeing tour than anything. Whilst walking through a part of campus, I saw a brick wall that was built ...
-1
votes
1answer
100 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
65 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
0answers
126 views

How to show $\pi^\pi$

I was being given a task to show $\pi$ and $\pi^\pi$ in the line of real numbers. I drew circle with diameter of $1$, then circumference of circle is equal to $\pi $. Then I drew function $f(x) = x^x$ ...
0
votes
2answers
113 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
51 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
59 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
111 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
votes
3answers
25 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
votes
0answers
58 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
66 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 ...
22
votes
17answers
961 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
75 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
124 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
38 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
74 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
176 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
78 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
123 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
93 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
126 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
19 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
92 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 ...
3
votes
2answers
129 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
votes
2answers
71 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
votes
0answers
43 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
73 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
129 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
102 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
113 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
votes
0answers
72 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
vote
0answers
42 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 ...
0
votes
2answers
51 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
57 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
431 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, $\phi$?