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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9
votes
6answers
197 views

When the approximation $\pi\simeq 3.14$ is NOT sufficent

It's common at schools to use $3.14$ as an appropriate approximation of $\pi$. However, here it's stated that for some purposes, $\pi$ should be approximated to $32$ decimal places. I need an example ...
1
vote
1answer
75 views

Pi for non mathematician

I've been long gone from math (shamefully) and have trouble using some quite familiar concepts... Consider the following picture in which I render two circles with radius 32 (...
1
vote
4answers
65 views

Why is it safe to approximate $2\pi r$ with regular polygons?

Considering this question: Is value of $\pi = 4$? I can intuitively see that when the number of sides of a regular polygon inscribed in a circle increases, its perimeter gets closer to the perimeter ...
8
votes
1answer
201 views

Help me ID this weird $\pi$ formula

I remembered, and managed to find, still gathering dust in a forgotten corner of the Internet, an old QuickBASIC program which, with a trick, can rapidly sum up a HUGE amount of terms of the famous ...
2
votes
4answers
190 views

Why does $e^{i\pi}=-1$? [duplicate]

I will first say that I fully understand how to prove this equation from the use of power series, what I am interested in though is why $e$ and $\pi$ should be linked like they are. As far as I know ...
8
votes
0answers
118 views

The “trick” functions in the “$\pi$ is transcendental” proofs

I was reading this paper and I wondered how did Hermite decide to define a function $$f(x)=\frac{x^{p-1}(x-1)^p\cdots (x-m)^p}{(p-1)!}$$ Are these functions only tricks or there is a deeper meaning?
0
votes
1answer
81 views

In the hyperbolic geometry, is there a range of $\pi$?

In Euclidean space, $\pi$ is the constant value $3.14159\dots$ But I tried to measure the value of $\pi$ and found that $\pi$ is not constant! So I wonder if there is a range of $\pi$. If so, is ...
6
votes
1answer
190 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
0
votes
0answers
60 views

Comparison of Pi with rational numbers

Is there a way to compare (i.e. to know if it is greater or less) Pi with any given rational number (given by a/b, where a and b are integers), and to know beforehand the maximum of steps (development ...
3
votes
3answers
129 views

Is it possible that $\pi$ is finite in other numerical bases?

In base $\pi$, the number $\pi$ is $1\cdot \pi^1 + 0\cdot \pi ^ 0 $, which is equal to $10$. So, is $\pi$ an irrational number in all bases or not?
1
vote
3answers
127 views

Using loop to approximate pi (Monte Carlo, MATLAB)

I've written the following code, based on a for loop to approximate the number pi using the Monte-Carlo-method for 100, 1000, 10000 and 100000 random points. ...
-2
votes
1answer
72 views

Why do sine and cosine functions intersect the multiples of pi at x-axis?

Why do sine and cosine functions intersect the multiples of pi at x-axis? Maybe a dumb question, but I can't figure out why is that... And since $\pi$ is a transcendental number, we can't find the ...
0
votes
1answer
32 views

Is there any combination of numbers which upon division gives the exact number of P?

In other words, there are (probably) infinite combination of numbers/operations which leads to irrational numbers. So I wonder, if there is one which gives exact number representation of P(π)? Do we ...
0
votes
4answers
76 views

there is any relation between $\pi$, $\sqrt{2}$ or a generic polygon?

I'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make use ...
0
votes
2answers
63 views

Is it true that the formulas of the volume and surface area of an n-dimensional sphere are best expressed in terms of $\eta = \frac{\pi}{2}$?

Someone told me that the formulas of the volume and surface area of an n-dimensional sphere get simplified a lot if we express them in terms of $\eta = \frac{\pi}{2}$ instead of $\pi$. . In terms of ...
3
votes
3answers
75 views

If $\pi $ is normal, can it be used as a random number generator?

If one day we finally prove the normality of $\pi $, would we be able to say that we have ourselves a sure-fire truly random number generator?
3
votes
2answers
106 views

Understanding proof that $\pi$ is irrational

Reading this: Simple proof that $\pi$ is irrational, I fail to understand the following part: Since $n!f(x)$ has integral coefficients and terms in $x$ of degree not less than $n$, $f(x)$ and ...
-3
votes
1answer
61 views

Why there are two different values of θ for same quadrant?

Let Sin θ = 1/2 is function. Let us find its solution set. sine is +ve in I and II quadrant with reference angle π/6 θ = π/6 (I quadrant) Now here is my problem. We can use π-θ and (π/2)+θ to find ...
1
vote
1answer
36 views

2 pi term in sinusoidal signal

My intuition is that the $2\pi$ term in the sinusoidal signal equation: $$x(t) = \sin(2\pi\,f\,t)$$ Is indicative of the fact that this signal can be described as movement around a circle, is that ...
0
votes
3answers
141 views

Is there any geometry where ratio of circle's circumference to its diameter is rational?

In Euclidean geometry, the ratio of the circumference of a circle to its diameter is an irrational number, 3.14159 and so on. But if you change to non-Euclidean geometries, you get other values for ...
2
votes
1answer
174 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
119 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
52 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
65 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 ...
8
votes
5answers
262 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
133 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 ...
1
vote
3answers
336 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
338 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
202 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
100 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
30 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 ...
5
votes
3answers
1k 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
126 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 ...
3
votes
3answers
116 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
52 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 ...
0
votes
1answer
148 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
91 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
167 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
57 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
94 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
143 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
60 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
85 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
106 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 ...
33
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
116 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
133 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
42 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
407 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
230 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\% ...