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
5answers
235 views

Is $22/7$ an often used approximation for $\pi$?

It is $\pi$-day and the internet is full of stories about $\pi$. One story mentions that "an approximation -- $22/7$ -- is used in many calculations." I have never actually used $22/7$ as an ...
4
votes
6answers
227 views

How can never ending decimal numbers represent finite lengths? e.g. pi(π), $\sqrt{2}$

Recently, I was in a discussion with a colleague that, whether the πd really can represent the accurate perimeter of a circle or not. To clarify that doubt, I came ...
9
votes
0answers
121 views

Infinitely nested radical expansions for real numbers

Conjecture. For any real number $x \in (0,1]$ there exists a unique expansion in the form $x=-2+\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}$ with $a_k$ being natural numbers from the set $(2,3,4,5,6)$. ...
1
vote
0answers
85 views

A double inequality for $\frac{\pi}{2}$

Approximating $\frac{\pi}{2}$ from above Since $$\left(\frac{\pi}{2}\right)^9\approx 58.220897$$ the root $$58^\frac{1}{9}\approx 1.5701$$ is not far from $$\frac{\pi}{2}\approx 1.570796$$ This ...
1
vote
1answer
70 views

An algorithm for creating a circle on a discrete plane and a limit for $\pi$

I know there is a well known algorithm which uses the circle equation to approximate it with pixels. However, I wanted to approach this problem from the most basic principles. So we start with a ...
-1
votes
2answers
70 views

Interesting Ways to Find the value of Pi [closed]

Is there any interesting ways to find pi? Thanks.
15
votes
2answers
257 views

What's the formula for this series for $\pi$?

These continued fractions for $\pi$ were given here, $$\small \pi = \cfrac{4} {1+\cfrac{1^2} {2+\cfrac{3^2} {2+\cfrac{5^2} {2+\ddots}}}} = \sum_{n=0}^\infty \frac{4(-1)^n}{2n+1} = \frac{4}{1} - \...
0
votes
1answer
62 views

The irrationality of $\pi/e$ is listed as open yet the infinite product formula for it seems to suggest a way to prove it.

And the formula of all rational products seems to suggest that taking some n as n approaches infinity, the formula will have an always increasing amount of uncancelled primes(so provably non ...
0
votes
1answer
74 views

Value Of $\pi$ obtained using limits!

What i thought was simple, a circle can be formed by increasing the number of sides of regular polygon( like pentagons, hexagons, etc ) up to infinity by keeping the distance between the center and ...
1
vote
0answers
43 views

Is there a simple way to prove pi is irrational using math, which is known understandable for someone doing HL Maths (Advance High School Maths)?

In school I have learned from early on that pi is irrational, but is there a simply proof which I could understand to show that this is the case. I am doing HL Maths (IB, with discrete as the option ...
0
votes
1answer
45 views

What is the relation between the gaussian integral and the volume of the n-ball?

Even if I've red other threads treating this question, it's still obscure to me what deeply relates the multiple Guassian integral $\int e^{-x^2} = \sqrt \pi$ and the area of a $n$-ball. Someone ...
2
votes
2answers
397 views

Pi's Recursiveness [duplicate]

I don't know if this will make sense, but: If $\pi$ is infinite and contains all strings of numbers including those of infinite length, then it must contain $\sqrt2$, and if $\sqrt2$ is infinite and ...
8
votes
2answers
108 views

Why do ratios of these Fibonacci-type sequences approach $\pi$?

Define $A_n$ by $A_1=12$, $A_2=18$, and $A_n=A_{n-1}+A_{n-2}$ for $n\ge3$. Similarly define $B_n$ by $B_1=5$, $B_2=5$, and $B_n=B_{n-1}+B_{n-2}$ for $n\ge3$. Terms of $A_n$: $12, 18, 30, 48, 78,\dots$...
6
votes
2answers
150 views

Why isn't there a formula for $\zeta(k)=\sum_{n=1}^\infty\frac{1}{n^k}$ involving $\pi$ when $k$ is odd?

Now we know that $\sum \frac{1}{n}=\text{divergent}, \sum \frac{1}{n^2}=\frac{\pi^2}{6}$ but now this for $\sum \frac{1}{n^3}=1.20....$ and again $\sum \frac{1}{n^4}=\frac{\pi^2}{90}$ .Now somewhere ...
0
votes
0answers
91 views

Why is Gelfond's constant transcendental?

I have seen a proof of $\pi$ being transcendental by conclude that transcendental number powered by algebraic number must be transcendental and algebraic number powered by algebraic number must be ...
1
vote
1answer
24 views

Formula for cycloid?

Is there a formula for cycloid? My approximation is $((2\times(x\div(\pi\div2)))-(x\div(\pi\div2))^2)^.626$.
0
votes
0answers
63 views

Show that $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$ [duplicate]

$(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$ How can we show this equivalent without using a calculator?
2
votes
0answers
43 views

What is wrong with my statement?

So with Euler identity I used $2 \pi(\tau)$ instead and got $e^{i2\pi} = 1$, and took natural logarithm $\ I 2\pi = 0$? But is see online the answer is $6.28....i$.
0
votes
1answer
35 views

Need an Ambiguous example

Suppose I have a floating-point number with $m$ where $(m > 0)$ digits after the decimal point. Now if I want to round it up to $d$ where $(0 ≤ d < m)$ digits after the decimal point, sometimes ...
37
votes
1answer
691 views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ $$u_2=v_2=...
0
votes
2answers
90 views

Use two formulas to approximate $\pi$

Hi guys. Any hints on part c? How are we going to approximate $\pi$ by computing $P_k$ and $p_k$
1
vote
2answers
107 views

Finding sine of an angle in degrees without $\pi$

The following is found using a combination of: (a) a polygon with an infinite number of sides is a circle, (b) the perimeter of that polygon is the circumference of the circle that it becomes (of ...
9
votes
0answers
235 views

Interpretation of $\frac{22}{7}-\pi$

Integral and series proofs that $\frac{22}{7}>\pi$ We can prove that $\frac{22}{7}$ exceeds $\pi$ by using Dalzell integral $$\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi$$ or its ...
8
votes
0answers
122 views

Limit approximation for $\pi$ in the four fours puzzle?

The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:...
5
votes
3answers
85 views

How to show $\lim\limits_{n\rightarrow \infty}\left(\raise{5pt}\frac{2^{4n}}{n\binom {2n} {n}^{2}}\right)=\pi $?

How to show this is true? $$\lim_{n\rightarrow \infty}\left(\raise{3pt}\frac{2^{4n}}{n\binom {2n} {n}^{2}}\right)=\pi $$
1
vote
1answer
57 views

Geometric interpretation of Leibniz formula for $\pi$

We know $\pi=4(\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}....)$. I'm wondering, is there a geometric interpretation of this identity. Can we prove this identity by finding a different way to ...
3
votes
2answers
112 views

Elementary proof that $\pi$ is irrational

I'm trying to understand the first proof in this page. So we have $$S=\frac{\pi }{4}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{2k-1}=S_{n}+R_{n}$$ where $S_{n}=\sum_{k=1}^{n}\frac{(-1)^{k-1}}{2k-1}$ and $...
-3
votes
2answers
154 views

A series of positive terms to prove $\pi>\frac{333}{106}$

This is a consequence of the answer to that question. A proof that $\pi > \frac{333}{106}$ is given by the series of positive terms $$\pi-\frac{333}{106} \\ =\frac{48}{371} \sum_{k=0}^\infty \frac{...
1
vote
1answer
255 views

Series and integrals for inequalities and approximations to $\pi$

Fundamentals Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral $$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ (see Why do we need an integral to prove ...
9
votes
3answers
592 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
13
votes
2answers
427 views

Is $\pi^k$ any closer to its nearest integer than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to its nearest ...
6
votes
1answer
191 views

Is there a series to show $22\pi^4>2143\,$?

This extends this post. I. For $\pi^3$: $$\pi^6-31^2 =\sum_{k=0}^\infty\left(-\frac{63}{(2k+2)^6}+\frac{31^2}{(2k+3)^6}\right) =\sum_{k=0}^\infty P_1(k)\tag1$$ As pointed out by J. Lafont, when ...
4
votes
1answer
133 views

Why is “$\pi^2= g $” where $g$ is the gravitational constant?

Some months ago a professor of mine showed us a 'proof' of why $g\approx 9.8 ~\text{m}/\text{s}^2$ (the gravitational acceleration at the surface of the Earth) is 'equal' to $\pi^2\approx9.86\dots$ ...
6
votes
0answers
230 views

A monotonically increasing series for $\pi^6-961$ to prove $\pi^3>31$

This question is motivated by Why is $\pi$ so close to $3$?, Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$. I. $\pi$ and $\pi^2$ There are series with all terms positive for $\...
4
votes
1answer
56 views

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could then the $\pi$ also be an integer?

$\pi$ is dependent on properties of geometry, assuming that we define it as $C/d$. Could there be a geometry where $\pi$ is a rational number or an integer?
0
votes
1answer
70 views

What do people mean by “finding the end of $\pi$” [closed]

So I have been wondering. I have heard many times statements like "if we find the end of $\pi$ then we might be in a virtual reality" or "new computer can calculate $X$ digits of $\pi$". My question:...
1
vote
1answer
20 views

How does the speed of convergence of these formulae for calculating PI compare with the best algorithms?

I came across some series many years ago for calculating PI. I found that the first member of that series has been known for a long time in the math world. It is the set of series defined by: $$ \pi^...
-1
votes
2answers
317 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
4
votes
1answer
239 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that $$3<\pi&...
1
vote
1answer
86 views

How to express an angle in terms of pi

I have the complex number $z = 5 + 6i$ in polar form $$z = \sqrt{61} (\cos \theta + i\sin \theta)$$ and $$\theta = \tan^{-1}\left(\frac{6}{5}\right) = 0.87605805059 \text{ rad}$$ But I need that ...
2
votes
1answer
195 views

A series related to $\pi\approx 2\sqrt{1+\sqrt{2}}$

This question follows a suggestion by Tito Piezas in Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$? Q: Is there a series by Ramanujan that justifies the approximation $\pi\...
1
vote
1answer
79 views

Simpler derivation to $\pi$ [closed]

I'm an amateur in mathematics, being in 9th grade. I have been trying to derive $\pi$. During this I reached a limit to find the value of $\pi$. $$\lim_{x \to 0} \frac{180\sin x}{x}$$ Where $x$ is in ...
0
votes
1answer
270 views

Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$?

The approximation $$\pi\approx\frac{22}{7}=3+\frac{1}{7}$$ suggests that the closest integer to $\frac{1}{\left(\pi-3\right)}$ is $7$. However, $$ \frac{1}{\left(\pi-3\right)^2}\approx49.879 $$ is ...
1
vote
1answer
92 views

Infinite tetration of $i$

Proof Euler's identity; $$e^{i\pi} + 1 = 0$$ can be manipulated in order to obtain the result: $$e^{i\pi} = -1$$ Raising both sides of the equality to the power of $i$ gives, after simplification:...
0
votes
1answer
59 views

Getting theta of Line Equation

Please forgive my lack of knowledge, which i think it's one of those basic formula related to Trigonometry. Let's look at visual example: https://drive.google.com/file/d/0BwoMn9VKDw-tSVNRMm9RWGp3dTg/...
3
votes
0answers
315 views

generalized 2-terms Machin's formula (an efficient way to compute $\pi$)

Looking at Machin's formulas in this post Machin's formulas and cousins, and digging a bit, I've finally computed the next formulas, allowing to generate an infinite number of 2-terms Machins's ...
6
votes
1answer
125 views

Why does this sequence converge to $\pi$?

Over at our friends at codegolf.SE, I asked a question about programs that seemed to converge to $\pi$, but didn't actually do that. One of the answers (by soktinpk...
0
votes
1answer
199 views

Does Pi contain itself? [duplicate]

Alright, recently there was a question on 9gag whether the digits of $\pi$ may contain $\pi$ itself here's the original. One user had - in my opinion - a really plausible answer: Here's his answer. ...
0
votes
2answers
143 views

Prove that $\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\cdots =\frac{\pi^2}{8}$ [closed]

Prove that $$ \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+ \cdots =\frac{\pi^2}{8}$$ and $$\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2} +\cdots =\frac{\pi^2}{24}.$$ I do ...
6
votes
0answers
255 views

Rational series representation of $e^\pi$

This question is related to Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$? by Tito Piezas III. Andrew Fraker (2014) found an almost-integer which is equivalent to the following ...