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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16
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0answers
372 views
+50

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
7
votes
1answer
2k views

Is 4 the second or third digit of pi

If someone says that they know 10 digits of pi, does that mean that they know ten digits starting with the 3 in 3.14 or with the 1 in 3.14?
1
vote
1answer
91 views

Show that the integral can not exceed $\frac{\pi^2}{96}$

Show that $$ \int_{-\infty}^{\infty}\int_{-x}^{x}\int_{-y}^{y}\int_{-z}^{z}e^{-(x^2+y^2+z^2+w^2)}\dfrac{|zw|}{(1+x^2)(1+y^2)} \,dw \,dz\,dy\,dx\le\frac{\pi^2}{96} $$ I am not understanding how $\pi$ ...
1
vote
1answer
50 views

Are these two functions equivalent?

I'm working my way through some 'Graphs of trigonometric functions' on khanacademy.org and came across something that I found to be a little confusing, and I wanted to know if my intuition is correct ...
0
votes
2answers
46 views

Measure the length of a wrapping

I'm interested in learning how to find the length of a wrapping. Let's say I'm going to be wrapping some flat fabric webbing around a pole. I'd like to find the amount (length of fabric) i'll need ...
0
votes
1answer
55 views

Proving $\pi$ irrational: help with Lambert's proof. “Circularity”?

This expression is irrational. $$\tan(x)=\frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-...}}}$$ But then he used the fact that $\tan{\frac{\pi}{4}}=1$, so $\frac{\pi}4$ is irrational. But how can we use ...
8
votes
1answer
190 views

Surprising behavior of Leibniz formula for Pi (as Euler product)

I wrote a program to compute successive approximations of Pi using the following Euler product: π/4 = (3/4)*(5/4)*(7/8)*(11/12)*(13/12)... in which the ...
42
votes
4answers
9k views

False proof: $\pi = 4$, but why?

Note: Over the course of this summer, I have taken both Geometry and Precalculus, and I am very excited to be taking Calculus 1 next year (Sophomore for me). In this question, I will use things that I ...
0
votes
2answers
66 views

Probability of finding a 3 digit number six times within the first 1000 digits of pi

What is the probability of six (or more) appearances of a $3$ digit string within the first $1,000$ digits of $\pi$ (assuming $\pi$ was random). Not including rep digits, which may complicate the ...
0
votes
0answers
32 views

Finding sequences in $\pi$ [duplicate]

I recently came across some programs which were able to calculate exactly, when a particular sequence of digits appears the first time in the decimal expansion of $\pi$. This made me wondering, if it ...
4
votes
0answers
60 views

How is it that $\pi$ appears in so many formulas that seem to be in no way geometric. [duplicate]

When I first saw: $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\mp\cdots.$$ I was puzzled that such an expression could have anything to do with circles. There are tons more and ...
-2
votes
2answers
82 views

Does this picture represent anything mathematically? [closed]

This image here shows a beautiful fractal-like image. Does this map some sort of function, each number corresponding to a section/colour? Or is this just pretty art? Thanks!
0
votes
2answers
41 views

Pi irrationality repetition limits [duplicate]

I am not a mathematician at all and I had a thought about Pi that I can't work out. Pi is irrational, with an infinite sequence of numbers A recurring number is infinite Would it be theoretically ...
0
votes
5answers
85 views

Euler's Identity in Degrees

Since we have a simple conversion method for converting from radians to degrees, $\frac{180}{\pi}$ or vice versa, could we apply this to Euler's Identity, $e^{i\pi}=-1$ and traditionally in radians, ...
11
votes
1answer
88 views

Continued fraction estimation of error in Leibniz series for $\pi$.

The following arctan formula for $\pi$ is quite well known $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\tag{1}$$ and bears the name of Madhava-Gregory-Leibniz series after ...
3
votes
0answers
75 views

Help with proving that $\pi$ is irrational

I was trying to prove that $\pi$ is irrational, just to see if I could do it. So far, I've tried to do this by using the fact that the sum $$S=\sum\limits_{k=1}^\infty ...
1
vote
1answer
91 views

What does this infinite product come out to?

$$1\cdot \frac{1}{2}\cdot 3\cdot \frac{1}{4}\cdot 5\cdot \frac{1}{6}\cdots$$ What does this product come out to? It does diverge, but products like this tend to have values $\lt \infty$. Here is what ...
2
votes
2answers
86 views

Find a function $f(x)$ in an integral

(Related question here). Is there a way to calculate the function $f(x)$ in this integral in terms of $x$ without using $a,b,c$: $$\int_{a}^{b} f(x)dx=c$$ Two examples $\rightarrow$ how do find ...
23
votes
0answers
260 views

Is there an integral for $\pi^4-\frac{2143}{22}$?

In Ramanujan's Notebooks, Vol 4, p.48 (and a related one in Quarterly Journal of Mathematics, XLV, 1914) there are various approximations, including the close (by just $10^{-7}$), $$\pi^4 \approx ...
2
votes
1answer
86 views

is true that $\sum_{n=0}^\infty B_n(-1)^n=\frac{\pi^2}{6}$?

Is true that $\sum_{n=0}^\infty B_n(-1)^n=\frac{\pi^2}{6}$? (where $B_n$is the Bernoulli number) If this is true, how can I prove it?
3
votes
0answers
93 views

Convergence to $\frac{1}{\pi}$

Mathematicians of all times found approximations for the value of $\pi$ using infinite sums. But I was asking to myself: is there any infinite sum that approximates $\frac{1}{\pi}$?
0
votes
1answer
32 views

The probability of a number appearing in an approximation of an irrational number?

I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 ...
21
votes
3answers
528 views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...
0
votes
1answer
71 views

Is there an O(1) operation to find the Nth digit of Pi?

I'm afraid that 10 minutes of googling isn't finding references to a paper that I thought existed. I remember seeing some years ago a reference to a paper that claims to have proven an algorithm that ...
1
vote
1answer
50 views

Randomness in pi and other irrational numbers [duplicate]

This is a post I read about pi while looking for stuff about tau -which is two times as much as pi. This makes me wonder, why does only pi contain such randomness? Don't other non-repeating and ...
-4
votes
1answer
105 views

How to approximate $0.714286$ as a fraction of $\pi$? [closed]

I'm doing an exercise that tells me that the answer must be a multiple of pi, like $12\pi$ or $\dfrac23\pi$. I need to approximate $0.714286$ as a fraction of $\pi$. How do I achieve this?
3
votes
4answers
734 views

Alternate method to calculate an infinite string of numbers that's not $\pi$, and contains any string

So, rather than using $\pi$, is there any way that isn't overly complicated, (and can be calculated on a computer without taking a year) in which I could generate an infinite string of numbers that ...
0
votes
1answer
71 views

Gosper Formula for inv $\pi$, properties.

I need to understand very good how the properties of this formula $\frac{4}{\pi} = \frac{5}{4} + \sum_{N \geq 1} \left[ 2^{-12N + 1} \times(42N + 5)\times {\binom {2N-1} {N}}^3 \right] $ Taken from ...
3
votes
0answers
43 views

$\tau$-ists and the History of Radian Measure?

Recently, I have been reading about the $\tau$ vs $\pi$ debate. One of the arguments for $\tau$ was that $1\tau$ radian is the whole circle, thus fractions of $\tau$ correspond to the fractions of the ...
8
votes
0answers
76 views

Generalizing Bellard's “exotic” formula for $\pi$ to $m=11$

Bellard's "exotic" pi formula has the form, $$a\pi+b = \sum_{n=1}^\infty \dfrac{P(n)}{{\displaystyle \binom{mn}{2n}2^{n-1}}}$$ where $a,b,m$ are integers and he uses $m=7$. However, it seems there ...
1
vote
1answer
26 views

Limit(s) of a Sequence from the decimal expansion of $\pi$

I found a statement in a book concerning the decimal expansion of $\pi$ that I do not really understand. The statement is my problem number 2, where problem number 1 really looks like a reference ...
22
votes
5answers
521 views

Why is it that this gives a good approximation of $\pi$?

At the end of a Physics examination, I decided to play around with my calculator, as I always do when I have time left over, and found that: $$\frac{1}{100} \cdot 11^{\ln(11)} \approx ...
14
votes
2answers
147 views

Bellard's exotic formula for $\pi$

In his webpage, Fabrice Bellard mentions an exotic formula for $\pi$ as follows $$\pi = \frac{1}{740025}\left(\sum_{n = 1}^{\infty}\dfrac{3P(n)}{{\displaystyle \binom{7n}{2n}2^{n - 1}}} - ...
-6
votes
4answers
412 views

Is $\pi \cdot 7$ actually $22$? [closed]

The value of $\pi$ is $\frac {22}{7}$. Now if I multiply, $\pi$ by $7$, it gives 22 (as $\frac{22}{7} \cdot 7 = 22$). But, a when we multiply a rational number with a irrational number, we are ...
4
votes
3answers
157 views

How do mathematicians invented and introduced $\pi$ term in the case of circle?

This is basic question. Since childhood I am mugging the mathematical formulae areas of square, rectangle and circle etc. Now,it is possible for me to understand formula of area of square I.e. ...
1
vote
1answer
79 views

What would a base $\pi$ number system look like?

Imagine if we used a base $\pi$ number system, what would it look like? Wouldn't it make certain problems more intuitive (eg: area and volume calculations simpler in some way)? This may seem like a ...
11
votes
0answers
90 views

First order definition of $\pi$

$e$ has a very short first-order definition. It's the only constant that makes this true: $$\forall x,e^x\ge x+1$$ What about $\pi$? What's the shortest first-order definition we could give it? I'm ...
3
votes
1answer
53 views

Can we show that the decimal expansion of $\pi$ doesn't occur in the decimal expansion of the Champernowne constant?

This question was inspired by an answer and some comments to this question. Recall that the Champernowne constant is obtained by concatenating all natural numbers written in base 10 and then put ...
11
votes
2answers
837 views

Decimal Expansion of Pi

Sorry if this has been asked before, but I have a query about the notion that the decimal expansion of pi contains every possible string of numbers (please note that I am only a "casual" maths ...
1
vote
2answers
47 views

Number system and PI

Ok, we all use the decimal system with numbers from 0 to 9. And we have PI with an infinite number of decimals. We also have a boolean system or hexadecimal. Is there any decimal system where PI has ...
0
votes
0answers
44 views

Extensions of the algebraic numbers

I have two extensions of the algebraic numbers, and I'd like to know whether they are equivalent. Definition 1. $x \in \mathbb E_1$ iff $x$ is a root of $e^{P(x)} + Q(x)$ with $P, Q$ some ...
0
votes
0answers
35 views

Euler's Equation [duplicate]

I'm new around here, I'm fourteen, and I am in Ninth Grade. Can somebody tell me what Euler's equation exactly is, and why it's important, and what we can use it for? The whole $e^{i\pi} = -1 $ thing. ...
2
votes
1answer
45 views

What's the fastest known running time for a spigot algorithm for computing an arbitrary digit of $\pi$?

That is, for the fastest known algorithm for doing so, how many steps will it compute the $n^{\text{th}}$ digit of $\pi$ in? I know some people define running time as the number of steps it will take ...
0
votes
0answers
37 views

Calculating the speed of a rotating object on its own axis and its speed around a point

I'm seeking help calculating the speed of rotation of the blue object $s_2$ in relation to $s_1$ with a distance 'd' from point $A$. In other words, the goal is to keep the object always facing $A$ if ...
6
votes
1answer
152 views

Does this curiousity have any meaning?

If $\pi$ is calculated to the $360$-th digit after the decimal point, the last digits are $360$ : ...
1
vote
0answers
54 views

Integral tending to an integral for $\pi$

I am examining: $$\int_0^1 (1-ax)^{1/2} dx$$ If we differentiate: $$\dfrac{d}{dx} \left[\dfrac{-2(1-ax)^{3/2}}{3a}\right]$$ we get to the function in the integral. The idea now is consider various ...
0
votes
0answers
33 views

Need proof of integration of sine parametrized functions [duplicate]

Yesterday, i encountered an integral formula (actually it's a generalization, i think). This : $$\int_0^\pi x f(\sin x)\,dx = \frac{\pi}{2} \int_0^\pi f(\sin x)\,dx$$ For simple functions like ...
14
votes
0answers
221 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$: (Mathematica code) ...
28
votes
5answers
1k views

Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$?

I've been struggling for a while with the following problem: Which is bigger: $(\pi+1)^{\pi+1}$ or $\pi^{\pi+2}$? Needless to say software aid is not allowed. All manual calculations should be ...
5
votes
4answers
702 views

Number raised to power of irrational number

What is the consequence of raising a number to the power of irrational number? Ex: $2^\pi , 5^\sqrt2$ Does this mathematically makes sense? (Are there any problems in physics world where we ...