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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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 ...
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59 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 ...
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2answers
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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!
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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 ...
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5answers
80 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, ...
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1answer
82 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 ...
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71 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 ...
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1answer
86 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 ...
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2answers
82 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 ...
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151 views

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

Ramanujan in his Lost Notebook (p.16, and a related one in Quarterly Journal of Mathematics, XLV, 1914) gave the very close approximation (by just $10^{-7}$), $$\pi^4 \approx ...
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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?
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89 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 $\frac{1}{\pi}$?
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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 ...
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3answers
453 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 ...
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1answer
66 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 ...
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1answer
46 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 ...
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1answer
104 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?
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4answers
720 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 ...
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1answer
70 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 ...
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0answers
41 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 ...
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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 ...
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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 ...
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5answers
516 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 ...
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2answers
138 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}}} - ...
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4answers
386 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 ...
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3answers
150 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. ...
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1answer
74 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 ...
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76 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 ...
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1answer
52 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 ...
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2answers
779 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 ...
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46 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 ...
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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 ...
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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. ...
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1answer
44 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 ...
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0answers
30 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 ...
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1answer
150 views

Does this curiousity have any meaning?

If $\pi$ is calculated to the $360$-th digit after the decimal point, the last digits are $360$ : ...
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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 ...
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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 ...
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0answers
208 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) ...
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5answers
968 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 ...
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4answers
674 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 ...
6
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0answers
53 views

$\pi$ base $e$ or $\pi=\sum\limits_{n=-1}^{\infty} a_ne^{-n}$ where $a_n\in\{0,1,-1\}$

I was "playing with $\pi$" trying to look at it in different numeral systems and it's not so hard to obtain $\pi$ base $2$ or $3$ or even $\varphi=\frac{\sqrt{5}+1}{2}$, using Maclaurin series of ...
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8answers
439 views

Explain why $e^{i\pi} = -1$ to an $8^{th}$ grader?

I am an $8^{th}$ grader that is taking Algebra I. But nearly everyday I try to learn things outside of what I am learning in class. Quite a while ago I discovered that $e^{i\pi} = -1$. This ...
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2k views

How to convert $\pi$ to base 16?

According to this Wikipedia article $\pi$ is approximately 3.243F in base 16 (i.e. hexadecimal). Can someone explain this? (Note: I understand how to convert an integer to base 16) Thanks
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Deriving the value of $\pi$ from a dart board

I saw this on a website and it was pretty interesting: The circle inscribed in the square has a radius of $1$ and the square has a side length of $2$. This means that the area of the circle is: ...
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1answer
82 views

How To Calculate a Tangent In Degrees Without a Calculator

So the other day in my Geometry class, I was bored so I decided to try and calculate pi (which is one of many things I do when I am bored). During that class, I finally developed an equation to ...
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3answers
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Is it possible to intuitively explain, how the three irrational numbers $e$, $i$ and $\pi$ are related?

I read a bit about this equation: $e^{i\pi}=-1$ For someone knowing high school maths this perplexes me. How are these three irrational numbers so seemingly smoothly related to one another? Can this ...
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Does π start with two identical decimal sequences?

Let d$(x,y)$ be the sequence of decimals of π from the x:th one to the y:th one. My question: is there a number $n$ such that d$(1,n)$ = d$(n+1, 2n)$? I.e., does π start with two identical decimal ...
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145 views

How does atan(1) * 4 equal PI?

I needed the PI constant in C++, and I was lead to the answer that: const PI = atan(1) * 4 Note that despite involving code, I'm asking this from a mathematics ...
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1answer
32 views

ratio of “diameter” of a n-gon to perimeter

So say I have a regular polygon with n sides, and I bisect the an angle E such and the line (EF). Assume line segment EF has length b, while the polygons side length is s. What is $b/(n*s)$, and as n ...