Questions tagged [pi]

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.

$\pi$ is the ratio of a circle's circumference to its diameter. Its definition is modern analysis is (by Karl Weierstrass) $$ \pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}}. $$ An alternative definition, popularised by Landau is: Define $\frac{\pi}{2}$ as the smallest positive root of the cosine function.

It can also be given by the Gregory-Leibniz series (exhibits sublinear convergence) $$ \pi = 4 \sum_{j=0}^\infty \frac{(-1)^j}{2j+1}. $$ $\pi$ has the approximate numerical value $3.14159265358979323846\dots$, can be approximated by fractions, for example, $\frac{22}{7}, \frac{333}{106}, \frac{355}{113},\dots$, and is both irrational and transcendental.

It is part of Euler's famous identity:

$$e^{i\pi}+1=0.$$

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Collecting π infinite series: Mystery Series

I am new to this forum, but I am hoping I can ask for your help. I have recently been interested in finding and recording as many infinite series apporximating π as I can find. There is one in particular I have noted on an old copybook, but I have…
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Could $\pi$, raised to some power can give rational result?

For example $\sqrt 7$ is irrational but $\sqrt 7$ raised to power $2$ is rational. Similarly, is it possible that $\pi$ raised to some power (say $n$) could be rational ?
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Is there any way to finitely represent all the information in pi?

Of course, we can represent it as 10 in base pi but that won't be much useful. Think of pi as a length from 0 to some unique point on the real line. A length which cannot be finitely expressed in any integer base system because integer base systems…
Ryder Rude
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Is this a valid mathematical proof?

When the circumference of a circle (c) and the perimeter of a square (8h) are equal, we call it a squared circle. Radius (r) is equal to one. (Fig. 1) Next we simplify to the quarter squared circle which gives us our first triangle A. (Fig.…
Folding Circles
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How can you determine rigorously if $e$ or $\pi$ are points on the real line?

This question was a part of a discussion at an interview. QUESTION: How can you determine rigorously if $e$ or $\pi$ are points on the real line? MY OPINION: They should be, since they are defined to be real and irrational in nature. But then again…
SchrodingersCat
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What is the closest fraction (that isn't something like 31415.../1000...) that gets you pretty close to pi?

I'm just wondering what is the closest fraction (question for math nerds and geniuses) that isn't like pi/length-of-pi that gets you relatively close (like accurate to the 20th place) to pi? For example, 22/7 gets you "ok" close (it is 3.14285714),…
user3576467
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How to approximate $0.714286$ as a fraction of $\pi$?

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?
Afzal
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Can you precisely calculate $x^{\pi}$ (up to first 200 decimals) without a computer?

I have recently learned how to calculate $a^b$ for all $a,b \in \mathbb{Q}$. I have noticed that the $a$ in fraction form will always be $a\cdot 10^n/10^n$ where $n$ is $a$’s decimal length. But if $a$ is a big number, it would be painful to write…
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How is Euler's Formula Wrong?

I figured out that if $x^{y} = z$, then $z^y = x^{y^2}$. Then we know Euler's Formula: $$e^{πi} = -1,\quad (e^π)^i = -1,\quad (e^{2π})^i = 1$$ Now, using the formula above, let $e^{2π}$ act as x, and let i act as y. Then, finally, let 1 act as z.…
Math Bob
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Geometry: Circumference of circle without $\pi$

I just found out a way to find out the circumference of the circle without using $\pi$: $$4\sqrt{(1.8626\cdot r)^2 - r^2}$$ It can calculate up to $2$ decimals equal to the answer got by using $\pi$. Please let me know if it works. Thanks.
SAD
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The Tau Manifesto

What really is the Tau Manifesto, and why is a large section of the math community in its favour? Wouldn't it be too much work and effort to edit, and republish the countless texts in mathematics? Lastly, what is it (proper mathematical reason)…
connected-subgroup
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What makes Pi special? What really makes it Irrational?

I tried to find what makes a number $\pi$ special. $22/7$, $355/113 \approx\pi$, which is an irrational number. Why is this constant is used for defining any cyclic function? Why is it that this constant (that we can calculate for a life time) is…
Aura
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Show that $\frac1{1^2}+\frac1{3^2}+\frac1{5^2}+\cdots =\frac{\pi^2}8$ if $\frac1{1^2}+\frac1{2^2}+\frac1{3^2} +\cdots =\frac{\pi^2}{6}$

This was a question on an exam yesterday. My professor always throws one question in that is above our level and this was the one. I had no idea what to do on the exam. I just wanted to see an answer and the mode of thinking behind said answer. I…
Lanous
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An integral and $\pi$

I want to give you this question as an enigma. Can you prove that the following integral is equal to $\pi$ ? $$\int_0^\infty \sqrt{\frac{256x^4}{x^{12}+6x^{10}+15x^8+35x^4+6x^2+1}}=\pi.$$ This can be proven without the use of complex numbers.
E. Joseph
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How many digits of $\pi$ are currently known?

How many digits of $\pi$ are currently known?
Yatin K
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