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Pi is defined as circumference/diameter, but it is an irrational number. And by definition an irrational number can't be defined by a fraction. So how is it that pi is circumference/diameter and on a side note: how are important irrational numbers like e normally found?

Martin Sleziak
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Airdish
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    Irrational numbers cannot be expressed as a fraction of integers. Thus, a circle cannot have a diameter and circumference which are both integers. – Plutoro Nov 10 '15 at 17:29
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    In practice this is no problem because we can use arbitary good rational approximations of $\pi$, a very good one is $\frac{355}{113}$ – Peter Nov 10 '15 at 17:30
  • So then how was pi originally found if it is not arbitrarily available? – Airdish Nov 10 '15 at 17:31
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    Archimedes was the first to calculate $\pi$ upto some digits. – Peter Nov 10 '15 at 17:31
  • How did he do it though? – Airdish Nov 10 '15 at 17:32
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    He approximated the circumfence of a circle with regular polygons inside ans outside the circle and took the middle value of the results. – Peter Nov 10 '15 at 17:33
  • Ah alright, but in formal terms, is there any expression that aptly defines pi and perhaps its significance? – Airdish Nov 10 '15 at 17:34
  • I know that Ramanajan came up with multiple series for 1/pi – Airdish Nov 10 '15 at 17:34
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    The definition is usually : The first positive root of $cos(x)$ is $\frac{\pi}{2}$ – Peter Nov 10 '15 at 17:35
  • By the way : $e$ can be either defined by the limit $$lim_{x\rightarrow \infty} (1+\frac{1}{x})^x$$ or by the sum $$\sum_{k=0}^\infty \frac{1}{k!}$$ – Peter Nov 10 '15 at 17:37
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    $\pi$ and $e$ are not only irrational, they are even transcendental : There is no polynomial with integer coefficients, such that $\pi$ or $e$ is a root of it. – Peter Nov 10 '15 at 17:40
  • A funny thing : The pythagoreer falsely believed that every constant in nature is expressible by a fraction, but $\sqrt{2}$ coming from the pythagorean equation $a^2+b^2=c^2$, when $a=b=1$ ironically is irrational showing that the diagonal of a unit square has no rational length. – Peter Nov 10 '15 at 17:44
  • $\pi$ is equal to the infinite series:$$4-\frac43+\frac45-\frac47+\dotsb$$where the denominators are the odd numbers and the signs alternate. There are several other infinite series. However, the usual definition of $\pi$ is $\frac{\text{circumference}}{\text{diameter}}$. (Since $\pi$ is irrational, no circle can have both the circumference and diameter be integers.) – Akiva Weinberger Nov 10 '15 at 17:54
  • (You need to know calculus to prove that $\pi$ is equal to the infinite series above. EDIT: Not necessarily; some Indian mathematicians proved it a few centuries before calculus was invented. But I think they used calculus-like ideas. Besides, you need calculus-like ideas to define what an infinite series means. It's not like you can just add them all up; you'd never finish.) – Akiva Weinberger Nov 10 '15 at 18:05

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