What do people mean by "finding the end of $\pi$"

Question

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: $\pi$ is irrational so how exactly does one find more digits to it? I understand that it is a ratio.... But it's not like we find $\pi$ by division (although we can estimate it)... Right? Is there like a large fraction, or some sort of sensor you can use in the real world?

What does it look like mathematically to calculate $\pi$ (formulas etc) and why does it get harder to calculate each new digit?

Thanks much!

More formulas than you can shake a stick at: https://en.wikipedia.org/wiki/Approximations_of_%CF%80 — , Feb 08 '16 at 02:00

Patrick Stevens · Accepted Answer · 2016-02-07T23:33:24.800

2

There is no "end of pi", at least in base 10. It just keeps going.

$\pi$ can be defined in various different mathematical ways which are entirely independent of the universe - for example, as twice the least positive root of $\cos(x) = 0$ - and so you don't need to measure things in real life to find pi. In fact, according to General Relativity, space is not flat, so drawing circles and then measuring circumference/diameter will not give us a "good" measure of pi (beyond some number of decimal places, anyway). Compare the fact that however well you draw a triangle on the surface of an orange, it could still have angles which sum to 270 degrees (if the triangle has surface area 1/8th of the total - thanks, Akiva Weinberger!).

Additionally, the uncertainty principle of quantum mechanics tells us that we cannot measure a circle in real life to better than a certain precision, even if space were completely flat.

$\pi$ has various infinite-sum representations, of which the most famous is probably $$\frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}$$

By truncating this sum to only finitely many terms, we can get successively better approximations to $\frac{\pi}{4}$. (It should be noted that this sum actually converges extremely slowly, and much better formulae exist. There is also a formula which directly calculates the $n$th binary digit of $\pi$ without calculating the previous binary digits.)

edited Feb 07 '16 at 23:33

answered Feb 07 '16 at 21:42

Patrick Stevens

36,135

That is soo cool! Also, What is the formula for the nth digit? – J.Doe Feb 07 '16 at 21:45
Would using base $\pi$ be considered cheating? ;-) – egreg Feb 07 '16 at 21:52
@egreg I think it would :P – Patrick Stevens Feb 07 '16 at 21:59
@J.Doe Look up the Bailey–Borwein–Plouffe formula - it's on Wikipedia, for instance. – Patrick Stevens Feb 07 '16 at 21:59
@PatrickStevens Rats! :-D – egreg Feb 07 '16 at 22:16
Wait, not all spherical triangles have their angles sum to $270^\circ$. Only those with (surface) area equal to one-eighth of the sphere are like that. – Akiva Weinberger Feb 07 '16 at 23:26
@AkivaWeinberger Oh, my bad. I'll fix it, thanks. – Patrick Stevens Feb 07 '16 at 23:32
Interestingly: Assume the sphere has radius $1$. Then this triangle has area $\pi/2$, and the sum of its angles is $3\pi/2$ radians (that is, $\pi/2$ more than you would find in a Euclidean triangle). In fact, if a spherical triangle has area $A$ (and is drawn on a sphere of radius $1$) then the sum of its angles is $\pi+A$ ($A$ more than you would find in a Euclidean triangle). The angle excess is exactly the area! – Akiva Weinberger Feb 07 '16 at 23:39

What do people mean by "finding the end of $\pi$"

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