# How is pi generated?

I was thinking about this recently since it was pi a few days ago :) How is pi, the number, generated in a way that can go on forever? How are extra characters discovered? Thanks.

We usually use an infinite series that converges to pi or some other iterative method.

For example arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ... up to infinity. If x=1 then arctan(1) = pi/4 so you can see that you could calculate pi in such a way. The problem is that this infinite series converges (approaches) pi very slowly. What this means is that you need to add a lot of terms before the number starts to look like pi.

Currently, there are much better infinite series to use for example: http://planetmath.org/encyclopedia/RamanujansFormulaForPi.html [Broken] is a much better infinite series to use to calculate pi.

Typically the computations are done, well, by computers that are able of doing calculations very fast. I think that currently pi has been calculated to like 10 billion digits or something ridiculous like that.

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Is there a way to calculate the number of the nth digit without expanding out pi?

^ Yes! That's what I was looking for. Thank you.