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I know that there was a guy that could get 100 decimal digits of $\pi$ before computers were able to get thousands.

How did the guy do that?

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At least two ways: Machin's formulae or the arithmetic-geometric mean. See this. – J. M. Dec 1 '10 at 12:45
Sorry.. duplicated :( – Diego Dec 1 '10 at 12:47
I'd love to accept both answers!!! – Diego Dec 1 '10 at 13:13
up vote 5 down vote accepted

According to Wikipedia, John Machin combined the formula $$\frac{\pi}{4}=4\cot^{-1}5-\cot^{-1}239$$ with the Taylor series expansion for the inverse tangent in order to compute $\pi$ to 100 decimal places.

A previous record was due to Abraham Sharp who used an arcsine series to find 72 decimal digits.

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This page on the chronology of pi contains many useful notes on how the pre-computer era calculations of $\pi$ were performed.

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See also Pi: A Source Book.

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Check this, you can find n-th PI digit without computing previous ones :) It was a big surprise when this formula was discovered.

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