Pi was easy to understand but harder to calculate hundreds of years ago.
So Archimedes might have used string and managed accuracy to two decimal places at best.
Since then number theory has progress very substantially.
For one thing Pi is always expressed as an infinite series that is chopped somewhere to give you the precision you need. There are alot of different formulae that equal Pi, but some converge alot faster than others.
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http://mathforum.org/library/drmath/view/52543.html
When Newton and Leibnitz developed calculus in the late seventeenth century, more formulas were discovered that could be used to compute pi. For example, there is a formula for the arctangent function:
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
If you substitute x = 1 and notice that arctan(1) is pi/4 you get a formula for pi. This is not useful because it takes too many terms to get any accuracy, but there are some related formulas that are very useful. The most famous of this is Machin's formula:
pi/4 = 4 arctan(1/5) - arctan(1/239)
This formula and similar ones were used to push the accuracy of approximations to pi to over 500 decimal places by the early eighteenth century (this was all hand calculation!)
In the twentieth century there have been two important developments: the invention of electronic computers and the discovery of much more powerful formulas for pi. For example, in 1910 the great Indian mathematician Ramanujan discovered a formula that in 1985 was used to compute pi to 17 million digits.
Other even better methods have been developed since, and computers aregetting ever more powerful. The current record is about 51 billion decimal places.
1/PI = (sqrt(8) / 9801) sumk=0~inf { [(4k)! (1103 + 26390k)] / [(k!)4 3964k] } (Ramanujan)
1/PI = (sqrt(10005) / 4270934400) sumk=0~inf { [(6k)! (13591409 + 545140134k)] / [(k!)3 (3k)! (-640320)3k] } (Chudnovsky)
1/PI = { 1 - sumk=0~inf [2k (ak2 - bk2)] } / 2M(a0, b0)2 (AGM)
y0 = sqrt(2) - 1, a0 = 6 - 4 sqrt(2)
yk = [1 - sqrt(sqrt(1 - yk-14))] / [1 + sqrt(sqrt(1 - yk-14))], ak = (1 + yk)4 ak-1 - 22k+1 yk (1 + yk + yk2)
1/PI = limk->inf ak (Borwein)
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The n sided polygon method mentioned above is the formulea as n heads to infinity of n * Sin[180/n] * Cos[180/n]. If you take the limit of this term as n gets big, you get Pi.
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Helaman Ferguson invented an algorithim in 2000 to calculated the nth digits of Pi - without calculating any digits before n - so if n is 1,000,000 you only start after the first million digits of Pi - an astounding feat!
Pi = Sum of from k = 0 -> infinity : 1/16^k [4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6)]
http://www.lbl.gov/Science-Articles/Archive/pi-algorithm.html
