Say you have a computer and some free time. What to do? You could calculate digits to . As you know, is irrational so its digits go on forever without repeating. Which means you can keep calculating digits ’till the cows come home. Which is what some people do!
It was recently announced by Alexander J. Yee and Shigeru Kondo that they have completely destroyed the old record of 2,699,999,990,000 digits set by Fabrice Bellard in 2009. Using a home computer (albeit a pretty high powered one!), over the course of 90 days they computed out 5,000,000,000 digits to and verified the correctness of the computations.
They did their caculation using the Chudnovsky formula:
The advantage of this particular series is that it converges very rapidly. So if you’d like a very good approximation to and, hence, (which is what we want), then you can calculate a relatively small partial sum for the right hand side of this equation.
By comparision, the above formula is closely related to a formula of the Indian mathematician Srinivasa Ramanujan:
where for each additional term of the series you compute, you obtain another 8 digits of !
To read about the history and the mathematics of approximating , check out Wikipedia’s article on the subject.
Of course, if you’re an engineer, then you are probably not so impressed. If your motto is “close enough is close enough”, then you probably remember that we told you on Pi Day last year that it only takes 39 digits of to compute the circumference of any circle that fits in the observable universe with error less than the diameter of a hydrogen atom!