June 14, 2013
The ratio of the circumference to the diameter of a circle, known as π, is constant regardless of the size of the circle; that fact has been known to mathematicians for about five thousand years; much of the history of mathematics is intertwined in the history of πi, as approximations of the ratio have improved over time. Much of the history of this blog is also intertwined in the calculation of π; one of the original ten exercises used a bounded spigot algorithm of Jeremy Gibbons to calculate π, and later we used an unbounded spigot algorithm also due to Gibbons; we studied the ancient approximation of 355/113 calculated by Archimedes; we studied two different Monte Carlo simulations of π; and we even had the Brent-Salamin approximation contributed by a reader in a comment.
The development of computers allows us to compute the digits of π to an astonishing accuracy; the current record, unless somebody has bettered it recently, is ten trillion digits. That record was set by the Chudnovsky brothers, two Russian mathematicians living in New York, using an algorithm they developed in 1987. The algorithm is based on a definition of π developed by Ramanujan, and is beautifully described by the two brothers.
Your task is to compute many digits of π using the Chudnovsky algorithm. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.