Finding an accurate approximation to has been one of the most noteworthy challenges in the history of mathematics. Srinivasa A. Ramanujan (1887–1920), a mathematical thinker of phenomenal abilities, discovered a mysterious infinite series for estimating the value of :
The series is known to be a specialization of a modular equation of order 58 .
This Demonstration gives numerical estimates for using the reciprocal of the series up to , which gives a correct approximation to 38 decimal places.
 S. Ramanujan, "Modular Equations and Approximations to ," The Quarterly Journal of Mathematics, 45, 1914 pp. 350–372.
 J. M. Borwein, P. B. Borwein and D. H. Bailey, "Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi," The American Mathematical Monthly, 96(3), 1989 pp. 201–219. doi:10.2307/2325206.