Ramanujan, Modular Equations, and Approximations to Pi or How to compute One Billion Digits of Pi***

D.H. Bailey
NASA Ames Research Center
Moffett Field
CA 94035

J.M. Borwein and P.B. Borwein
Math Dept., Simon Fraser Univ.
Burnaby, B.C.
V5A 1S6 Canada

Mathactivated text
Other available formats
Related links
Author biographies


This article will follow up one small thread of Ramanujan's work which has found a modern computational context, namely, one of his approaches to approximating pi. Our experience has been that as we have come to understand these pieces of Ramanujan's work, as they have become mathematically demystified, and as we have come to realize the intrinsic complexity of these results, we have come to realize how truly singular his abilities were. This article attempts to present a considerable amount of material and, of necessity, little is presented in detail. We have, however, given much more detail than Ramanujan provided. Our intention is that the circle of ideas will become apparent and that the finer points may be pursued through the indicated references.
***Previously appeared in the American Mathematical Monthly 96, (no. 3) 1989, 201 - 219.

Authors' Reflections:

Billions of Digits of Pi

The article underlying our contribution started life in 1986 when David Bailey and Jon and Peter Borwein began to write separate pieces on ``Pi and its computation'' for the MAA Monthly. With urging from Herb Wilf (the then editor) the authors joined forces and a joint paper issued forth that was published in 1989. Since 1989, while the computers have continued to improve and the computation of has continued unabated, the underlying mathematical and computational techniques being employed are still much as described in the printed piece.

We opted to use this article as our contribution to Organic Mathematics for several reasons: it is reasonably accessible; it clearly allows for ``enhancement''; it was complete as a written piece and so allowed us to concentrate on the key issues underlying the larger project. Moreover, it tells a 3,000 year story touching many themes (from transcendence theory, to elliptic functions, and the Fast Fourier Transform) and involving many great thinkers (Archimedes, Newton, Gauss, Euler, Ramanujan, Von Neumann, Hilbert). In sum then we felt that the article provided an exellent test-bed to try out what would work in a multi-modal mathematical environment.

A project like this presents several challenges for the mathematician as multi-media author. First, the technology is changing so quickly that tools uninvented when the project began are sure to be obsolete before it is finished. Second, it involves a super-highway to be sure but one running through terra incognita. At many levels one must balance the issues of form and function, speed and scope. The version presently before you looks little like the one of six months ago -- small changes to the interface can greatly enhance or reduce readability.

The opportunities are great. Within this article, you will see the bust of Ramanujan. You can read quotes from many sources and see part of the original documents from Hermite, Shanks, Von Neumann and others. You may use ``lattice basis reduction'' over the Web; plot and compute many fascinating functions. If you want to try more examples the opportunities will be there.

On the down side, you may plunge off into secondary links and get lost coming back. You may find the plethora of hypertext links and options distracting, and the present limitations of using Maple over the Internet frustrating.

Our hope is that this article and this collection will play some role in limning the possibilities and flagging the pot-holes of trying to use these new tools to engage, educate and excite students, users and producers of mathematics.