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**Mathematics by Experiment: Plausible Reasoning in the 21st Century**

Borwein, Jonathan, and David Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, 2003; x + 288 pp, $45. ISBN 1-56881-211-6.

This book presents "the rationale and historical context of experimental mathematics," together with "accessible examples of modern mathematics where intelligent computing plays a significant role." (A companion volume, Experimentation in Mathematics: Computational Paths to Discovery, offers additional examples.) Topics include prime numbers and the zeta function, normality of numbers, lots about pi "and its friends," high-precision arithmetic, and constructive approaches to many special functions. Experimental mathematics is dedicated to expanding mathematical knowledge rapidly, occasionally with a "temporary relaxation in rigor." As a sample, consider the Nilakantha-Gregory series

After 5 million terms, the series differs from [pi] in the 7th decimal place and periodically thereafter. "Such anomalous behavior begs explanation," and the authors proceed to explain exactly why and how this behavior occurs. They also include challenge problems, Internet sites, and enough mathematical connections to astonish anyone. (Note: An insert advises that some citations point to the wrong items in the references; those and other errata, plus other useful information, can be found at http : //www. expmath. info ).

Copyright Mathematical Association Of America Apr 2004

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