The term “experimental mathematics” refers to the use of computation to discover and explore mathematical concepts. It has a long tradition that includes the laborious calculations by hand of Gauss, Euler, and many other notables, but the current usage applies to computation mostly with modern software and computer-related investigations. The authors have assembled a collection of areas in which undergraduates, with a minimum of calculus and linear algebra, can explore to advantage.

The authors suggest that students could also learn mathematics as it is explored and discovered, to ask questions, and not just rely on how it is presented to them in print, or in the classroom. Only a minimum of computer programming skills is necessary provided that the student has access to a computing environment and access to documentation, online or otherwise. Computational approaches described here do not depend on any particular set of software. The authors use boxes called “algorithm” and “pseudocode” in the text to present steps within a computation.

The book has six main sections that provide a variety of mathematical topics that can be explored numerically and experimentally. These are: computing digits of pi, generalization of the Fibonacci sequence, iterated functions (starting with the Collatz sequence), primes, probabilistic simulation, and Markov chains (including Markov chain Monte Carlo sampling). Each section offers practice problems, exercises, and explorations – a total of 467 overall. This provides so many that it gives instructors and students plenty of opportunity to find problems of interest.

The introductory chapter on the digits of pi offers a setting for interesting questions, an introduction to computational ideas, and discussion of the concepts of conjecture and generalization. Things get progressively more challenging as the book progresses, but the authors succeed at keeping the material occasionally challenging but readily accessible to their target audience. The most challenging subject is probably RSA cryptography, and the authors suggest that it could be treated as an optional project.

The treatment of the Collatz sequence (the “3n+1 problem”) is particularly well done, and it was a nice touch to add a brief discussion of Terrence Tao’s theoretical result following the extensive numerical treatment. The discussion of Markov chain Monte Carlo methods, increasingly useful across many disciplines, is also quite good.

Each chapter has a small collection of suggestions for further reading, and an extensive
bibliography is also provided. The authors also supply online code files in Mathematica and Python on an AMS webpage for all examples in the text and all solutions to all the practice problems.

This book is the best I have seen at this level, and I’d highly recommend it.

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.