Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction (review)

In Number-Crunching, Paul Nahin attempts to explain and exemplify a number of mathematical problems, utilizing basic physics and high-speed computers, thus taming them from “roaring lions to purring cats.” This book is an addition to Nahin’s publications on mathematics and physics, and on science-fiction stories, which he successfully brings together and refers to throughout the book. His breadth of knowledge, as suggested by the list of references from which he draws and by anecdotes from his personal experience, is impressive. On occasion, the information in the reference section of a chapter and the extensive historical background provided therein can prove to be more interesting than the material covered in the chapter itself. The computer code presented is written in MATLAB. The code is adequately explained, but this may not suffice for someone with no prior programming experience to excel in programming. Yet it serves its purpose of illustrating the practicality of computer coding versus tedious analytical solutions that are often impossible. Algebraic derivations, calculus, trigonometry, differential equations, number theory and electronics are some of the topics visited and alternate between being the problem and being the tools for solving subsequent problems.

First we are introduced to Fermat and his famous last theorem, one of the “most difficult mathematical problems” in the Guinness Book of Records, of which the Pythagorean theorem is a special case. Fermat’s theorem was first conjectured in the 17th century. Various failed attempts and surprisingly innovative ways to prove it for the general case were suggested until the correct proof was finally given in 1995 by Wiles. A potentially mathematically unconvincing, yet otherwise ingenious, unpublished probabilistic approach by the physicist Feynman is also explained in the book.

The enchanting storytelling continues with De Morgan’s challenge to his students to calculate manually the roots of a third-degree equation to 100 digits, a time-consuming and tiresome task that an average computer today takes a few seconds to complete. The infinite resistor ladder problem, more thoroughly explained in one of Nahin’s earlier books, confirms the advantage of the computer’s solution over the manual solution in terms both of efficiency and accuracy.

The “modern electronic computer and its enormously powerful software” can come to the rescue even when analytical computers are not available. The Monte Carlo technique is used in computer simulations of complex physical and mathematical processes using random numbers. It is mostly useful when it is infeasible to compute theoretically exact answers. The “hot plate problem” presented is solved analytically, by iteration and using Monte Carlo methods. The mathematical derivations involved may not appeal to everyone. This is also true for the Fermi-Pasta-Ulam experiment, and one may fail to appreciate the surprise in its solution. Between the two problems, though, one should not miss the chronicle in Chapter 3 of the electronic computer, which begins in the 1940s. The first machine, charmingly described in the book, involved a rat’s nest of banana-plug cables and numerous switches that required manual adjustment for programming. It also occupied 128 cubic feet and required a 10-ton air conditioner, but at the time it was “a science fiction fantasy come true.”

We are then called to be astonished by the hanging mass problem, describing misbehaving and curious oscillators, requiring understanding of Euler’s wonderful identity, the law of conservation of energy, Cramer’s determinant rule and differential equations, even though we are spared the “algebraic horror” involved by using MATLAB. An inexperienced eye may interpret the behavior of the oscillators as chaotic, but chaos only enters the picture during the discussion on the Pythagorean three-body mass problem and while trying to determine the orbits of stars experiencing gravitational forces. Nahin goes through Newton’s two-body solution and Euler’s approach to the three-body problem. The exposition of Poincar...