Number Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction
by Paul J. Nahim
Princeton University Press, Princeton, NJ, 2011
400 pp., illus. 98 b/w. Trade & eBook, $29.95
ISBN: 978-0-691-14425-2; ISBN: 978-1-400-839582.
Reviewed by Irene Kaimi
School of Computing and Mathematics
Plymouth University, UK
In Number Crunching, Paul Nahim 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 Nahim’s publications on mathematics and physics, and science fiction stories, which he successfully brings together and refers to throughout the book. His breadth of knowledge, suggested by the list of references from which he draws and follows each chapter, 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 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 interchange between being the problem and the tools for solving subsequent problems.
We are first 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 A potentially mathematically unconvincing, yet otherwise ingenious, unpublished probabilistic approach by the physicist Feynman is explained in the book.
The enchanting storytelling continues with De Morgan's challenge to his students to calculate manually roots of a third degree equation to 100 digits, a time consuming and tiresome task, that an average computer nowadays takes a few seconds to complete. The infinite resistor ladder problem, more thoroughly explained in one of Nahim’s earlier books, confirms the advantage of the computer’s solution over the manual solution both in terms 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 an theoretical 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, in chapter 3 one should not miss the chronicle of the electronic computer, which started in the 1940s. The first machine, charmingly described in the book, involved a rat’s nest of banana-plug cables and numerous switches which required manual adjustment for programming. It also occupied 128 cubic feet, and required a ten-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 of the “algebraic horror” involved by using MATLAB. An inexperienced eye may interpret the behaviour 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. Nahim goes through Newton's two-body solution and Euler's approach to the three-body problem. The exposition of Poincaré's discovery of chaos, after submitting a flawed paper to the King Oscar II competition, is absorbing. MATLAB succeeds in producing the orbits only for restricted versions of the three body problem, which over three centuries since Newton’s death is still causing headaches to those attempting to solve it.
We revisit Newton towards the end of the book, when a mathematical physicist discovers a time machine and travels in time to meet great Newton. He then offers him the gift of a calculator, to save him from unnecessary manual calculations, only to be treated as devil’s employee and to be ignominiously dismissed. Other enthralling science fiction stories, old and new, show the depth of errors made and expected to be made in future predictions and how “the universe is not only queerer than we suppose, but queerer than we can suppose”.
For the mathematically inclined the book serves as a basis for further exploration into physics. More physically minded people will be fascinated by the elegance of the mathematics discussed and applied to physical problems. Both mathematicians and physicists will find the examples discussed as well as the challenging problems at the end of each chapter – with detailed solutions - exciting. The general audience can choose to ignore the extensive calculations and focus on the stimulating discussion that forms the main body of the book. Regardless of the reader’s level of technical understanding though, Nahim’s expertise and storytelling are highly enlightening.