I heartily recommend this book about exploring mathematics using computers to many people. It is worth its price, although the cost may limit its audience; a paperback version is available. The book is particularly worthwhile for mathematics teachers at all levels, for school and public libraries, for computer and other professionals to whom mathematics is fundamental, and for any individual interested in mathematics who has access to a computer.
The authors write in the preface that
the book is intended for anyone who has some mathematical knowledge and a little experience with programming… Some of the chapters assume no more mathematical knowledge than whole numbers, but for the most part we assume some calculus, and the rudiments of algebra (polynomials, equations) and trigonometry (sines, cosines and tangents). … readers with a Freshman calculus course behind them will… have little difficulty in following most of the mathematics here.
As for programming, the knowledge we assume is very small, and most programs are given full listings in the text. It seems to us that a very effective way to learn programming is to use it to solve interesting mathematical problems. We have regarded the mathematics as the pre-eminent interest, and have not tried too hard to make the sample programs beautiful or elegant, or even particularly efficient.
The authors use BASIC on microcomputers and recognize its variants. They “make only modest demands” on the powers of microcomputers. Readers with access to BASIC on other computing platforms may also use this book.
In the preface, the authors advocate the use of computers to do mathematics.
Mathematics is, like all the sciences, an experimental subject. Sample computations and extensive calculations form the body of experimental evidence. Sifting through data, and using his experience and judgement, the mathematician may conjecture a new theorem. … With a microcomputer to hand, we can all be experimenters on a scale which was impossible in the past. …
We believe that the microcomputer can bring certain parts of mathematics alive in a way that it is not possible for plain text, or even a teacher, to do unaided. This comes about partly because of the possibility of experimentation, but also because facts and relationships can be illustrated swiftly and effectively, either by numerical calculation or through graphics.
Later in the preface, the authors summarize.
We hope that the material in this book conveys the authors’ conviction that mathematics is an exciting, beautiful and central subject. Throughout history mathematics has occupied the waking … hours of very many people. … In this book we have tried to show how a wonder of modern technology, the microcomputer, can be used to illuminate some small parts of this vast living monument to the endeavors of so many men and women, past and present.
The range of subjects includes whole numbers, greatest common factors, continued fractions, quadratic residues, prime numbers, and mathematical manipulation of large numbers. The quest for large primes plays a major role in public key cryptosystems, such as the RSA system, which “relies on the fact that it is much easier to find large primes than factor numbers” (p. 185). The authors provide routines for mathematics on large numbers while staying within the ordinary precision of microcomputers. For instance, to test a Mersenne number for primality, M127=2127−1, the authors show how to multiply and divide involving numbers up to 78 digits in length (p. 175).
A chapter discusses special numbers such as square roots, e as the base for natural logarithms, and c for measuring the perimeters and areas of circles. For c, the authors explore Archimedes’s method, Wallis’s product, Leibniz-Gregory series, and Euler’s series for c2.
In the last 135 pages, the authors explore differential equations and the iteration of real functions. They say that “Much of science is devoted to the problems of predicting the future behavior of some physical system or other. Often the underlying physical law will describe the rate at which the system evolves; what we require is a description of how it evolves” (p. 281).
The authors take a graphical approach, showing Euler’s line method and Runge-Kutta’s improvement. They introduce autonomous systems from physics, biology, and economics that involve two differential equations rather than one. They demonstrate those mathematical concepts using prey-predator systems and the pendulum. The authors also sample more general systems, such as chaotic differential equations.
On the iteration of real functions, the authors write that
the reader should be warned that quadratic iteration sequences have been the subject of much recent research, and are still far from being completely understood. This section is an order of magnitude more demanding than any other material in the book. We hope the extra effort is amply rewarded by an appreciation of some of the subtle and beautiful phenomena exposed here (p. 318).
The book ends by referring the reader to further works on fractal images and chaotic dynamical systems.
The book has a suitable table of contents and index, and ample references appear at the end of each chapter. While each chapter may be approached independently, most readers should probably work from the beginning to acquire the concepts and skills that will be used later. For those who select a specific topic, all material is cross-referenced. The graphics in the text relate directly to what the reader will see on the computer screen. The quality of the type, printing, paper, and binding is good.
Find this book, locate a microcomputer with BASIC, block out some time, and enjoy an educational treat.
