Methods of Mathematics Applied to Calculus, Probability, and Statistics

Richard Hamming introduces students to calculus and more in Methods of Mathematics. His work contains many interesting mathematical gems. At the same time, it will not replace a standard calculus textbook for most students.

In 800 pages and 25 chapters, Methods covers all the standard material contained within a first year calculus sequence. It deals with differential and integral calculus and includes interesting applications. It visits the basic theory of infinite series and differential equations as well. Moreover, it attends to functions of several variables, but not in a way that would substitute for a multivariable calculus course.

Hamming’s text is noteworthy for its treatment of probability along with some unusual and interesting mathematics. This begins with his discussions of induction (2.3) and the method of undetermined coefficients (2.5). Later chapters address probability, especially in the context of calculus. Near the end, he devotes a full chapter to Fourier series, including the discrete Fourier transform and the Gibbs phenomenon. The book sprinkles additional tidbits along the way such as Buffon’s needle problem, liberal use of Wallis integrals and linear difference equations.

Why not use such a lovely and endearing text in lieu of a current calculus textbook? On reason is that its treatment of topics is somewhat unorthodox. While it eventually covers a year of calculus, Hamming’s ordering of the material may be foreign to the average calculus instructor. In addition, despite its title, little specific discussion of statistics takes place other than in reference to the least squares approximation. Even the discussion of probability, notable for its treatment of moment generating functions, omits important well-known discrete distributions or fails to mention them by name (uniform, geometric, negative binomial, and hypergeometric).

Besides functioning as a mere math textbook, Methods offers advice to a novice scientist. The author directs to the reader to issues in the text that are more important than others and also shares his conception of “how mathematics is really done.” He wisely informs the reader that mathematical results are rarely discovered in the final form that students see.

However, in my opinion, some of his advice would only distract and confuse a mathematical neophyte. In particular, he unnecessarily injects notes of skepticism in his exposition. Calculus students often struggle with the notion of an area under a smooth curve. They should not worry about whether the area of a rectangle is “really” the product of its base and height (first line of 11.2). Similarly, should a young reader new to the concepts of irrational and complex numbers really ponder Gödel’s incompleteness theorem at the same moment (p.96)? Finally, as a mathematical physicist, I definitely believe that the author’s digression on quantum mechanics (p.73) should have been omitted.

Beyond the unusual arrangement of material and the “give-or-take” advice, a vaguely-defined audience is the text’s biggest problem. The publisher recommends the book for “advanced undergraduates and graduate students” or as a “practical reference for professionals”. Yet, the author clearly intends this as an introductory text; it is neither appropriate for graduate students nor as a practical reference. At the same time, the author never directly explains the text’s intended use (course textbook vs. self-study), its prerequisites (algebra/trigonometry) or the level of sophistication required.

After reading the text, I would recommend this book either for self-study or as a course supplement. As a self-study text this is clearly for the mathematically inclined. In addition, despite the author’s pedagogical decision to avoid the calculus of trigonometric functions until chapter 16 (p.469), liberal use of trigonometry begins much earlier. I suggest that a student jump quickly to section 16.1 to review trigonometry before even reading page 26 which essentially discusses a feature of Chebyshev polynomials as an example of mathematical induction.

While Methods should not supplant a textbook, its 800 pages are well-written and worth a slow read. For the most part (except for trigonometry) it is internally consistent. It contains many useful topics that a mathematics undergraduate might otherwise wait a year or two to encounter. At the same time, readers should be wary of the book’s pitfalls. They should remember to question what is written and form their own opinions. In addition, the current edition contains a number of simple (and not so simple) mathematical errors (e.g. p.225). I plan to keep a listing of errors on my website and all industrious readers are encouraged to send me any mistakes that they find. The current publisher (Dover) has been contacted and does not wish to maintain a list of errata at this time.

Dov Chelst is an associate professor at DeVry University in North Brunswick, New Jersey where he teaches a range of undergraduate statistics, mathematics and physics courses. His main interests are mathematical physics, complex variables and online instruction. In his (non-existent) spare time he cycles, reads midrashic literature, and tinkers with (Mandriva) Linux.