
- Author: Bettina Richmond and Thomas Richmond
- Series: Pure and Applied Undergraduate Texts
- Publisher: American Mathematical Society
- Publication Date: 02/18/2009
- Number of Pages: 424
- Format: Paperback
- Price: $89.00
- ISBN: 978-1-4704-7204-7
- Category: textbook
[Reviewed by Russell Jay Hendel, on 11/12/2024]
Reviewed by Russell Jay Hendel
I can do no better than to start this review with the authors’ own book review of their book presented in the preface; I will then add two comments afterwards. The authors explain that their book deliberately has more material than can be covered in one semester so as to be able to accommodate and address a variety of situations including level 200 courses, level 300 courses, a course on introductions to proof methods, as a source of individual projects, and to provide a resource for students beginning to read mathematics on their own for either broadening their knowledge or sheer enjoyment. The prerequisites for the course are mathematical maturity which may be obtained by one to two semesters of calculus or a course in linear algebra even though calculus and linear algebra are not used or assumed in the book.
The book has a variety of topics, not all of which are always found in textbooks on Discrete Mathematics. These include the divisibility tests, discovering and proving patterns in numbers, a continual emphasis on multiple proofs including algebraic, geometric, and combinatoric, chapters on the Fibonacci numbers, continued fractions, and geometry (including the theorems of Pick, Cotes, Helly, and results on tiling), and inclusion of results on infinite products and nested radicals in the chapter on series. Such a diversity is a welcome addition to any course providing optional readings and exposure to students.
To me, the strong point of the book is the 650 problems for which the best adjective I can use to describe them is sparkling. A partial list or sample of problem categories found would include (a) Pattern finding problems: arising from numerical sequences with requests to formulate conjectures and prove them; (b) Recurrence theory: including exercises walking through advanced topics such as characteristic polynomials with multiple roots, methods of conjecturing recursions from sequences, or proving recursions given initial conditions and underlying recursions; (c) Generating Functions: including a step by step walkthrough of partial fraction decompositions; (d) Research Articles: including presenting results from journals, and walking through the proofs in a step by step exercise; (e) Miscellaneous challenging problems: for example, proving Bernoulli’s inequality, proving by induction n!2 <(n2)!, testing the function a*b defined as (a+b)/2 for associativity, commutativity, and possession of a unit; and (f) awareness of multiple proof methods, for example, proving Fn-2 + Fn+1 = 2 Fn directly, by induction, geometrically, and using book theorems (Fn are the Fibonacci numbers).
My only reservation with the book is that it is too good. I would not use it for a level 200 course (especially for non-math majors) where I would prefer automated homework encouraging practice on algorithms.
Russell Jay Hendel holds a Ph.D. in theoretical mathematics, an Associateship from the Society of Actuaries, and a Doctor of Science in Jewish Studies. He teaches at Towson University which is a center of actuarial excellence. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art, and poetry.