Finite Mathematics: An Applied Approach

This textbook includes chapters on linear systems, matrix algebra, linear programming, finance, combinatorics, probability, statistics, game theory, and logic. However, these topics are presented without a unifying concept. Even though the author defines finite mathematics as “the area of mathematics that deals with the study of finite sets” on p. 327, there is no attempt to use this description to tie everything together. The subtitle “An Applied Approach” does not reflect the pedagogical style of the book. At the beginning of each section, specific computational and/or conceptual objectives are listed. When each idea is encountered, a reference is given to relevant problems in the exercise set. For the most part, applications are presented only after all the nuts and bolts are in place so that the ideas appear in context. Even though a large number of applications are included, they do not drive the presentation. Some are highly imaginative, and others are routine.

The reviews provided in the appendices are concise. In order for a student to be prepared fully for this book, he or she will need to have a strong background in College Algebra. For example, the chapter on Finance makes extensive use of exponents, logarithms, geometric sequences, and recursively defined sequences. However, there are significant portions of the book that can be processed by students with only a background in Intermediate Algebra. For example, the chapter on Probability is very accessible. Thus, in order to make appropriate use of this textbook, the instructor will need to select the material to include in his or her syllabus with care.

The instructional value of the book varies noticeably from chapter to chapter. This perception might be due in part to the differing prerequisites for each topic. I found the chapter on the geometric approach to linear programming to be well written. The technical concepts are presented in detail so that less sophisticated readers can follow along, and several nice applications are given. On the other hand, the chapter on the Simplex Method demands a considerable amount of “mathematical maturity” from the reader. The steps are given for all key computations, but there are relatively few intuitive justifications. A similar dichotomy can be seen in Chapters 2 and 10. Several matrix applications are developed in these chapters. In Section 2.7, the authors clearly demonstrate the role of the inverse matrix in the Cryptography application, but there is no hint of the role of the transpose matrices in the solution to the Least Squares problem. In Section 10.3, the entries of the fundamental matrix of an absorbing Markov chain are interpreted and several examples are given, but no intuition is given to support the conclusions. On the other hand, the distinctions between pure and mixed Game Theory strategies and the justifications for the expected payoff computations in Sections 10.4 – 10.6 are clear.

A nice feature of the book is the inclusion of examples that utilize graphing utilities and spreadsheets. The authors highlight relevant syntax details. In some chapters, mathematical questions from professional exams (CPA, CMA, and Actuary Exams) are reproduced. A glossary would have been helpful for some of these questions. There are several errors and omissions that mar the exposition. The definition of echelon form in Chapter 2 is imprecise in the sense that the authors do not consider the possibility that the first column could represent a free variable. The presentation of the Open Leontief Model reverses notation between p. 131 and p. 136. The inflation model on p. 278 actually computes depreciation by a given rate, not the purchasing power. Table 20 on p. 609 is presented without explanation, and the description of indirect proof on p. 611 is much too terse. The argument in Exercise 4 on p. 612 has no conclusion. Section C.5 is listed in the contents for Appendix C on p. 621, but it is missing from the book.

Because of some pedagogical features, graded exercise sets, and extensive coverage of applications, the book can be used successfully in finite mathematics courses. Overall, however, I consider the book to be mediocre because of the deficiencies outlined above.

 Jerry G. Ianni teaches at LaGuardia Community College.