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Mathematical Problems and Proofs:Combinatorics, Number Theory, and Geometry

Branislav Kisacanin
Publication Date: 
Number of Pages: 
[Reviewed by
Carl D. Mueller
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When I first set out to read this book in preparation for this review, I was looking for a textbook for an undergraduate discrete mathematics course. It turns out that this book was not appropriate for the course I had in mind. But, to be fair, it was not written as an undergraduate textbook. In the words of the author:

This book is written for those who enjoy seeing mathematical formulas and ideas, interesting problems, and elegant solutions.

More specifically it is written for talented high-school students who are hungry for more mathematics and undergraduates who would like to see illustrations of abstract mathematical concepts and to learn a bit about their historical origin.

It is written with that hope that many readers will learn how to read mathematical literature in general.

There is no question that this book will be well received by the target audience. The book contains an engaging mix of topics and includes some topics that are often not contained in a high school or college mathematics curriculum, but that build a student's appreciation and love for mathematics. Some examples are: proofs of the Pythagorean theorem and a discussion of Pythagorean triples; a discussion of the arithmetic, geometric, and harmonic means and the relationships among them; and several approaches to the Towers of Hanoi problem.

The book is divided into 4 chapters and a set of Appendixes. There is also a Key to Symbols that will be particularly useful to those encountering notations for the first time. The chapters each cover one general subject area: Chapter 1 deals with Set Theory, Chapter 2 with Combinatorics, Chapter 3 treats Number Theory, and Chapter 4 focuses on Geometry. The four appendixes deal with Mathematical Induction, Important Mathematical Constants, Great Mathematicians, and the Greek Alphabet. Following those are a nice set of references and a thorough index.

Each of the chapters is divided into three or four sections of content followed by a selection of problems. The examples/problems include detailed solutions. A feature that I find particularly nice is the fact that more than one solution for a problem is often provided. Too often mathematics books leave students thinking that there is only one right way to solve a problem. Students often stumble onto a clever way of solving something and are left to wonder whether their technique is valid. (Unfortunately there are some teachers who actively discourage using any method other than the one being taught in the current section of the official class text.) This book makes it clear that there are many ways of solving some problems. By seeing several solutions the students will learn not only this fact but also learn additional techniques which will be useful in solving other problems. According to the author, the problems are designed to illustrate theorems and ideas and to develop the reader's problem-solving ability and sense for elegant solutions. I think that he has done a good job at selecting problems and providing solutions and explanations which succeed at this task. In order to pique the reader's interest in this book, I will describe some of the content in detail.

The chapter on Set Theory is brief (only 16 pages) but manages to pack in a lot of material without overwhelming the reader. In addition to a very complete set of definitions and properties, the author includes many examples. He proves that the square root of 2 is irrational and that the irrational numbers are uncountable. He discusses equivalence classes and touches on algebraic and transcendental numbers. In short, he covers a selection of topics that will grab hold of an eager student's interest.

The chapter on Combinatorics is considerably longer at 52 pages. Again the author has chosen topics which do an excellent job of covering the basics of combinatorics as well as capturing the interest of the curious reader. Topics range from basic counting problems to discussions of Euler's phi function, generating functions (including material on the Fibonacci numbers), and probability distributions used in statistical physics (Maxwell-Boltzmann statistics, Bose-Einstein statistics, and Fermi-Dirac statistics)! Quite a large range of topics, but, once again, dealt with in a way which does not overwhelm the reader.

The chapters on Number Theory (42 pages) and Geometry (43 pages) again cover the basics in a clear way and with a nice selection of supporting examples and problems. Once again the topics which go beyond the basics are chosen to reach out to the reader and show him or her what it is that makes mathematics fun and interesting.

I was particularly impressed with the Appendix on Mathematical Induction. At 22 pages, it is longer than the entire chapter on Set Theory and it includes numerous well chosen examples and problems.

There are some peculiarities to this book. For example, the pigeonhole principle is referred to as Dirichlet's principle. While this is an appropriate name for the principle, it is almost universally referred to (at least in my experience) as the pigeonhole principle. I would think that at the very least the student should be introduced to the common name. But apart from a few rather minor complaints like this one, the book is excellent. I would highly recommend it for mathematics departments, particularly at the high school level. This book belongs in the hands of eager high school and undergraduate level mathematics enthusiasts who will benefit from its range of topics and its large set of over 150 thoroughly solved examples and problems.

Carl D. Mueller ([email protected]) is Associate Professor of Mathematics at Georgia Southwestern State University in Americus, GA.

Set Theory. Combinatorics. Number Theory. Geometry. Appendices. Bibliography. Index.