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Adventures in Mathematics

Martin A. Moskowitz
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Don Vestal
, on

The adventures in this book occur in the areas of algebra and geometry: it covers the notion of numbers (rational, real, complex, algebraic), algebra (groups, finite fields, and linear algebra), and geometry (two and three dimensional geometry and topology). It's a short adventure, with the book just squeaking over 100 pages. Thus, in that time, not much of the mathematics is fleshed out. There are a handful of examples and exercises, but barely enough to get your feet (or brain) wet.

On the one hand, the book does provide a summary of some of the interesting tidbits of algebra and geometry. There is a decent description of group actions and a pretty good development of some of the basics of Euclidean, elliptic, and hyperbolic geometry. On the other hand, after reading the book, I felt somewhat bewildered. Why are these particular topics covered? What is the point? I felt like I just had the appetizer and was ready for the main course, only to be told, "Okay, here's the check." According to the preface, the purpose of the book is "to present mathematics as … a vast panorama of ideas, whose history reflects some of the most noble thoughts of mankind." I didn't see much of that, however, due to some typos, some imprecise mathematics (the definition of algebraic number, the definition of an exact sequence), and the lack of direction. Of course, I'm looking at this as a mathematician. Perhaps a high school student or undergraduate would get more out of it, or at least appreciate it more.

Donald L. Vestal is Associate Professor of Mathematics at Missouri Western State University. His interests include number theory, combinatorics, and a deep admiration for the crime-fighting efforts of the Aqua Teen Hunger Force. He can be reached at

* What is a Number?:
* The Positive Integers
* The Integers
* The Rational Numbers
* The Real Numbers
* The Integers Revisited
* The Complex Numbers
* Polynomials and Other Analogues of the Integers
* The Algebra on a Dial
* Groups, Finite Fields and Linear Algebra:
* Introduction to Groups and Their Actions
* Applications to Finite Fields and the Theory of Numbers
* Linear Algebra
* Two- and Three-Dimensional Geometry and Topology:
* The Euclidean Case
* Elliptic and Hyperbolic Geometry
* Some Two- and Three-Dimensional Topology