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Title: Numbers (Graduate Texts in Mathematics/Readings in Mathematics Vol 123) by Heinz-Dieter Ebbinghaus, John H. Ewing ISBN: 0-387-97497-0 Publisher: Springer Verlag Pub. Date: February, 1991 Format: Paperback Volumes: 1 List Price(USD): $54.95

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Average Customer Rating: 5 (1 review)

Rating: 5
Summary: Very good introduction to the basics of numbers
Comment: This book is the best introduction to the basics of numbers (including all aspects) I know. Every chapter is composed with the same systematics. First a historical introduction is given describing which persons, when, and under which circumstances and ideas have discovered the respective number system. Then the mathematical definition of the number system is given followed by theorems (all of them being proved) and examples. This way one can better understand how the train of thoughts in number theory proceeded without loosing any mathematical accuracy in the presentation. Additionally it is the only book that covers in the same rigorous manner the whole field from the natural numbers to octonions.

The book consists of 14 chapters; the 14-th chapter is about set theory (the basic of all mathematics) and can be read (as suggested by the authors) as the beginning chapter if one wishes to set the mathematical foundation before beginning with the number systems. If, however, one wishes to start in the order number theory developed during history one should begin with chapter 1 and then proceed in the given order.

The first chapter defines the natural numbers using Peano axioms, integers and rational numbers. The real numbers follow in the second chapter; here all three possible definitions of the real numbers are given: Dedekind cuts, fundamental sequences and nesting of intervals. In the third chapter the complex numbers are presented in all their representations. The forth chapter formulates the fundamental theorem of algebra based on the complex numbers. The fifth chapter is devoted to the number pi whereas the sixth chapter describes the p-adic numbers where the prime numbers play a crucial role.

In the seventh chapter the hypercomplex numbers are being introduced. It begins with William Rowan Hamilton's quaternions as the first generalization of the complex numbers, describes all their representation and the anti-commutativity by the use of the commutator. The eigth chapter presents isomorphism theorems. The ninth chapter further generalizes the quaternions to octonions, defines the CAYLEY duplication process and underlines the anti-associativity of the octonions with the help of the associator. The tenth chapter presents composition algebras where the four division algebras (reals, complexes, quaternions, octonions) play a role. The amazing fact that only 4 division algebras exist is proved topologically in chapter eleven.

The last chapters are devoted to nonstandard numbers (chapter twelve) and CONWAY's definition of numbers via games (chapter thirteen). Chapter fourteen concludes (if not read as introduction) with set theory.

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