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Numbers (Graduate Texts in Mathematics / Readings in Mathematics) (v. 123)

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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.

Part A is crammed with information on the real and complex numbers and the fundamental theorem of algebra with much historical background. There are also two odd chapters with all sorts of information on pi and on p-adic numbers (which has nothing to do with anything else in the book). In part B the authors free themselves from the constraints of classical number systems and study more or less number-like algebras. In particular, the privileged role of R,C,H,O is linked to the existence n-square identities and the possible dimensions of division algebras. Part C treats some selected foundational topics: non-standard analysis, Conway's "games" approach to the reals, set theory.

One may wish that this book was "a lively story about one thread of mathematics--the concept of 'number'-- ... organized into a historical narrative that leads the reader from ancient Egypt to the late twentieth century" (English edition editor's preface). But this is hardly the case. I suppose it takes the combined efforts of eight authors to produce such a garbled and disorganised account, with so many dead-end side tracks, of a topic with such extraordinary inherent continuity, both historical and logical. Also, as in so many other modern books, the authors are primarily interested in algebra and foundations, and their perception of history is tilted accordingly. Their fear of getting their hands dirty with classical analysis means that they can only mention, not prove, the transcendence of pi, for instance.