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.

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