On Numbers and Games
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On Numbers and Games is a mathematics book by John Horton Conway. The book is a serious mathematics book, written by a pre-eminent mathamtician, and is directed at other mathematicians; although its topic is throughly serious, it is developed in a most playful and unpretentious manner. Many chapters are accessible to non-mathematicians.
The book is roughly divided into two parts: the first half, on numbers, the second half, on games (in the sense of combinatorial game theory). In the first part, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite (but critically important) variation of the Dedekind section. As such, the construction is rooted in axiomatic set theory, and is closely related to the Zermelo-Frankel axioms. Conway's use of the section is developed in greater detail in the article on surreal numbers.
Next, Conwaythen notes that, in this notation, the numbers in fact belong to a larger class, the class of all two-player games. The axioms for greater than and less than are seen to be a natural ordering on games, corresponding to which of the two players may win. The remainder of the book is devoted exploring a number of different (non-traditional, mathematically inspired) two-player games, such as nim, hackenbush, the map-coloring col and snort. The development includes their scoring, a review of Sprague–Grundy theory, and the inter-relationships to numbers, including their relationship to infinitessimals.
The book was first published by Academic Press Inc in 1976, ISBN 0121863506, and re-released by AK Peters in 2000 (ISBN 1568811276).