# On Numbers and Games

CRC Press, Dec 11, 2000 - Mathematics - 256 pages
ONAG, as the book is commonly known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Originally written to define the relation between the theories of transfinite numbers and mathematical games, the resulting work is a mathematically sophisticated but eminently enjoyable guide to game theory. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers, that includes both real numbers and ordinal numbers. These surreal numbers are applied in the author's mathematical analysis of game strategies. The additions to the Second Edition present recent developments in the area of mathematical game theory, with a concentration on surreal numbers and the additive theory of partizan games.

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#### LibraryThing Review

User Review  - kevinashley - LibraryThing

I read the first edition of this book as an undergraduate student of mathematics and, like many of my peers, was astonished at the effortless way in which an entire new class of numbers was defined ... Read full review

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My Favourite Mathematical book. I first tried to read this in 8th grade, and got me into the realm of mathematics. I owe Conway my life's dream. Thank you for writing this.

### Contents

 The Class No is a Field 11 The Real and Ordinal Numbers 19 The Structure of the General Surreal Number 25 Algebra and Analysis of Numbers 35 Number Theory in the Land of Oz 41 The Curious Field On₂ 46 Appendix to Part Zero 60 FIRST PART AND GAMES 65
 Simplifying Games 105 Impartial Games and the Game of Nim 118 How to Lose when you Must 132 Animating Functions Welters Game and Hackenbush Unrestrained 149 How to Play Several Games at Once in a Dozen Different Ways 169 Ups Downs and Bynumbers 184 The Long and the Short and the Small 201 Epilogue 221

 Playing Several Games at Once 67 Some Games are Already Numbers 77 On Games and Numbers 93