On Numbers and GamesONAG, 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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Review: On Numbers and Games
User Review  David Hildebrand  GoodreadsConway's approach to foundations is frustratingly poorly ordered and I urge readers to look at the appendix to section 0 first. After skimming the book once it becomes extremely lucid and insightful if not haphazard. Truly a mathematical classic. Read full review
Review: On Numbers and Games
User Review  David Hildebrand  GoodreadsConway's approach to foundations is frustratingly poorly ordered and I urge readers to look at the appendix to section 0 first. After skimming the book once it becomes extremely lucid and insightful if not haphazard. Truly a mathematical classic. Read full review
Contents
The Class No is a Field  15 
The Real and Ordinal Numbers  23 
The Structure of the General Surreal Number  29 
Algebra and Analysis of Numbers  39 
Number Theory in the Land of Oz  45 
The Curious Field On₂  50 
Appendix to Part Zero  64 
FIRST PART AND GAMES  69 
Simplifying Games  109 
Impartial Games and the Game of Nim  122 
How to Lose when you Must  136 
Animating Functions Welters Game and Hackenbush Unrestrained  153 
How to Play Several Games at Once in a Dozen Different Ways  173 
Ups Downs and Bynumbers  188 
The Long and the Short and the Small  205 
Epilogue  225 
Common terms and phrases
assertion atomic weight Chapter Class component games compute conjunctive compound corresponding defined definition deleted diagram disjunctive compound disjunctive sum dominated dyadic rationals enumber edges Elwyn Berlekamp equal equation field finite number follows formula G and H game G generalised Grundy number Grundy's game Hackenbush heaps impartial games inductive inequalities infinite infinitesimal integer Kayles Left and Right Left option legal move misere play negative numbers Nimaddition Nimheap Nimsum nodes normal play Norton number x obtain ONAG option of G ordinal numbers ordinal sum P P P P P P partizan games positive numbers Proof proper Class properties prove real numbers replace reversible moves rule second player win short game signexpansion Simon Norton simpler simplest form simplest number small games square suppose Surreal Numbers suspense number THEOREM theory thermograph trominoes Welter function winning strategy zero