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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LibraryThing Review
User Review  kevinashley  LibraryThingI 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
Review: On Numbers and Games
User Review  Shayne  GoodreadsDon't know as I can say that I fully understood this book, but I am certainly done with it. I can see how it has earned such a high position in the canon of game literature. Read full review
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 
Appendix  225 
Index  231 
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 remote stars 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