On Numbers and Games
{{Short description|1976 mathematics book by John Conway}}
{{infobox book
| name = On Numbers and Games
| title_orig =
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| image = On Numbers and Games.jpg
| caption = First edition
| author = John Horton Conway
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| country = United States
| language = English
| series =
| genre = Mathematics
| publisher = Academic Press, Inc.
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| media_type = Print
| pages = 238 pp.
| isbn = 0-12-186350-6
| isbn_note =
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On Numbers and Games is a mathematics book by John Horton Conway first published in 1976.{{cite journal|author=Fraenkel, Aviezri S.|author-link=Aviezri Fraenkel|title=Review: On numbers and games, by J. H. Conway; and Surreal numbers, by D. E. Knuth|journal=Bull. Amer. Math. Soc.|year=1978|volume=84|issue=6|pages=1328–1336|url=https://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14564-9/S0002-9904-1978-14564-9.pdf|doi=10.1090/s0002-9904-1978-14564-9|doi-access=free}} The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpretentious manner and many chapters are accessible to non-mathematicians. Martin Gardner discussed the book at length, particularly Conway's construction of surreal numbers, in his Mathematical Games column in Scientific American in September 1976.{{cite magazine |first=Martin |last=Gardner |url=http://www.scientificamerican.com/article/mathematical-games-1976-09/ |title=Mathematical Games |date=September 1976 |magazine=Scientific American |volume=235 |issue=3}}
The book is roughly divided into two sections: the first half (or Zeroth Part), on numbers, the second half (or First Part), on games. In the Zeroth Part, Conway provides axioms for arithmetic: addition, subtraction, multiplication, division and inequality. This allows an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals. The object to which these axioms apply takes the form {L|R}, which can be interpreted as a specialized kind of set; a kind of two-sided set. By insisting that L
In the First Part, Conway notes that, by dropping the constraint that L
The book was first published by Academic Press in 1976, {{isbn|0-12-186350-6}}, and a second edition was released by A K Peters in 2001 ({{isbn|1-56881-127-6}}).
Zeroth Part ... On Numbers
{{main|Surreal numbers}}
In the Zeroth Part, Chapter 0, Conway introduces a specialized form of set notation, having the form {L|R}, where L and R are again of this form, built recursively, terminating in {|}, which is to be read as an analog of the empty set. Given this object, axiomatic definitions for addition, subtraction, multiplication, division and inequality may be given. As long as one insists that L
The ordinal is built by transfinite induction. As with conventional ordinals, can be defined. Thanks to the axiomatic definition of subtraction, can also be coherently defined: it is strictly less than , and obeys the "obvious" equality Yet, it is still larger than any natural number.
The construction enables an entire zoo of peculiar numbers, the surreals, which form a field. Examples include , , , and similar.
First Part ... and Games
In the First Part, Conway abandons the constraint that L
is called 1, and the game {|0} is called -1. The game {0|0} is called * (star), and is the first game we find that is not a number.}, which is called 0. We consider a player who must play a turn but has no options to have lost the game. Given this game 0 there are now two possible sets of options, the empty set and the set whose only element is zero. The game {0
All numbers are positive, negative, or zero, and we say that a game is positive if Left has a winning strategy, negative if Right has a winning strategy, or zero if the second player has a winning strategy. Games that are not numbers have a fourth possibility: they may be fuzzy, meaning that the first player has a winning strategy. * is a fuzzy game.{{cite journal |first1=Dierk |last1=Schleicher |first2=Michael |last2=Stoll |title=An Introduction to Conway's Games and Numbers |journal=Moscow Math Journal |volume=6 |number=2 |year=2006 |pages=359–388|doi=10.17323/1609-4514-2006-6-2-359-388 |arxiv=math.CO/0410026 }}