Editing On Numbers and Games

{{Short description|1976 mathematics book by John Conway}}
{{infobox book <!-- See Wikipedia:WikiProject_Novels or Wikipedia:WikiProject_Books -->
| name          = On Numbers and Games
| title_orig    = 
| translator    = 
| image         = On Numbers and Games.jpg
| caption       = First edition
| author        = [[John Horton Conway]]
| cover_artist  = 
| country       = [[United States]]
| language      = [[English language|English]]
| series        = 
| genre         = [[Mathematics]]
| publisher     = [[Academic Press, Inc.]]
| release_date  = 
| media_type    = Print
| pages         = 238 pp.
| isbn          = 0-12-186350-6
| isbn_note     = 
| dewey= 
| congress= 
| oclc= 
| preceded_by   = 
| followed_by   = 
}}

'''''On Numbers and Games''''' is a [[mathematics]] book by [[John Horton Conway]] first published in 1976.<ref>{{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}}</ref> 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 number]]s, in his [[List of Martin Gardner Mathematical Games columns|Mathematical Games column]] in ''Scientific American'' in September 1976.<ref>{{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}}</ref>

The book is roughly divided into two sections: the first half (or ''Zeroth Part''), on [[number]]s, the second half (or ''First Part''), on [[combinatorial game theory|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 [[integer]]s, [[Real number|real]]s, the [[countable infinity]], and entire towers of infinite [[ordinal number|ordinals]]. The object to which these axioms apply takes the form {L|R}, which can be interpreted as a specialized kind of [[set (mathematics)|set]]; a kind of two-sided set. By insisting that L<R, this two-sided set resembles the [[Dedekind cut]]. The resulting construction yields a [[field (mathematics)|field]], now called the [[surreal number]]s. The ordinals are embedded in this field. The construction is rooted in [[axiomatic set theory]], and is closely related to the [[Zermelo–Fraenkel axioms]]. In the original book, Conway simply refers to this field as "the numbers". The term "[[surreal numbers]]" is adopted later, at the suggestion of [[Donald Knuth]].

In the ''First Part'', Conway notes that, by dropping the constraint that L<R, the axioms still apply and the construction goes through, but the resulting objects can no longer be interpreted as numbers. They can be interpreted as the [[class (set theory)|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 to exploring a number of different (non-traditional, mathematically inspired) two-player games, such as [[nim]], [[hackenbush]], and the [[map-coloring games]] [[col (game)|col]] and [[Snort (game)|snort]]. The development includes their scoring, a review of the [[Sprague–Grundy theorem]], and the inter-relationships to numbers, including their relationship to [[infinitesimal]]s.

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 (mathematics)|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<R (with this holding vacuously true when L or R are the empty set), then the resulting class of objects can be interpreted as numbers, the [[surreal number]]s. The {L|R} notation then resembles the [[Dedekind cut]].

The ordinal <math>\omega</math> is built by [[transfinite induction]]. As with conventional ordinals, <math>\omega+1</math> can be defined. Thanks to the axiomatic definition of subtraction, <math>\omega-1</math> can also be coherently defined: it is strictly less than <math>\omega</math>, and obeys the "obvious" equality <math>(\omega-1)+1=\omega.</math> Yet, it is still larger than any [[natural number]].

The construction enables an entire zoo of peculiar numbers, the surreals, which form a [[field (mathematics)|field]].  Examples include <math>\omega/2</math>, <math>1/\omega</math>, <math>\sqrt{\omega}=\omega^{1/2}</math>, <math>\omega^{1/\omega}</math> and similar.

== First Part ... and Games ==
In the First Part, Conway abandons the constraint that L<R, and then interprets the form {L|R} as a two-player game: a position in a contest between two players, '''Left''' and '''Right'''.  Each player has a [[set (mathematics)|set]] of games called ''options'' to choose from in turn.  Games are written {L|R} where L is the set of '''Left's''' options and R is the set of '''Right's''' options.<ref>Alternatively, we often list the elements of the sets of options to save on braces.  This causes no confusion as long as we can tell whether a singleton option is a game or a set of games.</ref> At the start there are no games at all, so the [[empty set]] (i.e., the set with no members) is the only set of options we can provide to the players.  This defines the game {|}, which is called [[zero game|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|} is called 1, and the game {|0} is called -1.  The game {0|0} is called [[star (game theory)|* (star)]], and is the first game we find that is not a number.

All numbers are [[sign (mathematics)|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 game|fuzzy]], meaning that the first player has a winning strategy.  * is a fuzzy game.<ref>{{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 }}</ref>

==See also==
* ''[[Winning Ways for Your Mathematical Plays]]''

==References==
<references />

[[Category:1976 non-fiction books]]
[[Category:Combinatorial game theory]]
[[Category:Mathematics books]]
[[Category:Systems of set theory]]
[[Category:John Horton Conway]]
[[Category:Academic Press books]]