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On Numbers and Games is a mathematics book by John Horton Conway. The book is a serious mathematics book, written by a preeminent mathematician, and is directed at other mathematicians. The material is, however, developed in a most playful and unpretentious manner and many chapters are accessible to non-mathematicians. Image File history File links Wiki_letter_w. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ...

The book is roughly divided into two parts: the first half (or Zeroth Part), on numbers, the second half (or First Part), on games. In the first part, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite (but critically important) variation of the Dedekind section. As such, the construction is rooted in axiomatic set theory, and is closely related to the Zermelo-Frankel axioms. Conway's use of the section is developed in greater detail in the article on surreal numbers. A number is an abstract entity that represents a count or measurement. ... Combinatorial game theory (CGT) is a mathematical theory that only studies two-player games which have a position which the players take turns changing in defined ways or moves to achieve a defined winning condition. ... In mathematics, an axiomatic theory is one based on axioms. ... In the mathematical field of set theory, there are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. ... The integers are commonly denoted by the above symbol. ... The word real has many different meanings. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... This article or section is in need of attention from an expert on the subject. ... Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ...

Conway then notes that, in this notation, the numbers in fact belong to a larger class, the 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, the map-coloring col and snort. The development includes their scoring, a review of Sprague–Grundy theory, and the inter-relationships to numbers, including their relationship to infinitesimals. In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... The feasible regions of linear programming are defined by a set of inequalities. ... For the socioeconomic meaning, see social inequality. ... Nim is a two-player mathematical game of strategy in which players take turns removing objects from distinct heaps. ... Hackenbush is a two-player partisan mathematical game that consists of several colored line segments connected to the ground. ... Several map-coloring games are studied in Combinatorial game theory. ... In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game is equivalent to a nimber. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...

The book was first published by Academic Press Inc in 1976, ISBN 0-12-186350-6, and re-released by AK Peters in 2000 (ISBN 1-56881-127-6). 1976 (MCMLXXVI) was a leap year starting on Thursday. ... This article is about the year 2000. ...

Synopsis

A game in the sense of Conway is a position in a contest between two players, Left and Right. Each player has a 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.^[1] 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 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), and is the first game we find that is not a number. This article is about sets in mathematics. ... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In combinatorial game theory, the zero game is the game where neither player has any legal options. ... Star, written as * or *1, is the value given to the combinatorial game {0 | 0}, where zero is the zero game. ...

All numbers are positive, negative, or zero, and we say that a game is positive if Left will win, negative if Right will win, or zero if the second player will win. Games that are not numbers have a fourth possibility: they may be fuzzy, meaning that the first player will win. * is a fuzzy game. A negative number is a number that is less than zero, such as −3. ... This article may be too technical for most readers to understand. ...

A more extensive introduction to On Numbers and Games is available online.^[2]

See also

• Winning Ways for your Mathematical Plays

Winning Ways for your Mathematical Plays (ISBN 1568811306) by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games. ...

References

1. ^ 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.
2. ^ Dierk Schleicher and Michael Stoll, An Introduction to Conway's Games and Numbers, Arxiv math.CO/0410026