On Numbers and Games
Encyclopedia
On Numbers and Games is a mathematics
book by John Horton Conway
. The book is a serious mathematics book, written by a pre-eminent 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.
The book is roughly divided into two sections: the first half (or Zeroth Part), on number
s, the second half (or First Part), on games
. In the first section, Conway provides an axiomatic
construction of numbers and ordinal arithmetic
, namely, the integer
s, real
s, 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 cut
. As such, the construction is rooted in axiomatic set theory, and is closely related to the Zermelo–Fraenkel axioms. Conway's use of the section is developed in greater detail in the Wikipedia article on surreal number
s.
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
, and the map-coloring games
col and snort. The development includes their scoring, a review of Sprague–Grundy theory, and the inter-relationships to numbers, including their relationship to infinitesimal
s.
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).
(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.
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 more extensive introduction to On Numbers and Games is available online.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
book by John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
. The book is a serious mathematics book, written by a pre-eminent 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.
The book is roughly divided into two sections: the first half (or Zeroth Part), on number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s, the second half (or First Part), on games
Combinatorial game theory
Combinatorial game theory is a branch of applied mathematics and theoretical computer science that studies sequential games with perfect information, that is, two-player games which have a position in which the players take turns changing in defined ways or moves to achieve a defined winning...
. In the first section, Conway provides an axiomatic
Axiomatic
* In mathematics, an "axiomatic" theory is one based on axioms* Axiomatic , a collection of short stories by Greg Egan* Axiomatic , a 2005 album by Australian band Taxiride...
construction of numbers and ordinal arithmetic
Ordinal arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the...
, namely, the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s, real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, the countable infinity, and entire towers of infinite ordinals
Ordinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
, using a notation that is essentially an almost trite (but critically important) variation of the Dedekind cut
Dedekind cut
In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element....
. As such, the construction is rooted in axiomatic set theory, and is closely related to the Zermelo–Fraenkel axioms. Conway's use of the section is developed in greater detail in the Wikipedia article on surreal number
Surreal number
In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number...
s.
Conway then notes that, in this notation, the numbers in fact belong to a larger class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
, 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
Nim
Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap....
, hackenbush
Hackenbush
Hackenbush is a two-player mathematical game that may be played on any configuration of colored line segments connected to one another by their endpoints and to the ground...
, and the map-coloring games
Map-coloring games
Several map-coloring games are studied in combinatorial game theory. The general idea is that we are given a map with regions drawn in but with not all the regions colored. Two players, Left and Right, take turns coloring in one uncolored region per turn, subject to various constraints...
col and snort. The development includes their scoring, a review of Sprague–Grundy theory, and the inter-relationships to numbers, including their relationship to infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s.
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).
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. At the start there are no games at all, so the empty setEmpty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
(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
Zero game
In combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, the first player automatically loses, and it is a second-player win. The zero game has a Sprague–Grundy value of zero...
. 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.
All numbers are positive, negative, or zero
Sign (mathematics)
In mathematics, the word sign refers to the property of being positive or negative. Every nonzero real number is either positive or negative, and therefore has a sign. Zero itself is signless, although in some contexts it makes sense to consider a signed 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
Fuzzy game
In combinatorial game theory, a fuzzy game is a game which is incomparable with the zero game: it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right; nor equal to 0 which would be a win for the second player to move...
, meaning that the first player will win. * is a fuzzy game.
A more extensive introduction to On Numbers and Games is available online.