In mathematics, the proper class of nimbers is introduced in combinatorial game theory, where they arise as the sizes of nim heaps. It is the proper class of ordinals endowed with a new nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... 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. ... Combinatorial game theory (CGT) is a mathematical theory that studies a certain kind of game. ... Nim is a two-player game of strategy in which players take turns removing objects from heaps, one or more objects at a time but only from a single heap. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

It can be shown that every impartial game is equivalent to a nim heap of a certain size. Nimber addition can be used to calculate the size of a single heap equivalent to a collection of heaps. It is defined recursively by In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. ...

α + β = mex{α ′ + β : α ′ < α, α + β ′ : β ′ < β},

where for a set S of ordinals, mex(S) is defined to be the "minimum excluded ordinal", i.e. mex(S) is the smallest ordinal which is not an element of S. For finite ordinals, the nim sum is easily evaluated on computer by taking the exclusive-or of the corresponding numbers (whereby the numbers are given their binary expansions, and the binary expansion of x xor y is evaluated bit-wise). In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... mex: minimum excludant The m stands for minimum and the ex stands for excludant. ... The binary numeral system represents numeric values using two symbols, typically 0 and 1. ... A bit (abbreviated b) is the most basic information unit used in computing and information theory. ...

Nimber multiplication is defined recursively by

α β = mex{α ′ β + α β ′ − α ′ β ′ : α ′ < α, β ′ < β} = mex{α ′ β + α β ′ + α ′ β ′ : α ′ < α, β ′ < β}.

Except for the fact that nimbers form a proper class and not a set, the class of nimbers determines an algebraically closed field of characteristic 2. The nimber additive identity is the ordinal 0, and the nimber multiplicative identity is the ordinal 1. In keeping with the characteristic being 2, the nimber additive inverse of the ordinal α is α itself. The nimber multiplicative inverse of the nonzero ordinal α is given by 1/α = mex(S), where S is the smallest set of ordinals (nimbers) such that 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. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... The word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). ...

1. 0 is an element of S;
2. if 0 < α ′ < α and β ′ is an element of S, then [1 + (α ′ − α) β ′ ]/α ′ is also an element of S.

For all natural numbers n, the set of nimbers less than form the Galois field of order . In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

Just as in the case of nimber addition, there is a means of computing the nimber product of finite ordinals. This is determined by the rules that

1. The nimber product of distinct Fermat 2-powers (numbers of the form ) is equal to their ordinary product;
2. The nimber square of a Fermat 2-power x is equal to 3x/2 as evaluated under the ordinary multiplication of natural numbers.

The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal , where ω is the smallest infinite ordinal. It follows that as a nimber, is transcendental over the field. In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...

References

J.H. Conway, On Numbers and Games, Academic Press Inc. (London) Ltd., 1976 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. ...