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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the surreal number system is a
totally ordered In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
containing the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s as well as infinite and infinitesimal numbers, respectively larger or smaller in
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s, the
Levi-Civita field In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member a can be constructed as a formal series of the form ...
, the
superreal number In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theo ...
s (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the maximal class surreal field.


History of the concept

Research on the Go endgame by
John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
led to the original definition and construction of the surreal numbers. Conway's construction was introduced in
Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
's 1974 book ''Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness''. In his book, which takes the form of a dialogue, Knuth coined the term ''surreal numbers'' for what Conway had called simply ''numbers''. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book ''
On Numbers and Games ''On Numbers and Games'' is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpre ...
''. A separate route to defining the surreals began in 1907, when Hans Hahn introduced Hahn series as a generalization of
formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
, and Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962 Norman Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α, and in 1987 he showed that taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers. If the surreals are considered as 'just' a proper class sized real closed field, Alling's 1962 paper handles the case of
strongly inaccessible In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of f ...
cardinals which can naturally be considered as proper classes by cutting off the cumulative hierarchy of the universe one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense. There is an important additional field structure on the surreals that isn't visible through this lens however, namely the notion of a 'birthday' and the corresponding natural description of the surreals as the result of a cut-filling process along their birthdays given by Conway. This additional structure has become fundamental to a modern understanding of the surreal numbers, and Conway is thus given credit for discovering the surreals as we know them today—Alling himself gives Conway full credit in a 1985 paper preceding his book on the subject.


Description

In the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers ''a'' and ''b'', or . (Both may hold, in which case ''a'' and ''b'' are equivalent and denote the same number.) Each number is formed from an ordered pair of subsets of numbers already constructed: given subsets ''L'' and ''R'' of numbers such that all the members of ''L'' are strictly less than all the members of ''R'', then the pair represents a number intermediate in value between all the members of ''L'' and all the members of ''R''. Different subsets may end up defining the same number: and may define the same number even if ''L'' ≠ ''L′'' and ''R'' ≠ ''R′''. (A similar phenomenon occurs when
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
are defined as quotients of integers: and are different representations of the same rational number.) So strictly speaking, the surreal numbers are
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of representations of form that designate the same number. In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: . This representation, where ''L'' and ''R'' are both empty, is called 0. Subsequent stages yield forms like : = 1 : = 2 : = 3 and : = −1 : = −2 : = −3 The integers are thus contained within the surreal numbers. (The above identities are definitions, in the sense that the right-hand side is a name for the left-hand side. That the names are actually appropriate will be evident when the arithmetic operations on surreal numbers are defined, as in the section below). Similarly, representations such as : = : = : = arise, so that the
dyadic rational In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in compute ...
s (rational numbers whose denominators are powers of 2) are contained within the surreal numbers. After an infinite number of stages, infinite subsets become available, so that any
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
''a'' can be represented by , where ''La'' is the set of all dyadic rationals less than ''a'' and ''Ra'' is the set of all dyadic rationals greater than ''a'' (reminiscent of a
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the r ...
). Thus the real numbers are also embedded within the surreals. There are also representations like : = ω : = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth.


Construction

Surreal numbers are constructed inductively as
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of pairs of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set. The construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.


Forms

A ''form'' is a pair of sets of surreal numbers, called its ''left set'' and its ''right set''. A form with left set ''L'' and right set ''R'' is written . When ''L'' and ''R'' are given as lists of elements, the braces around them are omitted. Either or both of the left and right set of a form may be the empty set. The form with both left and right set empty is also written .


Numeric forms and their equivalence classes

Construction Rule :A form is ''numeric'' if the intersection of ''L'' and ''R'' is the empty set and each element of ''R'' is greater than every element of ''L'', according to the order relation ≤ given by the comparison rule below. The numeric forms are placed in equivalence classes; each such equivalence class is a ''surreal number''. The elements of the left and right set of a form are drawn from the universe of the surreal numbers (not of ''forms'', but of their ''equivalence classes''). Equivalence Rule : Two numeric forms ''x'' and ''y'' are forms of the same number (lie in the same equivalence class) if and only if both ''x'' ≤ ''y'' and ''y'' ≤ ''x''. An ordering relationship must be antisymmetric, i.e., it must have the property that ''x'' = ''y'' (i. e., ''x'' ≤ ''y'' and ''y'' ≤ ''x'' are both true) only when ''x'' and ''y'' are the same object. This is not the case for surreal number ''forms'', but is true by construction for surreal ''numbers'' (equivalence classes). The equivalence class containing is labeled 0; in other words, is a form of the surreal number 0.


Order

The recursive definition of surreal numbers is completed by defining comparison: Given numeric forms ''x'' = and ''y'' = , ''x'' ≤ ''y'' if and only if both: *There is no ''xL'' ∈ ''XL'' such that ''y'' ≤ ''xL''. That is, every element in the left part of ''x'' is strictly smaller than ''y''. *There is no ''yR'' ∈ ''YR'' such that ''yR'' ≤ ''x''. That is, every element in the right part of ''y'' is strictly larger than ''x''. Surreal numbers can be compared to each other (or to numeric forms) by choosing a numeric form from its equivalence class to represent each surreal number.


Induction

This group of definitions is recursive, and requires some form of
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable via ''finite induction'' are the dyadic fractions; a wider universe is reachable given some form of
transfinite induction Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC. Induction by cases Let P(\alpha) be a property defined for ...
.


Induction rule

* There is a generation ''S''0 = , in which 0 consists of the single form . * Given any
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
''n'', the generation ''S''''n'' is the set of all surreal numbers that are generated by the construction rule from subsets of \cup_ S_i. The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no ''Si'' with ''i'' < 0, the expression \cup_ S_i is the empty set; the only subset of the empty set is the empty set, and therefore ''S''0 consists of a single surreal form lying in a single equivalence class 0. For every finite ordinal number ''n'', ''Sn'' is
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...
ed by the ordering induced by the comparison rule on the surreal numbers. The first iteration of the induction rule produces the three numeric forms < < (the form is non-numeric because 0≤0). The equivalence class containing is labeled 1 and the equivalence class containing is labeled −1. These three labels have a special significance in the axioms that define a ring; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels. For every ''i'' < ''n'', since every valid form in ''S''''i'' is also a valid form in ''S''''n'', all of the numbers in ''Si'' also appear in ''Sn'' (as supersets of their representation in ''Si''). (The set union expression appears in our construction rule, rather than the simpler form ''S''''n''−1, so that the definition also makes sense when ''n'' is a limit ordinal.) Numbers in ''Sn'' that are a superset of some number in ''Si'' are said to have been ''inherited'' from generation ''i''. The smallest value of α for which a given surreal number appears in ''S''α is called its ''birthday''. For example, the birthday of 0 is 0, and the birthday of −1 is 1. A second iteration of the construction rule yields the following ordering of equivalence classes: : = = = : < = : < = : < = = = : < = : < = : < = = = Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow: # ''S''2 contains four new surreal numbers. Two contain extremal forms: contains all numbers from previous generations in its right set, and contains all numbers from previous generations in its left set. The others have a form that partitions all numbers from previous generations into two non-empty sets. # Every surreal number ''x'' that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers ''other than'' ''x'' from previous generations into a left set (all numbers less than ''x'') and a right set (all numbers greater than ''x''). # The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set. The informal interpretations of and are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of and are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled and −. These labels will also be justified by the rules for surreal addition and multiplication below. The equivalence classes at each stage ''n'' of induction may be characterized by their ''n''-''complete forms'' (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains ''every'' number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels: : −2 < −1 < − < 0 < < 1 < 2. The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number is therefore equivalent to ; one can establish that these are forms of 3 by using the ''birthday property'', which is a consequence of the rules above.


Birthday property

A form ''x'' = occurring in generation ''n'' represents a number inherited from an earlier generation ''i'' < ''n'' if and only if there is some number in ''Si'' that is greater than all elements of ''L'' and less than all elements of the ''R''. (In other words, if ''L'' and ''R'' are already separated by a number created at an earlier stage, then ''x'' does not represent a new number but one already constructed.) If ''x'' represents a number from any generation earlier than ''n'', there is a least such generation ''i'', and exactly one number ''c'' with this least ''i'' as its birthday that lies between ''L'' and ''R''; ''x'' is a form of this ''c''. In other words, it lies in the equivalence class in ''Sn'' that is a superset of the representation of ''c'' in generation ''i''.


Arithmetic

The addition, negation (additive inverse), and multiplication of surreal number ''forms'' ''x'' = and ''y'' = are defined by three recursive formulas.


Negation

Negation of a given number ''x'' = is defined by :-x = - \ = \, where the negation of a set ''S'' of numbers is given by the set of the negated elements of ''S'': :-S = \. This formula involves the negation of the surreal ''numbers'' appearing in the left and right sets of ''x'', which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form. This only makes sense if the result is the same, irrespective of the choice of form of the operand. This can be proved inductively using the fact that the numbers occurring in ''XL'' and ''XR'' are drawn from generations earlier than that in which the form ''x'' first occurs, and observing the special case: :-0 = - \ = \ = 0.


Addition

The definition of addition is also a recursive formula: : x + y = \ + \ = \, where :X + y = \ , x + Y = \. This formula involves sums of one of the original operands and a surreal number drawn from the left or right set of the other. It can be proved inductively with the special cases: : 0 + 0 = + = = 0 : ''x'' + 0 = x + = = = ''x'' : 0 + ''y'' = + ''y'' = = = ''y'' For example: : + = + = = , which by the birthday property is a form of 1. This justifies the label used in the previous section.


Multiplication

Multiplication can be defined recursively as well, beginning from the special cases involving 0, the
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
1, and its additive inverse -1: :\begin xy & = \ \ \\ & = \left\ \\ \end The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression X_R y + x Y_R - X_R Y_R that appears in the left set of the product of ''x'' and ''y''. This is understood as the set of numbers generated by picking all possible combinations of members of X_R and Y_R, and substituting them into the expression. For example, to show that the square of is : :\begin \frac \cdot \frac &= \left\ \cdot \left\\\ &= \left\ \\ &= \left\ = \left\ = \frac. \end


Division

The definition of division is done in terms of the reciprocal and multiplication: :\frac xy = x \left( \frac 1y \right) where :\frac 1y = \left\ for positive ''y''. Only positive ''yL'' are permitted in the formula, with any nonpositive terms being ignored (and ''yR'' are always positive). This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets of ''y'', but also recursion in that the members of the left and right sets of itself. 0 is always a member of the left set of , and that can be used to find more terms in a recursive fashion. For example, if ''y'' = 3 = , then we know a left term of will be 0. This in turn means = is a right term. This means :\frac2=\frac14 is a left term. This means :\frac2 = \frac 38 will be a right term. Continuing, this gives :\frac13 = \left\ For negative ''y'', is given by :\frac1y=-\left(\frac1\right) If ''y'' = 0, then is undefined.


Consistency

It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that: * Addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday ''n'' will eventually be expressed entirely in terms of operations on numbers with birthdays less than ''n''; * Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday ''n'' will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than ''n''; * As long as the operands are well-defined surreal number forms (each element of the left set is less than each element of the right set), the results are again well-defined surreal number forms; * The operations can be extended to ''numbers'' (equivalence classes of forms): the result of negating ''x'' or adding or multiplying ''x'' and ''y'' will represent the same number regardless of the choice of form of ''x'' and ''y''; and * These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a field, with additive identity 0 = and multiplicative identity 1 = . With these rules one can now verify that the numbers found in the first few generations were properly labeled. The construction rule is repeated to obtain more generations of surreals: : ''S0'' = : ''S1'' = : ''S2'' = : ''S3'' = : ''S4'' =


Arithmetic closure

For each
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
(finite ordinal) ''n'', all numbers generated in ''Sn'' are dyadic fractions, i.e., can be written as an irreducible fraction , where ''a'' and ''b'' are
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s and 0 ≤ ''b'' < ''n''. The set of all surreal numbers that are generated in some ''Sn'' for finite ''n'' may be denoted as ''S'' = \cup_ S_n. One may form the three classes :\begin S_ &= \ \\ S_ &= \ \\ S_ &= \ \end of which ''S'' is the union. No individual ''Sn'' is closed under addition and multiplication (except ''S''0), but ''S'' is; it is the subring of the rationals consisting of all dyadic fractions. There are infinite ordinal numbers β for which the set of surreal numbers with birthday less than β is closed under the different arithmetic operations. For any ordinal α, the set of surreal numbers with birthday less than β = ωα (using powers of ω) is closed under addition and forms a group; for birthday less than ωωα it is closed under multiplication and forms a ring; and for birthday less than an (ordinal) epsilon number εα it is closed under multiplicative inverse and forms a field. The latter sets are also closed under the exponential function as defined by Kruskal and Gonshor. However, it is always possible to construct a surreal number that is greater than any member of a set of surreals (by including the set on the left side of the constructor) and thus the collection of surreal numbers is a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
. With their ordering and algebraic operations they constitute an
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
, with the caveat that they do not form a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
. In fact it is the biggest ordered field, in that every ordered field is a subfield of the surreal numbers. The class of all surreal numbers is denoted by the symbol \mathbb.


Infinity

Define ''S''ω as the set of all surreal numbers generated by the construction rule from subsets of ''S''. (This is the same inductive step as before, since the ordinal number ω is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can only be performed in a set theory that allows such a union.) A unique infinitely large positive number occurs in ''S''ω: : \omega = \ = \. ''S''ω also contains objects that can be identified as the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s. For example, the ω-complete form of the fraction is given by: : \tfrac = \. The product of this form of with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1. Not only do all the rest of the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s appear in ''S''ω; the remaining finite
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s do too. For example, : \pi = \. The only infinities in ''S''ω are ω and −ω; but there are other non-real numbers in ''S''ω among the reals. Consider the smallest positive number in ''S''ω: : \varepsilon = \ = \ = \. This number is larger than zero but less than all positive dyadic fractions. It is therefore an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
number, often labeled ε. The ω-complete form of ε (respectively -ε) is the same as the ω-complete form of 0, except that 0 is included in the left (respectively right) set. The only "pure" infinitesimals in ''S''ω are ε and its additive inverse -ε; adding them to any dyadic fraction ''y'' produces the numbers ''y'' ± ε, which also lie in ''S''ω. One can determine the relationship between ω and ε by multiplying particular forms of them to obtain: : ω · ε = . This expression is only well-defined in a set theory which permits transfinite induction up to ''S''ω2. In such a system, one can demonstrate that all the elements of the left set of ω''S''ω·''S''ωε are positive infinitesimals and all the elements of the right set are positive infinities, and therefore ω''S''ω·''S''ωε is the oldest positive finite number, 1. Consequently, = ω. Some authors systematically use ω−1 in place of the symbol ε.


Contents of ''S''ω

Given any ''x'' = in ''S''ω, exactly one of the following is true: * ''L'' and ''R'' are both empty, in which case ''x'' = 0; * ''R'' is empty and some integer ''n''≥0 is greater than every element of ''L'', in which case ''x'' equals the smallest such integer ''n''; * ''R'' is empty and no integer ''n'' is greater than every element of ''L'', in which case ''x'' equals +ω; * ''L'' is empty and some integer ''n''≤0 is less than every element of ''R'', in which case ''x'' equals the largest such integer ''n''; * ''L'' is empty and no integer ''n'' is less than every element of ''R'', in which case ''x'' equals −ω; * ''L'' and ''R'' are both non-empty, and: ** Some dyadic fraction ''y'' is "strictly between" ''L'' and ''R'' (greater than all elements of ''L'' and less than all elements of ''R''), in which case ''x'' equals the oldest such dyadic fraction ''y''; ** No dyadic fraction ''y'' lies strictly between ''L'' and ''R'', but some dyadic fraction y \in L is greater than or equal to all elements of ''L'' and less than all elements of ''R'', in which case ''x'' equals ''y'' + ε; ** No dyadic fraction ''y'' lies strictly between ''L'' and ''R'', but some dyadic fraction y \in R is greater than all elements of ''L'' and less than or equal to all elements of ''R'', in which case ''x'' equals ''y'' − ε; ** Every dyadic fraction is either greater than some element of ''R'' or less than some element of ''L'', in which case ''x'' is some real number that has no representation as a dyadic fraction. ''S''ω is not an algebraic field, because it is not closed under arithmetic operations; consider ω+1, whose form :\omega + 1 = \ + \ = \ does not lie in any number in ''S''ω. The maximal subset of ''S''ω that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities ±ω, the infinitesimals ±ε, and the infinitesimal neighbors ''y'' ± ε of each nonzero dyadic fraction ''y''. This construction of the real numbers differs from the
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the r ...
s of standard analysis in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in ''S''ω with its forms in previous generations. (The ω-complete forms of real elements of ''S''ω are in one-to-one correspondence with the reals obtained by Dedekind cuts, under the proviso that Dedekind reals corresponding to rational numbers are represented by the form in which the cut point is omitted from both left and right sets.) The rationals are not an identifiable stage in the surreal construction; they are merely the subset ''Q'' of ''S''ω containing all elements ''x'' such that ''x'' ''b'' = ''a'' for some ''a'' and some nonzero ''b'', both drawn from ''S''. By demonstrating that ''Q'' is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element of ''Q'' is reachable from ''S'' by a finite series (no longer than two, actually) of arithmetic operations ''including multiplicative inversion'', one can show that ''Q'' is strictly smaller than the subset of ''S''ω identified with the reals. The set ''S''ω has the same
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
as the real numbers ''R''. This can be demonstrated by exhibiting surjective mappings from ''S''ω to the closed unit interval ''I'' of ''R'' and vice versa. Mapping ''S''ω onto ''I'' is routine; map numbers less than or equal to ε (including −ω) to 0, numbers greater than or equal to 1 − ε (including ω) to 1, and numbers between ε and 1 − ε to their equivalent in ''I'' (mapping the infinitesimal neighbors ''y''±ε of each dyadic fraction ''y'', along with ''y'' itself, to ''y''). To map ''I'' onto ''S''ω, map the (open) central third (, ) of ''I'' onto = 0; the central third (, ) of the upper third to = 1; and so forth. This maps a nonempty open interval of ''I'' onto each element of ''S'', monotonically. The residue of ''I'' consists of the
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
''2''ω, each point of which is uniquely identified by a partition of the central-third intervals into left and right sets, corresponding precisely to a form in ''S''ω. This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday ω.


Transfinite induction

Continuing to perform transfinite induction beyond ''S''ω produces more ordinal numbers α, each represented as the largest surreal number having birthday α. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is ω+1 = . There is another positive infinite number in generation ω+1: : ω − 1 = . The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled ω − 1 on the basis that it coincides with the sum of and . Similarly, there are two new infinitesimal numbers in generation ω + 1: : 2ε = ε + ε = and : = ε · = . At a later stage of transfinite induction, there is a number larger than ω + ''k'' for all natural numbers ''k'': : 2ω = ω + ω = This number may be labeled ω + ω both because its birthday is ω + ω (the first ordinal number not reachable from ω by the successor operation) and because it coincides with the surreal sum of ω and ω; it may also be labeled 2ω because it coincides with the product of and . It is the second limit ordinal; reaching it from ω via the construction step requires a transfinite induction on :\bigcup_ S_ This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required. Note that the ''conventional'' addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals 1 + ω equals ω, but the surreal sum is commutative and produces 1 + ω = ω + 1 > ω. The addition and multiplication of the surreal numbers associated with ordinals coincides with the natural sum and natural product of ordinals. Just as 2ω is bigger than ω + ''n'' for any natural number ''n'', there is a surreal number that is infinite but smaller than ω − ''n'' for any natural number ''n''. That is, is defined by : = where on the right hand side the notation ''x'' − ''Y'' is used to mean . It can be identified as the product of ω and the form of . The birthday of is the limit ordinal ω2.


Powers of ω and the Conway normal form

To classify the "orders" of infinite and infinitesimal surreal numbers, also known as archimedean classes, Conway associated to each surreal number ''x'' the surreal number * ω''x'' = , where ''r'' and ''s'' range over the positive real numbers. If ''x'' < ''y'' then ω''y'' is "infinitely greater" than ω''x'', in that it is greater than ''r'' ω''x'' for all real numbers ''r''. Powers of ω also satisfy the conditions * ω''x'' ω''y'' = ω''x''+''y'', * ω−''x'' = , so they behave the way one would expect powers to behave. Each power of ω also has the redeeming feature of being the ''simplest'' surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal number ''x'' there will always exist some positive real number ''r'' and some surreal number ''y'' so that ''x'' − ''r''ω''y'' is "infinitely smaller" than ''x''. The exponent ''y'' is the "base ω logarithm" of ''x'', defined on the positive surreals; it can be demonstrated that logω maps the positive surreals onto the surreals and that :logω(''xy'') = logω(''x'') + logω(''y''). This gets extended by transfinite induction so that every surreal number has a "normal form" analogous to the
Cantor normal form 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 ...
for ordinal numbers. This is the Conway normal form: Every surreal number ''x'' may be uniquely written as : ''x'' = ''r''0ω''y''0 + ''r''1ω''y''1 + ..., where every ''r''α is a nonzero real number and the ''y''αs form a strictly decreasing sequence of surreal numbers. This "sum", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number. (Zero corresponds of course to the case of an empty sequence, and is the only surreal number with no leading exponent.) Looked at in this manner, the surreal numbers resemble a power series field, except that the decreasing sequences of exponents must be bounded in length by an ordinal and are not allowed to be as long as the class of ordinals. This is the basis for the formulation of the surreal numbers as a Hahn series.


Gaps and continuity

In contrast to the real numbers, a (proper) subset of the surreal numbers does not have a least upper (or lower) bound unless it has a maximal (minimal) element. Conway defines a gap as such that every element of ''L'' is less than every element of ''R'', and ''L'' ∪ ''R'' = \mathbb; this is not a number because at least one of the sides is a proper class. Though similar, gaps are not quite the same as
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the r ...
s, but we can still talk about a completion \mathbb_\mathfrak of the surreal numbers with the natural ordering which is a (proper class-sized)
linear continuum In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set ''S'' of more than one element that is densely ordered, i.e., between any t ...
. For instance there is no least positive infinite surreal, but the gap :∞ = is greater than all real numbers and less than all positive infinite surreals, and is thus the least upper bound of the reals in \mathbb_\mathfrak. Similarly the gap \mathbb = is larger than all surreal numbers. (This is an
esoteric Western esotericism, also known as esotericism, esoterism, and sometimes the Western mystery tradition, is a term scholars use to categorise a wide range of loosely related ideas and movements that developed within Western society. These ideas ...
pun A pun, also known as paronomasia, is a form of word play that exploits multiple meanings of a term, or of similar-sounding words, for an intended humorous or rhetorical effect. These ambiguities can arise from the intentional use of homophoni ...
: In the general construction of ordinals, α "is" the set of ordinals smaller than α, and we can use this equivalence to write in the surreals; \mathbb denotes the class of ordinal numbers, and because \mathbb is cofinal in \mathbb we have by extension.) With a bit of set-theoretic care, \mathbb can be equipped with a topology where the
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s are unions of open intervals (indexed by proper sets) and continuous functions can be defined. An equivalent of
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
s can be defined as well, although they have to be indexed by the class of ordinals; these will always converge, but the limit may be either a number or a gap that can be expressed as :\sum_ r_\alpha \omega^ with ''a''α decreasing and having no lower bound in \mathbb. (All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as ∞ and \mathbb).


Exponential function

Based on unpublished work by Kruskal, a construction (by transfinite induction) that extends the real
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
exp(''x'') (with base ''e'') to the surreals was carried through by Gonshor.


Other exponentials

The powers of ω function is also an exponential function, but does not have the properties desired for an extension of the function on the reals. It will, however, be needed in the development of the base-''e'' exponential, and it is this function that is meant whenever the notation ω''x'' is used in the following. When ''y'' is a dyadic fraction, the
power function Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
, may be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively. Its values are completely determined by the basic relation , and where defined it necessarily agrees with any other
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
that can exist.


Basic induction

The induction steps for the surreal exponential are based on the series expansion for the real exponential, :\exp x = \sum_ \frac more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For ''x'' positive these are denoted 'x''sub>''n'' and include all partial sums; for ''x'' negative but finite, 'x''sub>2''n''+1 denotes the odd steps in the series starting from the first one with a positive real part (which always exists). For ''x'' negative infinite the odd-numbered partial sums are strictly decreasing and the 'x''sub>2''n''+1 notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction. The relations that hold for real are then : and : and this can be extended to the surreals with the definition : This is well-defined for all surreal arguments (the value exists and does not depend on the choice of ''z''''L'' and ''z''''R'').


Results

Using this definition, the following holds: * exp is a strictly increasing positive function, * exp satisfies * exp is a surjection (onto \mathbb_+) and has a well-defined inverse, * exp coincides with the usual exponential function on the reals (and thus ) * For ''x'' infinitesimal, the value of the formal power series (
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
) of exp is well defined and coincides with the inductive definition ** When ''x'' is given in Conway normal form, the set of exponents in the result is well-ordered and the coefficients are finite sums, directly giving the normal form of the result (which has a leading 1) ** Similarly, for ''x'' infinitesimally close to 1, log ''x'' is given by power series expansion of * For positive infinite ''x'', exp ''x'' is infinite as well ** If ''x'' has the form ωα (α > 0), exp ''x'' has the form ωωβ where β is a strictly increasing function of α. In fact there is an inductively defined bijection whose inverse can also be defined inductively ** If ''x'' is "pure infinite" with normal form where all , then ** Similarly, for , the inverse is given by * Any surreal number can be written as the sum of a pure infinite, a real and an infinitesimal part, and the exponential is the product of the partial results given above ** The normal form can be written out by multiplying the infinite part (a single power of ω) and the real exponential into the power series resulting from the infinitesimal ** Conversely, dividing out the leading term of the normal form will bring any surreal number into the form , for , where each factor has a form for which a way of calculating the logarithm has been given above; the sum is then the general logarithm *** While there is no general inductive definition of log (unlike for exp), the partial results are given in terms of such definitions. In this way, the logarithm can be calculated explicitly, without reference to the fact that it's the inverse of the exponential. * The exponential function is much greater than any finite power ** For any positive infinite ''x'' and any finite ''n'', exp(''x'')/''x''''n'' is infinite ** For any integer ''n'' and surreal ''x'' > ''n''2, exp(''x'') > ''x''''n''. This stronger constraint is one of the Ressayre axioms for the real exponential field * exp satisfies all the Ressayre axioms for the real exponential field ** The surreals with exponential is an elementary extension of the real exponential field ** For εβ an ordinal epsilon number, the set of surreal numbers with birthday less than εβ constitute a field that is closed under exponentials, and is likewise an elementary extension of the real exponential field


Examples

The surreal exponential is essentially given by its behaviour on positive powers of ω, i.e., the function ''g(a)'', combined with well-known behaviour on finite numbers. Only examples of the former will be given. In addition, holds for a large part of its range, for instance for any finite number with positive real part and any infinite number that is less than some iterated power of ω ( for some number of levels). * exp ω = ωω * exp ω1/ω = ω and log ω = ω1/ω * exp (ω · log ω) = exp (ω · ω1/ω) = ωω(1 + 1/ω) ** This shows that the "power of ω" function is not compatible with exp, since compatibility would demand a value of ωω here * exp ε0 = ωωε0 + 1 * log ε0 = ε0 / ω


Exponentiation

A general exponentiation can be defined as , giving an interpretation to expressions like . Again it is essential to distinguish this definition from the "powers of ω" function, especially if ω may occur as the base.


Surcomplex numbers

A surcomplex number is a number of the form , where ''a'' and ''b'' are surreal numbers and is the square root of .Surreal vectors and the game of Cutblock
James Propp, August 22, 1994.
The surcomplex numbers form an
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
field (except for being a proper class),
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the
algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of the field generated by extending the
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
by a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map f ...
of algebraically independent transcendental elements. Up to field
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, this fact characterizes the field of surcomplex numbers within any fixed set theory.


Games

The definition of surreal numbers contained one restriction: each element of L must be strictly less than each element of R. If this restriction is dropped we can generate a more general class known as games. All games are constructed according to this rule: ; Construction rule : If ''L'' and ''R'' are two sets of games then is a game. Addition, negation, and comparison are all defined the same way for both surreal numbers and games. Every surreal number is a game, but not all games are surreal numbers, e.g. the game is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a field, but the class of games does not. The surreals have a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
: given any two surreals, they are either equal, or one is greater than the other. The games have only a
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
: there exist pairs of games that are neither equal, greater than, nor less than each other. Each surreal number is either positive, negative, or zero. Each game is either positive, negative, ''
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
'', or ''
fuzzy Fuzzy or Fuzzies may refer to: Music * Fuzzy (band), a 1990s Boston indie pop band * Fuzzy (composer) (born 1939), Danish composer Jens Vilhelm Pedersen * ''Fuzzy'' (album), 1993 debut album by the Los Angeles rock group Grant Lee Buffalo * "Fu ...
'' (incomparable with zero, such as ). A move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a
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 ...
for the first player to move. If ''x'', ''y'', and ''z'' are surreals, and ''x''=''y'', then ''x'' ''z''=''y'' ''z''. However, if ''x'', ''y'', and ''z'' are games, and ''x''=''y'', then it is not always true that ''x'' ''z''=''y'' ''z''. Note that "=" here means equality, not identity.


Application to combinatorial game theory

The surreal numbers were originally motivated by studies of the game Go, and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized ''Game'' for the mathematical object , and the lowercase ''game'' for recreational games like
Chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
or Go. We consider games with these properties: * Two players (named ''Left'' and ''Right'') *
Deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
(the game at each step will completely depend on the choices the players make, rather than a random factor) * No hidden information (such as cards or tiles that a player hides) * Players alternate taking turns (the game may or may not allow multiple moves in a turn) * Every game must end in a finite number of moves * As soon as there are no legal moves left for a player, the game ends, and that player loses For most games, the initial board position gives no great advantage to either player. As the game progresses and one player starts to win, board positions will occur in which that player has a clear advantage. For analyzing games, it is useful to associate a Game with every board position. The value of a given position will be the Game , where L is the set of values of all the positions that can be reached in a single move by Left. Similarly, R is the set of values of all the positions that can be reached in a single move by Right. The zero Game (called 0) is the Game where L and R are both empty, so the player to move next (L or R) immediately loses. The sum of two Games G = and H = is defined as the Game G + H = where the player to move chooses which of the Games to play in at each stage, and the loser is still the player who ends up with no legal move. One can imagine two chess boards between two players, with players making moves alternatively, but with complete freedom as to which board to play on. If G is the Game , -G is the Game , i.e. with the role of the two players reversed. It is easy to show G - G = 0 for all Games G (where G - H is defined as G + (-H)). This simple way to associate Games with games yields a very interesting result. Suppose two perfect players play a game starting with a given position whose associated Game is ''x''. We can classify all Games into four classes as follows: * If x > 0 then Left will win, regardless of who plays first. * If x < 0 then Right will win, regardless of who plays first. * If x = 0 then the player who goes second will win. * If x , , 0 then the player who goes first will win. More generally, we can define G > H as G - H > 0, and similarly for <, = and , , . The notation G , , H means that G and H are incomparable. G , , H is equivalent to G−H , , 0, i.e. that G > H, G < H and G = H are all false. Incomparable games are sometimes said to be ''confused'' with each other, because one or the other may be preferred by a player depending on what is added to it. A game confused with zero is said to be
fuzzy Fuzzy or Fuzzies may refer to: Music * Fuzzy (band), a 1990s Boston indie pop band * Fuzzy (composer) (born 1939), Danish composer Jens Vilhelm Pedersen * ''Fuzzy'' (album), 1993 debut album by the Los Angeles rock group Grant Lee Buffalo * "Fu ...
, as opposed to positive, negative, or zero. An example of a fuzzy game is star (*). Sometimes when a game nears the end, it will decompose into several smaller games that do not interact, except in that each player's turn allows moving in only one of them. For example, in Go, the board will slowly fill up with pieces until there are just a few small islands of empty space where a player can move. Each island is like a separate game of Go, played on a very small board. It would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game. This doesn't appear to be easy to do. For example, there might be two subgames where whoever moves first wins, but when they are combined into one big game, it is no longer the first player who wins. Fortunately, there is a way to do this analysis. The following theorem can be applied: :If a big game decomposes into two smaller games, and the small games have associated Games of ''x'' and ''y'', then the big game will have an associated Game of ''x''+''y''. A game composed of smaller games is called the
disjunctive sum In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. The sum game finishes when there ...
of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the disjunctive sum of the addends. Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their
disjunctive sum In the mathematics of combinatorial games, the sum or disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. The sum game finishes when there ...
. From this he invented the concept of a Game and the addition operator for it. From there he moved on to developing a definition of negation and comparison. Then he noticed that a certain class of Games had interesting properties; this class became the surreal numbers. Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.


Alternative realizations

Alternative approaches to the surreal numbers complement Conway's exposition in terms of games.


Sign expansion


Definitions

In what is now called the ''sign-expansion'' or ''sign-sequence'' of a surreal number, a surreal number is a function whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
is an ordinal and whose
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
is . This is equivalent to Conway's L-R sequences. Define the binary predicate "simpler than" on numbers by ''x'' is simpler than ''y'' if ''x'' is a
proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of ''y'', ''i.e.'' if dom(''x'') < dom(''y'') and ''x''(α) = ''y''(α) for all α < dom(''x''). For surreal numbers define the binary relation < to be lexicographic order (with the convention that "undefined values" are greater than −1 and less than 1). So ''x'' < ''y'' if one of the following holds: * ''x'' is simpler than ''y'' and ''y''(dom(''x'')) = + 1; * ''y'' is simpler than ''x'' and ''x''(dom(''y'')) = − 1; * there exists a number ''z'' such that ''z'' is simpler than ''x'', ''z'' is simpler than ''y'', ''x''(dom(''z'')) = − 1 and ''y''(dom(''z'')) = + 1. Equivalently, let δ(''x'',''y'') = min( ∪ ), so that ''x'' = ''y'' if and only if δ(''x'',''y'') = dom(''x'') = dom(''y''). Then, for numbers ''x'' and ''y'', ''x'' < ''y'' if and only if one of the following holds: * δ(''x'',''y'') = dom(''x'') ∧ δ(''x'',''y'') < dom(''y'') ∧ ''y''(δ(''x'',''y'')) = + 1; * δ(''x'',''y'') < dom(''x'') ∧ δ(''x'',''y'') = dom(''y'') ∧ ''x''(δ(''x'',''y'')) = − 1; * δ(''x'',''y'') < dom(''x'') ∧ δ(''x'',''y'') < dom(''y'') ∧ ''x''(δ(''x'',''y'')) = − 1 ∧ ''y''(δ(''x'',''y'')) = + 1. For numbers ''x'' and ''y'', ''x'' ≤ ''y'' if and only if ''x'' < ''y'' ∨ ''x'' = ''y'', and ''x'' > ''y'' if and only if ''y'' < ''x''. Also ''x'' ≥ ''y'' if and only if ''y'' ≤ ''x''. The relation < is transitive, and for all numbers ''x'' and ''y'', exactly one of ''x'' < ''y'', ''x'' = ''y'', ''x'' > ''y'', holds (law of trichotomy). This means that < is a
linear order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
(except that < is a proper class). For sets of numbers, ''L'' and ''R'' such that ∀''x'' ∈ ''L'' ∀''y'' ∈ ''R'' (''x'' < ''y''), there exists a unique number ''z'' such that * ∀''x'' ∈ ''L'' (''x'' < ''z'') ∧ ∀''y'' ∈ ''R'' (''z'' < ''y''), * For any number ''w'' such that ∀''x'' ∈ ''L'' (''x'' < ''w'') ∧ ∀''y'' ∈ ''R'' (''w'' < ''y''), ''w'' = ''z'' or ''z'' is simpler than ''w''. Furthermore, ''z'' is constructible from ''L'' and ''R'' by transfinite induction. ''z'' is the simplest number between ''L'' and ''R''. Let the unique number ''z'' be denoted by σ(''L'',''R''). For a number ''x'', define its left set ''L''(''x'') and right set ''R''(''x'') by * ''L''(''x'') = ; * ''R''(''x'') = , then σ(''L''(''x''),''R''(''x'')) = ''x''. One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals. However, similar definitions can be made that eliminate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule ∀''g'' ∈ dom ''f'' (∀''h'' ∈ dom ''g'' (''h'' ∈ dom ''f'' )) and whose range is . "Simpler than" is very simply defined now—''x'' is simpler than ''y'' if ''x'' ∈ dom ''y''. The total ordering is defined by considering ''x'' and ''y'' as sets of ordered pairs (as a function is normally defined): Either ''x'' = ''y'', or else the surreal number ''z'' = ''x'' ∩ ''y'' is in the domain of ''x'' or the domain of ''y'' (or both, but in this case the signs must disagree). We then have ''x'' < ''y'' if ''x''(''z'') = − or ''y''(''z'') = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom ''f'' in order of simplicity (i.e., inclusion), and then write down the signs that ''f'' assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is .


Addition and multiplication

The sum ''x'' + ''y'' of two numbers, ''x'' and ''y'', is defined by induction on dom(''x'') and dom(''y'') by ''x'' + ''y'' = σ(''L'',''R''), where * ''L'' = ∪, * ''R'' = ∪. The additive identity is given by the number 0 = , ''i.e.'' the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number ''x'' is the number − ''x'', given by dom(− ''x'') = dom(''x''), and, for α < dom(''x''), (− ''x'')(α) = − 1 if ''x''(α) = + 1, and (− ''x'')(α) = + 1 if ''x''(α) = − 1. It follows that a number ''x'' is
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
if and only if 0 < dom(''x'') and ''x''(0) = + 1, and ''x'' is negative if and only if 0 < dom(''x'') and ''x''(0) = − 1. The product ''xy'' of two numbers, ''x'' and ''y'', is defined by induction on dom(''x'') and dom(''y'') by ''xy'' = σ(''L'',''R''), where * ''L'' = ∪ , * ''R'' = ∪ . The multiplicative identity is given by the number 1 = , ''i.e.'' the number 1 has domain equal to the ordinal 1, and 1(0) = + 1.


Correspondence with Conway's realization

The map from Conway's realization to sign expansions is given by ''f''() = σ(''M'',''S''), where ''M'' = and ''S'' = . The inverse map from the alternative realization to Conway's realization is given by ''g''(''x'') = , where ''L'' = and ''R'' = .


Axiomatic approach

In another approach to the surreals, given by Alling, explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the axiomatic approach to the reals, these axioms guarantee uniqueness up to isomorphism. A triple \langle \mathbb, \mathrm, b \rangle is a surreal number system if and only if the following hold: * < is a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
over \mathbb * ''b'' is a function from \mathbb onto the class of all ordinals (''b'' is called the "birthday function" on \mathbb). * Let ''A'' and ''B'' be subsets of \mathbb such that for all ''x'' ∈ ''A'' and ''y'' ∈ ''B'', ''x'' < ''y'' (using Alling's terminology, 〈 ''A'',''B'' 〉 is a "Conway cut" of \mathbb). Then there exists a unique ''z'' ∈ \mathbb such that ''b(z)'' is minimal and for all ''x'' ∈ ''A'' and all ''y'' ∈ ''B'', ''x'' < ''z'' < ''y''. (This axiom is often referred to as "Conway's Simplicity Theorem".) * Furthermore, if an ordinal ''α'' is greater than ''b(x)'' for all ''x'' ∈ ''A'', ''B'', then ''b(z)'' ≤ ''α''. (Alling calls a system that satisfies this axiom a "full surreal number system".) Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms. Given these axioms, Alling derives Conway's original definition of ≤ and develops surreal arithmetic.


Simplicity hierarchy

A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich, The difference from the usual definition of a tree is that the set of ancestors of a vertex is well-ordered, but may not have a maximal element (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation.


Hahn series

Alling also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of Hahn series with real coefficients on the value group of surreal numbers themselves (the series representation corresponding to the normal form of a surreal number, as defined above). This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory. This isomorphism makes the surreal numbers into a valued field where the valuation is the additive inverse of the exponent of the leading term in the Conway normal form, e.g., ν(ω) = -1. The valuation ring then consists of the finite surreal numbers (numbers with a real and/or an infinitesimal part). The reason for the sign inversion is that the exponents in the Conway normal form constitute a reverse well-ordered set, whereas Hahn series are formulated in terms of (non-reversed) well-ordered subsets of the value group.


Relation to hyperreals

Philip Ehrlich has constructed an isomorphism between Conway's maximal surreal number field and the maximal
hyperreals In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...
in von Neumann–Bernays–Gödel set theory.


See also

*
Hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
*
Non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...


Notes


References


Further reading

*
Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
's original exposition: ''Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness'', 1974, . More information can be found a
the book's official homepage
* An update of the classic 1976 book defining the surreal numbers, and exploring their connections to games: John Conway, ''On Numbers And Games'', 2nd ed., 2001, . * An update of the first part of the 1981 book that presented surreal numbers and the analysis of games to a broader audience: Berlekamp, Conway, and Guy, ''Winning Ways for Your Mathematical Plays'', vol. 1, 2nd ed., 2001, . *
Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...
, ''Penrose Tiles to Trapdoor Ciphers,'' W. H. Freeman & Co., 1989, , Chapter 4. A non-technical overview; reprint of the 1976 ''Scientific American'' article. * Polly Shulman
"Infinity Plus One, and Other Surreal Numbers"
''
Discover Discover may refer to: Art, entertainment, and media * ''Discover'' (album), a Cactus Jack album * ''Discover'' (magazine), an American science magazine Businesses and brands * DISCover, the ''Digital Interactive Systems Corporation'' * D ...
'', December 1995. * A detailed treatment of surreal numbers: Norman L. Alling, ''Foundations of Analysis over Surreal Number Fields'', 1987, . * A treatment of surreals based on the sign-expansion realization: Harry Gonshor, ''An Introduction to the Theory of Surreal Numbers'', 1986, . * A detailed philosophical development of the concept of surreal numbers as a most general concept of number:
Alain Badiou Alain Badiou (; ; born 17 January 1937) is a French philosopher, formerly chair of Philosophy at the École normale supérieure (ENS) and founder of the faculty of Philosophy of the Université de Paris VIII with Gilles Deleuze, Michel Fouc ...
, ''Number and Numbers'', New York: Polity Press, 2008, (paperback), (hardcover). * The surreal numbers are studied in the context of
homotopy type theory In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory a ...
in section 11.6.


External links


Hackenstrings, and the 0.999... ?= 1 FAQ, by A. N. Walker
an archive of the disappeared original
A gentle yet thorough introduction by Claus Tøndering

Good Math, Bad Math: Surreal Numbers
a series of articles about surreal numbers and their variations
Conway's Mathematics after Conway
survey of Conway's accomplishments in the AMS Notices, with a section on surreal numbers {{DEFAULTSORT:Surreal Number Combinatorial game theory Mathematical logic Infinity Real closed field John Horton Conway Nonstandard analysis