Surreal number

In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.
The surreals share many properties with the reals, including the usual arithmetic operations ; as such, they form an ordered field. If formulated in set theory">Set (mathematics)">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 functions, the Levi-Civita field, the superreal numbers can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

History of the concept

Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth'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.
A separate route to defining the surreals began in 1907, when Hans Hahn introduced Hahn series as a generalization of formal power series, and Felix Hausdorff introduced certain ordered sets called -sets for ordinals 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 and, in 1987, he showed that taking 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 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 is not 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

Notation

In the context of surreal numbers, an ordered pair of sets and, which is written as in many other mathematical contexts, is instead written including the extra space adjacent to each brace. When a set is empty, it is often simply omitted. When a set is explicitly described by its elements, the pair of braces that encloses the list of elements is often omitted. When a union of sets is taken, the operator that represents that is often a comma. For example, instead of, which is common notation in other contexts, we typically write.

Outline of construction

In the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers and, or. Each number is formed from an ordered pair of subsets of numbers already constructed: given subsets and of numbers such that all the members of are strictly less than all the members of, then the pair represents a number intermediate in value between all the members of and all the members of.
Different subsets may end up defining the same number: and may define the same number even if and. Each surreal number is an equivalence class of representations of the form that designate the same number, noting that each equivalence class is a proper class rather than a set.
In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: . This representation, where and are both empty, is called 0. Subsequent stages yield forms like
and
The integers are thus contained within the surreal numbers. Similarly, representations such as
arise, so that the dyadic rationals are contained within the surreal numbers.
After an infinite number of stages, infinite subsets become available, so that any real number can be represented by
where is the set of all dyadic rationals less than and
is the set of all dyadic rationals greater than . Thus the real numbers are also embedded within the surreals.
There are also representations like
where is a transfinite number greater than all integers and is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations 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 or and so forth.

Construction

Surreal numbers are constructed inductively as equivalence classes 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 and right set is written. When and 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
The numeric forms are placed in equivalence classes; each such equivalence class is a surreal number. The elements of the left and right sets of a form are drawn from the universe of the surreal numbers.
Equivalence rule
An ordering relationship must be antisymmetric, i.e., it must have the property that only when and are the same object. This is not the case for surreal number forms, but is true by construction for surreal numbers.
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 and, if and only if both:

There is no such that. That is, every element in the left part of is strictly smaller than.
There is no such that. That is, every element in the right part of is strictly larger than.

Surreal numbers can be compared to each other 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 to define the universe of objects 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.

Induction rule

There is a generation, in which 0 consists of the single form.
Given any ordinal number, the generation is the set of all surreal numbers that are generated by the construction rule from subsets of.

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 with, the expression is the empty set; the only subset of the empty set is the empty set, and therefore consists of a single surreal form lying in a single equivalence class 0.
For every finite ordinal number, is well-ordered 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 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, the multiplicative identity, and the additive inverse of 1. The arithmetic operations defined below are consistent with these labels.
For every, since every valid form in is also a valid form in, all of the numbers in also appear in . Numbers in that are a superset of some number in are said to have been inherited from generation. The smallest value of for which a given surreal number appears in 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:

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 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 from previous generations into a left set and a right set.
The equivalence class of a number depends on only 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 of induction may be characterized by their -complete forms. 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:
The third observation extends to all surreal numbers with finite left and right sets. 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 occurring in generation represents a number inherited from an earlier generation if and only if there is some number in that is greater than all elements of and less than all elements of the. If represents a number from any generation earlier than, there is a least such generation, and exactly one number with this least as its birthday that lies between and ; is a form of this. In other words, it lies in the equivalence class in that is a superset of the representation of in generation.

Arithmetic

The addition, negation, and multiplication of surreal number forms and are defined by three recursive formulas.

Negation

Negation of a given number is defined by
where the negation of a set of numbers is given by the set of the negated elements of :
This formula involves the negation of the surreal numbers appearing in the left and right sets of, 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 makes sense only 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 and are drawn from generations earlier than that in which the form first occurs, and observing the special case:

Addition

The definition of addition is also a recursive formula:
where
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:
For example:
which by the birthday property is a form of 1. This justifies the label used in the previous section.

Subtraction

Subtraction is defined with addition and negation:

Multiplication

Multiplication can be defined recursively as well, beginning from the special cases involving 0, the multiplicative identity 1, and its additive inverse −1:
The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression that appears in the left set of the product of and. This is understood as, the set of numbers generated by picking all possible combinations of members of and, and substituting them into the expression.
For example, to show that the square of is :

Division

The definition of division is done in terms of the reciprocal and multiplication:
where
for positive. Only positive are permitted in the formula, with any nonpositive terms being ignored. This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets of, 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, then we know a left term of will be 0. This in turn means is a right term. This means
is a left term. This means
will be a right term. Continuing, this gives
For negative, is given by
If, 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 will eventually be expressed entirely in terms of operations on numbers with birthdays less than ;
Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than ;
As long as the operands are well-defined surreal number forms, the results are again well-defined surreal number forms;
The operations can be extended to numbers : the result of negating or adding or multiplying and will represent the same number regardless of the choice of form of and ; and
These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a field, with additive identity and multiplicative identity.

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:

Arithmetic closure

For each natural number , all numbers generated in are dyadic fractions, i.e., can be written as an irreducible fraction, where and are integers and.
The set of all surreal numbers that are generated in some for finite may be denoted as. One may form the three classes
of which is the union. No individual is closed under addition and multiplication, but is; it is the subring of the rationals consisting of all dyadic fractions.
There are infinite ordinal numbers for which the set of surreal numbers with birthday less than is closed under the different arithmetic operations. For any ordinal, the set of surreal numbers with birthday less than is closed under addition and forms a group; for birthday less than it is closed under multiplication and forms a ring; and for birthday less than an epsilon number 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 and thus the collection of surreal numbers is a proper class. With their ordering and algebraic operations they constitute an ordered field, with the caveat that they do not form a set. In fact, it is a very special ordered field: the biggest one, in that every ordered field is a subfield of the surreal numbers. The class of all surreal numbers is denoted by the symbol.

Infinity

Define as the set of all surreal numbers generated by the construction rule from subsets of. A unique infinitely large positive number occurs in :
also contains objects that can be identified as the rational numbers. For example, the -complete form of the fraction is given by:
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 numbers appear in ; the remaining finite real numbers do too. For example,
The only infinities in are and ; but there are other non-real numbers in among the reals. Consider the smallest positive number in :
This number is larger than zero but less than all positive dyadic fractions. It is therefore an infinitesimal number, often labeled. The -complete form of is the same as the -complete form of 0, except that 0 is included in the left set. The only "pure" infinitesimals in are and its additive inverse ; adding them to any dyadic fraction produces the numbers, which also lie in.
One can determine the relationship between and by multiplying particular forms of them to obtain:
This expression is well-defined only in a set theory which permits transfinite induction up to. In such a system, one can demonstrate that all the elements of the left set of are positive infinitesimals and all the elements of the right set are positive infinities, and therefore is the oldest positive finite number, 1. Consequently,. Some authors systematically use in place of the symbol.

Contents of ''S''

Given any in, exactly one of the following is true:

and are both empty, in which case ;
is empty and some integer is greater than every element of, in which case equals the smallest such integer ;
is empty and no integer is greater than every element of, in which case equals ;
is empty and some integer is less than every element of, in which case equals the largest such integer ;
is empty and no integer is less than every element of, in which case equals ;
and are both non-empty, and:
* Some dyadic fraction is "strictly between" and , in which case equals the oldest such dyadic fraction ;
* No dyadic fraction lies strictly between and, but some dyadic fraction is greater than or equal to all elements of and less than all elements of, in which case equals ;
* No dyadic fraction lies strictly between and, but some dyadic fraction is greater than all elements of and less than or equal to all elements of, in which case equals ;
* Every dyadic fraction is either greater than some element of or less than some element of, in which case is some real number that has no representation as a dyadic fraction.

is not an algebraic field, because it is not closed under arithmetic operations; consider, whose form
does not lie in any number in. The maximal subset of that is closed under arithmetic operations is the field of real numbers, obtained by leaving out the infinities, the infinitesimals, and the infinitesimal neighbors of each nonzero dyadic fraction.
This construction of the real numbers differs from the Dedekind cuts of standard analysis in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in with its forms in previous generations. The rationals are not an identifiable stage in the surreal construction; they are merely the subset of containing all elements such that for some and some nonzero, both drawn from. By demonstrating that 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 is reachable from by a finite series of arithmetic operations including multiplicative inversion, one can show that is strictly smaller than the subset of identified with the reals.
The set has the same cardinality as the real numbers. This can be demonstrated by exhibiting surjective mappings from to the closed unit interval of and vice versa. Mapping onto is routine; map numbers less than or equal to to 0, numbers greater than or equal to to 1, and numbers between and to their equivalent in . To map onto, map the central third of onto ; the central third of the upper third to ; and so forth. This maps a nonempty open interval of onto each element of, monotonically. The residue of consists of the Cantor set, 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. This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday.

Transfinite induction

Continuing to perform transfinite induction beyond produces more ordinal numbers, each represented as the largest surreal number having birthday. The first such ordinal is. There is another positive infinite number in generation :
The surreal number is not an ordinal; the ordinal is not the successor of any ordinal. This is a surreal number with birthday, which is labeled on the basis that it coincides with the sum of and. Similarly, there are two new infinitesimal numbers in generation :
At a later stage of transfinite induction, there is a number larger than for all natural numbers :
This number may be labeled both because its birthday is and because it coincides with the surreal sum of and ; it may also be labeled because it coincides with the product of and. It is the second limit ordinal; reaching it from via the construction step requires a transfinite induction on
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 equals, but the surreal sum is commutative and produces. The addition and multiplication of the surreal numbers associated with ordinals coincides with the natural sum and natural product of ordinals.
Just as is bigger than for any natural number, there is a surreal number that is infinite but smaller than for any natural number. That is, is defined by
where on the right hand side the notation is used to mean. It can be identified as the product of and the form of. The birthday of is the limit ordinal.

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 the surreal number

where and range over the positive real numbers. If then is "infinitely greater" than, in that it is greater than for all real numbers. Powers of also satisfy the conditions

so they behave the way one would expect powers to behave.
Each power of 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 there will always exist some positive real number and some surreal number so that is "infinitely smaller" than. The exponent is the "base logarithm" of, defined on the positive surreals; it can be demonstrated that maps the positive surreals onto the surreals and that
This gets extended by transfinite induction so that every surreal number has a "normal form" analogous to the Cantor normal form for ordinal numbers. This is the Conway normal form: Every surreal number may be uniquely written as
where every is a nonzero real number and the s 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.
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 subset of the surreal numbers does not have a least upper bound unless it has a maximal element. Conway defines a gap as such that every element of is less than every element of, and ; 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 cuts, but we can still talk about a completion of the surreal numbers with the natural ordering which is a linear continuum.
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. Similarly the gap is larger than all surreal numbers.
With a bit of set-theoretic care, can be equipped with a topology where the open sets are unions of open intervals and continuous functions can be defined. An equivalent of Cauchy sequences 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
with decreasing and having no lower bound in..

Exponential function

Based on unpublished work by Kruskal, a construction that extends the real exponential function to the surreals was carried through by Gonshor.

Other exponentials

The powers of 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- exponential, and it is this function that is meant whenever the notation is used in the following.
When is a dyadic fraction, the power function, 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 that can exist.

Basic induction

The induction steps for the surreal exponential are based on the series expansion for the real exponential,
more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For positive these are denoted and include all partial sums; for negative but finite, denotes the odd steps in the series starting from the first one with a positive real part. For negative infinite the odd-numbered partial sums are strictly decreasing and the 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.

Results

Using this definition, the following hold:

is a strictly increasing positive function,
satisfies
is a surjection and has a well-defined inverse,
coincides with the usual exponential function on the reals
For infinitesimal, the value of the formal power series of is well defined and coincides with the inductive definition
* When 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
* Similarly, for infinitesimally close to, is given by power series expansion of
For positive infinite, is infinite as well
* If has the form , has the form where is a strictly increasing function of. In fact there is an inductively defined bijection whose inverse can also be defined inductively
* If 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 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 , 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 and any finite, is infinite
* For any integer and surreal,. This stronger constraint is one of the Ressayre axioms for the real exponential field
satisfies all the Ressayre axioms for the real exponential field
* The surreals with exponential is an elementary extension of the real exponential field
* For an ordinal epsilon number, the set of surreal numbers with birthday less than 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, i.e., the function, 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 .

and
* This shows that the "power of " function is not compatible with, since compatibility would demand a value of here

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 " function, especially if may occur as the base.

Surcomplex numbers

A surcomplex number is a number of the form, where and are surreal numbers and is the square root of. The surcomplex numbers form an algebraically closed field, isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, 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 must be strictly less than each element of. 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 and 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 [star (game theory)|] 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: given any two surreals, they are either equal, or one is greater than the other. The games have only a partial order: 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, or fuzzy.
A move in a game involves the player whose move it is choosing a game from those available in or 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 for the first player to move.
If,, and are surreals, and, then. However, if,, and are games, and, then it is not always true that. 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 or Go.
We consider games with these properties:

Two players
Deterministic
No hidden information
Players alternate taking turns
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 is the set of values of all the positions that can be reached in a single move by Left. Similarly, is the set of values of all the positions that can be reached in a single move by Right.
The zero Game is the Game where and are both empty, so the player to move next immediately loses. The sum of two Games and is defined as the Game 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 alternately, but with complete freedom as to which board to play on. If is the Game, is the Game, i.e. with the role of the two players reversed. It is easy to show for all Games .
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. We can classify all Games into four classes as follows:

If then Left will win, regardless of who plays first.
If then Right will win, regardless of who plays first.
If then the player who goes second will win.
If then the player who goes first will win.

More generally, we can define as, and similarly for, and.
The notation means that and are incomparable. is equivalent to, i.e. that, and 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, 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:
A game composed of smaller games is called the disjunctive sum 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. 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 the original exposition by Conway 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 is an ordinal and whose codomain is. This notion has been introduced by Conway himself in the equivalent formulation of L-R sequences.
Define the binary predicate "simpler than" on numbers by: is simpler than if is a proper subset of, i.e. if and for all.
For surreal numbers define the binary relation to be lexicographic order. So if one of the following holds:

is simpler than and ;
is simpler than and ;
there exists a number such that is simpler than, is simpler than, and.

Equivalently, let,
so that if and only if. Then, for numbers and, if and only if one of the following holds:
For numbers and, if and only if, and if and only if. Also if and only if.
The relation is transitive, and for all numbers and, exactly one of,,, holds. This means that is a linear order.
For sets of numbers and such that, there exists a unique number such that

For any number such that, or is simpler than.

Furthermore, is constructible from and by transfinite induction. is the simplest number between and. Let the unique number be denoted by.
For a number, define its left set and right set by
then.
One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's original 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 class of functions whose domain is a subset of the surreals satisfying the transitivity rule and whose range is. "Simpler than" is very simply defined now: is simpler than if. The total ordering is defined by considering and as sets of ordered pairs : Either, or else the surreal number is in the domain of or the domain of . We then have if or . Converting these functions into sign sequences is a straightforward task; arrange the elements of in order of simplicity, and then write down the signs that 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 of two numbers and is defined by induction on and by, where

The additive identity is given by the number, i.e. the number is the unique function whose domain is the ordinal, and the additive inverse of the number is the number, given by, and, for, if, and if.
It follows that a number is positive if and only if and, and is negative if and only if and.
The product of two numbers, and, is defined by induction on and by, where
The multiplicative identity is given by the number, i.e. the number has domain equal to the ordinal, and.

Correspondence with Conway's realization

The map from Conway's realization to sign expansions is given by, where and.
The inverse map from the alternative realization to Conway's realization is given by, where and.

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 is a surreal number system if and only if the following hold:

is a total order over
is a function from onto the class of all ordinals.
Let and be subsets of such that for all and, . Then there exists a unique such that is minimal and for all and all,.
Furthermore, if an ordinal is greater than for all, then.

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 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 ; 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. Ehrlich additionally constructed an isomorphism between Conway's maximal surreal number field and the maximal hyperreals in von Neumann–Bernays–Gödel set theory.

Hahn series

Alling also proves that the field of surreal numbers is isomorphic to the field of Hahn series with real coefficients on the value group of surreal numbers themselves. This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.
Note that the support of the Hahn series must be a set, not a proper class; for instance, the Hahn series summed over all ordinals has no surreal counterpart.
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.,. The valuation ring then consists of the finite surreal numbers. 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 well-ordered subsets of the value group.