Inmathematics, thesurreal number system is atotally orderedproper class containing not only thereal numbers but alsoinfinite andinfinitesimal numbers, respectively larger or smaller inabsolute value than any positive real number. Research on theGo endgame byJohn Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced inDonald Knuth's 1974 bookSurreal 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 (addition, subtraction, multiplication, and division); as such, they form anordered field.^[a] If formulated invon 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, therational functions, theLevi-Civita field, thesuperreal numbers (including thehyperreal numbers) can be realized as subfields of the surreals.^[1] The surreals also contain alltransfiniteordinal numbers; the arithmetic on them is given by thenatural operations. It has also been shown (invon Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field isisomorphic to the maximal class surreal field.

Research on theGo endgame byJohn Horton Conway led to the original definition and construction of the surreal numbers.^[2] Conway's construction was introduced inDonald Knuth's 1974 bookSurreal 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 termsurreal numbers for what Conway had called simplynumbers.^[3] Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 bookOn Numbers and Games.

A separate route to defining the surreals began in 1907, whenHans Hahn introducedHahn series as a generalization offormal power series, andFelix Hausdorff introduced certain ordered sets calledη_α-sets for ordinalsα 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 ordinalsα and, in 1987, he showed that takingα to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.^[4]

If the surreals are considered as 'just' a proper-class-sized real closed field, Alling's 1962 paper handles the case ofstrongly inaccessible cardinals which can naturally be considered as proper classes by cutting off thecumulative 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.^{[citation needed]} 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.^[5]

In the context of surreal numbers, anordered pair of setsL andR, which is written as(L,R) in many other mathematical contexts, is instead written{L |R } 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(L₁ ∪L₂ ∪ {0, 1, 2}, ∅), which is common notation in other contexts, we typically write{L₁,L₂, 0, 1, 2 | }.

In the Conway construction,^[6] the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbersa andb,a ≤b orb ≤a. (Both may hold, in which casea andb are equivalent and denote the same number.) Each number is formed from an ordered pair of subsets of numbers already constructed: given subsetsL andR of numbers such that all the members ofL are strictly less than all the members ofR, then the pair{L |R } represents a number intermediate in value between all the members ofL and all the members ofR.

Different subsets may end up defining the same number:{L |R } and{L′ |R′ } may define the same number even ifL ≠L′ andR ≠R′. (A similar phenomenon occurs whenrational numbers are defined as quotients of integers:⁠1/2⁠ and⁠2/4⁠ are different representations of the same rational number.) Each surreal number is anequivalence class of representations of the form{L |R } that designate the same number, noting that each equivalence class is aproper 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, whereL andR are both empty, is called 0. Subsequent stages yield forms like

and

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 thedyadic rationals (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 anyreal numbera can be represented by{L_a |R_a },whereL_a is the set of all dyadic rationals less thana andR_a is the set of all dyadic rationals greater thana (reminiscent of aDedekind cut). Thus the real numbers are also embedded within the surreals.

There are also representations like

whereω is a transfinite number greater than all integers andε 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 about2ω orω − 1 and so forth.

Surreal numbers areconstructed inductively asequivalence classes ofpairs 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.

Aform is a pair of sets of surreal numbers, called itsleft set and itsright set. A form with left setL and right setR is written{L |R }. WhenL andR 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{ | }.

Construction rule

The numeric forms are placed in equivalence classes; each such equivalence class is asurreal number. The elements of the left and right sets of a form are drawn from the universe of the surreal numbers (not offorms, but of theirequivalence classes).

Equivalence rule

Anordering relationship must beantisymmetric, i.e., it must have the property thatx =y (i. e.,x ≤y andy ≤x are both true) only whenx andy are the same object. This is not the case for surreal numberforms, but is true by construction for surrealnumbers (equivalence classes).

The equivalence class containing{ | } is labeled 0; in other words,{ | } is a form of the surreal number 0.

The recursive definition of surreal numbers is completed by defining comparison:

Given numeric formsx = {X_L |X_R } andy = {Y_L |Y_R },x ≤y if and only if both:

• There is nox_L ∈X_L such thaty ≤x_L. That is, every element in the left part ofx is strictly smaller thany.
• There is noy_R ∈Y_R such thaty_R ≤x. That is, every element in the right part ofy is strictly larger thanx.

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.

This group of definitions isrecursive, and requires some form ofmathematical induction to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable viafinite induction are thedyadic fractions; a wider universe is reachable given some form oftransfinite induction.

• There is a generationS₀ = { 0 }, in which 0 consists of the single form{ | }.
• Given anyordinal numbern, the generationS_n 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 noS_i withi < 0, the expression is the empty set; the only subset of the empty set is the empty set, and thereforeS₀ consists of a single surreal form{ | } lying in a single equivalence class 0.

For every finite ordinal numbern,S_n iswell-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{ | 0 } < { | } < { 0 | } (the form{ 0 | 0 } is non-numeric because0 ≤ 0). The equivalence class containing{ 0 | } is labeled 1 and the equivalence class containing{ | 0 } is labeled −1. These three labels have a special significance in the axioms that define aring; 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 everyi <n, since every valid form inS_i is also a valid form inS_n, all of the numbers inS_i also appear inS_n (as supersets of their representation inS_i). (The set union expression appears in our construction rule, rather than the simpler formS_n−1, so that the definition also makes sense whenn is alimit ordinal.) Numbers inS_n that are a superset of some number inS_i are said to have beeninherited from generationi. The smallest value ofα for which a given surreal number appears inS_α is called itsbirthday. 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:

1. S₂ contains four new surreal numbers. Two contain extremal forms:{ | −1, 0, 1 } contains all numbers from previous generations in its right set, and{ −1, 0, 1 | } 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.
2. Every surreal numberx that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbersother thanx from previous generations into a left set (all numbers less thanx) and a right set (all numbers greater thanx).
3. 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{ 1 | } and{ | −1 } 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{ 0 | 1 } and{ −1 | 0 } are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled⁠1/2⁠ and −⁠1/2⁠. These labels will also be justified by the rules for surreal addition and multiplication below.

The equivalence classes at each stagen of induction may be characterized by theirn-complete forms (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form containsevery 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. (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{ 1, 2 | 5, 8 } is therefore equivalent to{ 2 | 5 }; one can establish that these are forms of 3 by using thebirthday property, which is a consequence of the rules above.

A formx = {L |R } occurring in generationn represents a number inherited from an earlier generationi <n if and only if there is some number inS_i that is greater than all elements ofL and less than all elements of theR. (In other words, ifL andR are already separated by a number created at an earlier stage, thenx does not represent a new number but one already constructed.) Ifx represents a number from any generation earlier thann, there is a least such generationi, and exactly one numberc with this leasti as its birthday that lies betweenL andR;x is a form of thisc. In other words, it lies in the equivalence class inS_n that is a superset of the representation ofc in generationi.

The addition, negation (additive inverse), and multiplication of surreal numberformsx = {X_L |X_R } andy = {Y_L |Y_R } are defined by three recursive formulas.

Negation of a given numberx = {X_L |X_R } is defined by$${\displaystyle -x=-\{X_L\mid X_R\}=\{-X_R\mid -X_L\},}$$where the negation of a setS of numbers is given by the set of the negated elements ofS:$${\displaystyle -S=\{-s:s\in S\}.}$$

This formula involves the negation of the surrealnumbers appearing in the left and right sets ofx, 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 inX_L andX_R are drawn from generations earlier than that in which the formx first occurs, and observing the special case:$${\displaystyle -0=-\{\mid \}=\{\mid \}=0.}$$

The definition of addition is also a recursive formula:$${\displaystyle x+y=\{X_L\mid X_R\}+\{Y_L\mid Y_R\}=\{X_L+y,x+Y_L\mid X_R+y,x+Y_R\},}$$where

$${\displaystyle X+y=\{x^{\prime }+y:x^{\prime }\in X\},x+Y=\{x+y^{\prime }:y^{\prime }\in Y\}}$$

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:$${\displaystyle 0+0=\{\mid \}+\{\mid \}=\{\mid \}=0}$$$${\displaystyle x+0=x+\{\mid \}=\{X_L+0\mid X_R+0\}=\{X_L\mid X_R\}=x}$$$${\displaystyle 0+y=\{\mid \}+y=\{0+Y_L\mid 0+Y_R\}=\{Y_L\mid Y_R\}=y}$$For example:

⁠1/2⁠ +⁠1/2⁠ = { 0 | 1 } + { 0 | 1 } = {⁠1/2⁠ |⁠3/2⁠ },

which by the birthday property is a form of 1. This justifies the label used in the previous section.

Subtraction is defined with addition and negation:$${\displaystyle x-y=\{X_L\mid X_R\}+\{-Y_R\mid -Y_L\}=\{X_L-y,x-Y_R\mid X_R-y,x-Y_L\}.}$$

Multiplication can be defined recursively as well, beginning from the special cases involving 0, themultiplicative 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$X_{R}y+x{Y}_R-X_R{Y}_R$ that appears in the left set of the product ofx andy. This is understood as$\left\{x^{\prime }y+x{y}^{\prime }-x^{\prime }{y}^{\prime }:x^{\prime }\in X_R,y^{\prime }\in Y_R\right\}$, 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 of⁠1/2⁠ is⁠1/4⁠:

⁠1/2⁠ ⋅⁠1/2⁠ = { 0 | 1 } ⋅ { 0 | 1 } = { 0 |⁠1/2⁠ } =⁠1/4⁠.

The definition of division is done in terms of the reciprocal and multiplication:

$${\displaystyle \frac{x}{y}=x\cdot \frac{1}{y}}$$

where^[6]: 21

$${\displaystyle \frac{1}{y}=\left\{0,\frac{1+(y_R-y){\left(\frac{1}{y}\right)}_L}{y_R},\frac{1+\left(y_L-y\right){\left(\frac{1}{y}\right)}_R}{y_L}|\frac{1+(y_L-y){\left(\frac{1}{y}\right)}_L}{y_L},\frac{1+(y_R-y){\left(\frac{1}{y}\right)}_R}{y_R}\right\}}$$

for positivey. Only positivey_L are permitted in the formula, with any nonpositive terms being ignored (andy_R are always positive). This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets ofy, but also recursion in that the members of the left and right sets of⁠1/y⁠ itself. 0 is always a member of the left set of⁠1/y⁠, and that can be used to find more terms in a recursive fashion. For example, ify = 3 = { 2 |}, then we know a left term of⁠1/3⁠ will be 0. This in turn means⁠1 + (2 − 3)0/2⁠ =⁠1/2⁠ is a right term. This means$${\displaystyle \frac{1+(2-3)\left(\frac{1}{2}\right)}{2}=\frac{1}{4}}$$is a left term. This means$${\displaystyle \frac{1+(2-3)\left(\frac{1}{4}\right)}{2}=\frac{3}{8}}$$will be a right term. Continuing, this gives$${\displaystyle \frac{1}{3}=\left\{0,\frac{1}{4},\frac{5}{16},\dots |\frac{1}{2},\frac{3}{8},\dots \right\}}$$

For negativey,⁠1/y⁠ is given by$${\displaystyle \frac{1}{y}=-\left(\frac{1}{-y}\right)}$$

Ify = 0, then⁠1/y⁠ is undefined.

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 birthdayn will eventually be expressed entirely in terms of operations on numbers with birthdays less thann;
• Multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthdayn will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less thann;
• 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 tonumbers (equivalence classes of forms): the result of negatingx or adding or multiplyingx andy will represent the same number regardless of the choice of form ofx andy; and
• These operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of afield, with additive identity0 = { | } and multiplicative identity1 = { 0 | }.

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 = { 0 }

S₁ = { −1 < 0 < 1 }

S₂ = { −2 < −1 < −⁠1/2⁠ < 0 <⁠1/2⁠ < 1 < 2 }

S₃ = { −3 < −2 < −⁠3/2⁠ < −1 < −⁠3/4⁠ < −⁠1/2⁠ < −⁠1/4⁠ < 0 <⁠1/4⁠ <⁠1/2⁠ <⁠3/4⁠ < 1 <⁠3/2⁠ < 2 < 3 }

S₄ = { −4 < −3 < ... < −⁠1/8⁠ < 0 <⁠1/8⁠ <⁠1/4⁠ <⁠3/8⁠ <⁠1/2⁠ <⁠5/8⁠ <⁠3/4⁠ <⁠7/8⁠ < 1 <⁠5/4⁠ <⁠3/2⁠ <⁠7/4⁠ < 2 <⁠5/2⁠ < 3 < 4 }

For eachnatural number (finite ordinal)n, all numbers generated inS_n aredyadic fractions, i.e., can be written as anirreducible fraction⁠a/2^b⁠, wherea andb areintegers and0 ≤b <n.

The set of all surreal numbers that are generated in someS_n for finiten may be denoted as$S_{\ast }=\bigcup _{n\in N}{S}_n$. One may form the three classesof whichS_∗ is the union. No individualS_n is closed under addition and multiplication (exceptS₀), butS_∗ is; it is the subring of the rationals consisting of all dyadic fractions.

There are infinite ordinal numbersβ for which the set of surreal numbers with birthday less thanβ is closed under the different arithmetic operations.^[7] For any ordinalα, the set of surreal numbers with birthday less thanβ =ω^α (usingpowers ofω) is closed under addition and forms a group; for birthday less thanω^ωα it is closed under multiplication and forms a ring;^[b] and for birthday less than an (ordinal)epsilon numberε_α 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.^{[7][8]: ch. 10 [7]}

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 aproper class. With their ordering and algebraic operations they constitute anordered field, with the caveat that they do not form aset. In fact, it is a very special ordered field: the biggest one, in that every ordered field is a subfield of the surreal numbers.^[1] The class of all surreal numbers is denoted by the symbol$\mathbb{N}\mathbb{o}$.

DefineS_ω as the set of all surreal numbers generated by the construction rule from subsets ofS_∗. (This is the same inductive step as before, since the ordinal numberω 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 be performed only in a set theory that allows such a union.) A unique infinitely large positive number occurs inS_ω:$${\displaystyle \omega =\{S_{\ast }\mid \}=\{1,2,3,4,\dots \mid \}.}$$S_ω also contains objects that can be identified as therational numbers. For example, theω-complete form of the fraction⁠1/3⁠ is given by:The product of this form of⁠1/3⁠ 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 therational numbers appear inS_ω; the remaining finitereal numbers do too. For example,$${\displaystyle \pi =\left\{3,\frac{25}{8},\frac{201}{64},\dots \mid 4,\frac{7}{2},\frac{13}{4},\frac{51}{16},\dots \right\}.}$$

The only infinities inS_ω areω and−ω; but there are other non-real numbers inS_ω among the reals. Consider the smallest positive number inS_ω:$${\displaystyle \epsilon =\{S_{-}\cup S_0\mid S_{+}\}=\left\{0\mid 1,\frac{1}{2},\frac{1}{4},\frac{1}{8},\dots \right\}=\{0\mid y\in S_{\ast }:y>0\}}$$This number is larger than zero but less than all positive dyadic fractions. It is therefore aninfinitesimal number, often labeledε. Theω-complete form ofε (respectively−ε) is the same as theω-complete form of 0, except that 0 is included in the left (respectively right) set. The only "pure" infinitesimals inS_ω areε and its additive inverse−ε; adding them to any dyadic fractiony produces the numbersy ±ε, which also lie inS_ω.

One can determine the relationship betweenω andε by multiplying particular forms of them to obtain:

ω ·ε = {ε ·S₊ |ω ·S₊ +S_∗ +ε ·S_∗ }.

This expression is well-defined only in a set theory which permits transfinite induction up toS_ω². In such a system, one can demonstrate that all the elements of the left set ofωS_ω ·‍S_ωε are positive infinitesimals and all the elements of the right set are positive infinities, and thereforeωS_ω ·‍S_ωε is the oldest positive finite number, 1. Consequently,⁠1/ε⁠ =ω. Some authors systematically useω⁻¹ in place of the symbolε.

Given anyx = {L |R } inS_ω, exactly one of the following is true:

• L andR are both empty, in which casex = 0;
• R is empty and some integern ≥ 0 is greater than every element ofL, in which casex equals the smallest such integern;
• R is empty and no integern is greater than every element ofL, in which casex equals+ω;
• L is empty and some integern ≤ 0 is less than every element ofR, in which casex equals the largest such integern;
• L is empty and no integern is less than every element ofR, in which casex equals−ω;
• L andR are both non-empty, and:
  • Some dyadic fractiony is "strictly between"L andR (greater than all elements ofL and less than all elements ofR), in which casex equals the oldest such dyadic fractiony;
  • No dyadic fractiony lies strictly betweenL andR, but some dyadic fraction$y\in L$ is greater than or equal to all elements ofL and less than all elements ofR, in which casex equalsy +ε;
  • No dyadic fractiony lies strictly betweenL andR, but some dyadic fraction$y\in R$ is greater than all elements ofL and less than or equal to all elements ofR, in which casex equalsy −ε;
  • Every dyadic fraction is either greater than some element ofR or less than some element ofL, in which casex 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; considerω + 1, whose form$${\displaystyle \omega +1=\{1,2,3,4,...\mid \}+\{0\mid \}=\{1,2,3,4,\dots ,\omega \mid \}}$$does not lie in any number inS_ω. The maximal subset ofS_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 neighborsy ±ε of each nonzero dyadic fractiony.

This construction of the real numbers differs from theDedekind cuts ofstandard analysis in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction inS_ω with its forms in previous generations. (Theω-complete forms of real elements ofS_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 subsetQ ofS_ω containing all elementsx such thatxb =a for somea and some nonzerob, both drawn fromS_∗. By demonstrating thatQ is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element ofQ is reachable fromS_∗ by a finite series (no longer than two, actually) of arithmetic operationsincluding multiplicative inversion, one can show thatQ is strictly smaller than the subset ofS_ω identified with the reals.

The setS_ω has the samecardinality as the real numbersR. This can be demonstrated by exhibiting surjective mappings fromS_ω to the closed unit intervalI ofR and vice versa. MappingS_ω ontoI is routine; map numbers less than or equal toε (including−ω) to 0, numbers greater than or equal to1 −ε (includingω) to 1, and numbers betweenε and1 −ε to their equivalent inI (mapping the infinitesimal neighborsy±ε of each dyadic fractiony, along withy itself, toy). To mapI ontoS_ω, map the (open) central third (⁠1/3⁠,⁠2/3⁠) ofI onto{ | } = 0; the central third (⁠7/9⁠,⁠8/9⁠) of the upper third to{ 0 | } = 1; and so forth. This maps a nonempty open interval ofI onto each element ofS_∗, monotonically. The residue ofI consists of theCantor set2^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{L |R } inS_ω. This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthdayω.

Continuing to performtransfinite induction beyondS_ω produces more ordinal numbersα, each represented as the largest surreal number having birthdayα. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal isω + 1 = {ω | }. There is another positive infinite number in generationω + 1:

The surreal numberω − 1 is not an ordinal; the ordinalω is not the successor of any ordinal. This is a surreal number with birthdayω + 1, which is labeledω − 1 on the basis that it coincides with the sum ofω = { 0, 1, 2, 3, 4, ... | } and−1 = { | 0 }. Similarly, there are two new infinitesimal numbers in generationω + 1:

At a later stage of transfinite induction, there is a number larger thanω +k for all natural numbersk:

This number may be labeledω +ω both because its birthday isω +ω (the first ordinal number not reachable fromω by the successor operation) and because it coincides with the surreal sum ofω andω; it may also be labeled2ω because it coincides with the product ofω = { 1, 2, 3, 4, ... | } and2 = { 1 | }. It is the second limit ordinal; reaching it fromω via the construction step requires a transfinite induction onThis involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.

Note that theconventional addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals1 +ω equalsω, but the surreal sum is commutative and produces1 +ω =ω + 1 >ω. The addition and multiplication of the surreal numbers associated with ordinals coincides with thenatural sum and natural product of ordinals.

Just as2ω is bigger thanω +n for any natural numbern, there is a surreal number⁠ω/2⁠ that is infinite but smaller thanω −n for any natural numbern. That is,⁠ω/2⁠ is defined by

where on the right hand side the notationx −Y is used to mean{x −y :y ∈Y }. It can be identified as the product ofω and the form{ 0 | 1 } of⁠1/2⁠. The birthday of⁠ω/2⁠ is the limit ordinalω2.

To classify the "orders" of infinite and infinitesimal surreal numbers, also known asarchimedean classes, Conway associated to each surreal numberx the surreal number

• ω^x = { 0,rω^x_L |sω^x_R },

wherer ands range over the positive real numbers. Ifx <y thenω^y is "infinitely greater" thanω^x, in that it is greater thanrω^x for all real numbersr. Powers ofω also satisfy the conditions

• ω^xω^y =ω^x+y,
• ω^−x =⁠1/ω^x⁠,

so they behave the way one would expect powers to behave.

Each power ofω also has the redeeming feature of being thesimplest surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal numberx there will always exist some positive real numberr and some surreal numbery so thatx −rω^y is "infinitely smaller" thanx. The exponenty is the "baseω logarithm" ofx, defined on the positive surreals; it can be demonstrated thatlog_ω 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 theCantor normal form for ordinal numbers. This is the Conway normal form: Every surreal numberx may be uniquely written as

where everyr_α is a nonzero real number and they_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 apower 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 aHahn series.

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^[6] a gap as{L |R } such that every element ofL is less than every element ofR, and$L\cup R=\mathbb{N}\mathbb{o}$; this is not a number because at least one of the sides is a proper class. Though similar, gaps are not quite the same asDedekind cuts,^[c] but we can still talk about a completion${\mathbb{N}\mathbb{o}}_{\mathfrak{D}}$ of the surreal numbers with the natural ordering which is a (proper class-sized)linear continuum.^[9]

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${\mathbb{N}\mathbb{o}}_{\mathfrak{D}}$. Similarly the gap$\mathbb{O}\mathbb{n}=\{\mathbb{N}\mathbb{o}\mid \}$ is larger than all surreal numbers. (This is anesoteric pun: In the general construction of ordinals,α "is" the set of ordinals smaller thanα, and we can use this equivalence to writeα = {α | } in the surreals;$\mathbb{O}\mathbb{n}$ denotes the class of ordinal numbers, and because$\mathbb{O}\mathbb{n}$ iscofinal in$\mathbb{N}\mathbb{o}$ we have$\{\mathbb{N}\mathbb{o}\mid \}=\{\mathbb{O}\mathbb{n}\mid \}=\mathbb{O}\mathbb{n}$ by extension.)

With a bit of set-theoretic care,^[d]$\mathbb{N}\mathbb{o}$ can be equipped with a topology where theopen sets are unions of open intervals (indexed by proper sets) and continuous functions can be defined.^[9] An equivalent ofCauchy 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$${\displaystyle \sum _{\alpha \in \mathbb{N}\mathbb{o}}{r}_{\alpha }{\omega }^{a_{\alpha }}}$$witha_α decreasing and having no lower bound in$\mathbb{N}\mathbb{o}$. (All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as∞ and$\mathbb{O}\mathbb{n}$).^[9]

Based on unpublished work byKruskal, a construction (bytransfinite induction) that extends the realexponential functionexp(x) (with basee) to the surreals was carried through by Gonshor.^{[8]: ch. 10}

Thepowers ofω 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 notationω^x is used in the following.

Wheny is a dyadic fraction, thepower function$x\in \mathbb{N}\mathbb{o}$,x ↦x^y 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 relationx^y+z =x^y ·x^z, and where defined it necessarily agrees with any otherexponentiation that can exist.

The induction steps for the surreal exponential are based on the series expansion for the real exponential,$${\displaystyle \exp x=\sum _{n\ge 0}\frac{x^n}{n!}}$$more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. Forx positive these are denoted[x]_n and include allpartial sums; forx negative but finite,[x]_2n+1 denotes the odd steps in the series starting from the first one with a positive real part (which always exists). Forx negative infinite the odd-numbered partial sums are strictly decreasing and the[x]_2n+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 realx <y are then

and

and this can be extended to the surreals with the definition

$${\displaystyle \exp z=\{0,\exp z_L\cdot [z-z_L{]}_n,\exp z_R\cdot [z-z_R{]}_{2n+1}\mid \exp z_R/[z_R-z{]}_n,\exp z_L/[z_L-z{]}_{2n+1}\}.}$$

This is well-defined for all surreal arguments (the value exists and does not depend on the choice ofz_L andz_R).

Using this definition, the following hold:^[e]

• exp is a strictly increasing positive function,x <y ⇒ 0 < expx < expy
• exp satisfiesexp(x +y) = expx · expy
• exp is asurjection (onto${\mathbb{N}\mathbb{o}}_{+}$) and has a well-defined inverse,log = exp^–1
• exp coincides with the usual exponential function on the reals (and thusexp 0 = 1, exp 1 =e)
• Forx infinitesimal, the value of the formal power series (Taylor expansion) ofexp is well defined and coincides with the inductive definition
  • Whenx 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 leading1)
  • Similarly, forx infinitesimally close to1,logx is given by power series expansion ofx – 1
• For positive infinitex,expx is infinite as well
  • Ifx has the formω^α (α > 0),expx has the formω^ωβ whereβ is a strictly increasing function ofα. In fact there is an inductively defined bijection$g:{\mathbb{N}\mathbb{o}}_{+}\to \mathbb{N}\mathbb{o}:\alpha \mapsto \beta$ whose inverse can also be defined inductively
  • Ifx is "pure infinite" with normal formx = Σ_α<βr_αω^a_α where alla_α > 0, thenexpx =ω^{Σ_α<βr_αωg(a_α)}
  • Similarly, forx =ω^{Σ_α<βr_αωb_α}, the inverse is given bylogx = Σ_α<βr_αω^g–1(b_α)
• 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 ofω) 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(ω^{Σ_γ<δt_γωb_γ})·r·(1 + Σ_α<βs_αω^a_α), fora_α < 0, 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 oflog (unlike forexp), 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 infinitex and any finiten,exp(x)/xⁿ is infinite
  • For any integern and surrealx >n²,exp(x) >xⁿ. This stronger constraint is one of the Ressayre axioms for the realexponential field^[7]
• exp satisfies all the Ressayre axioms for the real exponential field^[7]
  • The surreals with exponential is anelementary extension of the real exponential field
  • Forε_β an ordinal epsilon number, the set of surreal numbers with birthday less thanε_β constitute a field that is closed under exponentials, and is likewise an elementary extension of the real exponential field

The surreal exponential is essentially given by its behaviour on positive powers ofω, i.e., the function⁠${\displaystyle g(a)}$⁠, combined with well-known behaviour on finite numbers. Only examples of the former will be given. In addition,⁠${\displaystyle g(a)=a}$⁠ 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 ofω (ω^ω··ω for some number of levels).

• expω =ω^ω
• expω^1/ω =ω andlogω =ω^1/ω
• exp (ω · logω) = exp (ω ·ω^1/ω) =ω^{ω1 + 1/ω}
  • This shows that the "power ofω" function is not compatible withexp, since compatibility would demand a value ofω^ω here
• expε₀ =ω^{ωε₀ + 1}
• logε₀ =ε₀ /ω

A general exponentiation can be defined asx^y = exp(y · logx), giving an interpretation to expressions like2^ω = exp(ω · log 2)=ω^{log 2 ·ω}. Again it is essential to distinguish this definition from the "powers ofω" function, especially ifω may occur as the base.

Asurcomplex number is a number of the forma +bi, wherea andb are surreal numbers andi is the square root of−1.^[10][11] The surcomplex numbers form analgebraically closed field (except for being a proper class),isomorphic to thealgebraic closure of the field generated by extending therational numbers by aproper class ofalgebraically independenttranscendental elements.Up to fieldisomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.^[6]: Th.27

The definition of surreal numbers contained one restriction: each element ofL must be strictly less than each element ofR. If this restriction is dropped we can generate a more general class known asgames. All games are constructed according to this rule:

Construction rule

IfL andR are two sets of games then{L |R } 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{0 |0 } 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 afield, but the class of games does not. The surreals have atotal order: given any two surreals, they are either equal, or one is greater than the other. The games have only apartial 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, orfuzzy (incomparable with zero, such as{1 | −1}).

A move in a game involves the player whose move it is choosing a game from those available inL (for the left player) orR (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 afuzzy game for the first player to move.

Ifx,y, andz are surreals, andx =y, thenxz =yz. However, ifx,y, andz are games, andx =y, then it is not always true thatxz =yz. Note that "=" here means equality, not identity.

The surreal numbers were originally motivated by studies of the gameGo,^[2] and there are numerous connections between popular games and the surreals. In this section, we will use a capitalizedGame for the mathematical object{L |R }, and the lowercasegame for recreational games likeChess orGo.

We consider games with these properties:

• Two players (namedLeft andRight)
• Deterministic (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{L|R}, whereL 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 (called0) is the Game whereL andR are both empty, so the player to move next (L orR) immediately loses. The sum of two GamesG = { L1 | R1 } andH = { L2 | R2 } is defined as the GameG + H = { L1 + H, G + L2 | R1 + H,G + R2 } 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. IfG is the Game{L | R},−G is the Game{−R | −L}, i.e. with the role of the two players reversed. It is easy to showG − G = 0 for all GamesG (whereG − H is defined asG + (−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 isx. We can classify all Games into four classes as follows:

• Ifx > 0 then Left will win, regardless of who plays first.
• Ifx < 0 then Right will win, regardless of who plays first.
• Ifx = 0 then the player who goes second will win.
• Ifx || 0 then the player who goes first will win.

More generally, we can defineG > H asG − H > 0, and similarly for<,= and||.

The notationG || H means thatG andH are incomparable.G || H is equivalent toG − H || 0, i.e. thatG > H,G < H andG = H are all false. Incomparable games are sometimes said to beconfused 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 befuzzy, as opposed topositive, negative, or zero. An example of a fuzzy game isstar (*).

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 thedisjunctive 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 analyzingGo endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of theirdisjunctive 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 approaches to the surreal numbers complement the original exposition by Conway in terms of games.

In what is now called thesign-expansion orsign-sequence of a surreal number, a surreal number is afunction whosedomain is anordinal and whosecodomain is{ −1, +1 }.^{[8]: ch. 2} This notion has been introduced by Conway himself in the equivalent formulation of L-R sequences.^[6]

Define the binary predicate "simpler than" on numbers by:x is simpler thany ifx is aproper subset ofy, i.e. ifdom(x) <dom(y) andx(α) =y(α) for allα < 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 than1). Sox <y if one of the following holds:

• x is simpler thany andy(dom(x)) = +1;
• y is simpler thanx andx(dom(y)) = −1;
• there exists a numberz such thatz is simpler thanx,z is simpler thany,x(dom(z)) = −1 andy(dom(z)) = +1.

Equivalently, letδ(x, y) = min({ dom(x), dom(y)} ∪ {α :α < dom(x) ∧α < dom(y) ∧x(α) ≠y(α) }),so thatx =y if and only ifδ(x, y) = dom(x) = dom(y). Then, for numbersx andy,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 numbersx andy,x ≤y if and only ifx <y ∨x =y, andx >y if and only ify <x. Alsox ≥y if and only ify ≤x.

The relation< istransitive, and for all numbersx andy, exactly one ofx <y,x =y,x >y, holds (law oftrichotomy). This means that< is alinear order (except that< is a proper class).

For sets of numbersL andR such that∀x ∈L ∀y ∈R (x <y), there exists a unique numberz such that

• ∀x ∈L (x <z) ∧ ∀y ∈R (z <y),
• For any numberw such that∀x ∈L (x <w) ∧ ∀y ∈R (w <y),w =z orz is simpler thanw.

Furthermore,z is constructible fromL andR by transfinite induction.z is the simplest number betweenL andR. Let the unique numberz be denoted byσ(L,‍R).

For a numberx, define its left setL(x) and right setR(x) by

• L(x) = {x|_α :α < dom(x) ∧x(α) = +1};
• R(x) = {x|_α :α < dom(x) ∧x(α) = −1},

thenσ(L(x), R(x)) =x.

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 (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule∀g ∈ domf (∀h ∈ domg (h ∈ domf )) and whose range is{ −, + }. "Simpler than" is very simply defined now:x is simpler thany ifx ∈ domy. The total ordering is defined by consideringx andy as sets of ordered pairs (as a function is normally defined): Eitherx =y, or else the surreal numberz =x ∩y is in the domain ofx or the domain ofy (or both, but in this case the signs must disagree). We then havex <y ifx(z) = − ory(z) = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements ofdomf  in order of simplicity (i.e., inclusion), and then write down the signs thatf assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is{ + }.

The sumx +y of two numbersx andy is defined by induction ondom(x) anddom(y) byx +y =σ(L,‍R), where

• L = {u +y :u ∈L(x) } ∪ {x +v :v ∈L(y) },
• R = {u +y :u ∈R(x) } ∪ {x +v :v ∈R(y) }.

The additive identity is given by the number0 = { }, i.e. the number0 is the unique function whose domain is the ordinal0, and the additive inverse of the numberx is the number−x, given bydom(−x) = dom(x), and, forα < dom(x),(−x)(α) = −1 ifx(α) = +1, and(−x)(α) = +1 ifx(α) = −1.

It follows that a numberx ispositive if and only if0 < dom(x) andx(0) = +1, andx isnegative if and only if0 < dom(x) andx(0) = −1.

The productxy of two numbers,x andy, is defined by induction ondom(x) anddom(y) byxy =σ(L,‍R), where

• L = {uy +xv −uv :u ∈L(x),v ∈L(y) } ∪ {uy +xv −uv :u ∈R(x),v ∈R(y) }
• R = {uy +xv −uv :u ∈L(x),v ∈R(y) } ∪ {uy +xv −uv :u ∈R(x),v ∈L(y) }

The multiplicative identity is given by the number1 = { (0, +1) }, i.e. the number1 has domain equal to the ordinal1, and1(0) = +1.

The map from Conway's realization to sign expansions is given byf ({L |R }) =σ(M,‍S), whereM = {f (x) :x ∈L } andS = {f (x) :x ∈R }.

Theinverse map from the alternative realization to Conway's realization is given byg(x) = {L |R }, whereL = {g(y) :y ∈L(x) } andR = {g(y) :y ∈R(x) }.

In another approach to the surreals, given by Alling,^[11] explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like theaxiomatic approach to the reals, these axioms guarantee uniquenessup to isomorphism.

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

• < is atotal order over$\mathbb{N}\mathbb{o}$
• b is a function from$\mathbb{N}\mathbb{o}$onto the class of all ordinals (b is called the "birthday function" on$\mathbb{N}\mathbb{o}$).
• LetA andB be subsets of$\mathbb{N}\mathbb{o}$ such that for allx ∈A andy ∈B,x <y (using Alling's terminology,〈A,B 〉 is a "Conway cut" of$\mathbb{N}\mathbb{o}$). Then there exists a unique$z\in \mathbb{N}\mathbb{o}$ such thatb(z) is minimal and for allx ∈A and ally ∈B,x <z <y. (This axiom is often referred to as "Conway's Simplicity Theorem".)
• Furthermore, if an ordinalα is greater thanb(x) for allx ∈A,B, thenb(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^[11] derives Conway's original definition of≤ and develops surreal arithmetic.

A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich.^[12] The difference from the usual definition of a tree is that the set of ancestors of a vertex iswell-ordered, but may not have amaximal 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. Ehrlich additionally constructed an isomorphism between Conway's maximal surreal number field and the maximalhyperreals invon Neumann–Bernays–Gödel set theory.^[12]

Alling^{[11]: th. 6.55, p. 246} also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field ofHahn 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 definedabove). 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${\displaystyle {\omega }^{-\alpha }}$ summed over all ordinalsα has no surreal counterpart.

This isomorphism makes the surreal numbers into avalued 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. Thevaluation 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.

• Hyperreal number
• Non-standard analysis

• Donald Knuth's original exposition:Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness, 1974,ISBN 0-201-03812-9. More information can be found atthe book's official homepage (archived).
• 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,ISBN 1-56881-127-6.
• 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,ISBN 1-56881-130-6.
• Martin Gardner,Penrose Tiles to Trapdoor Ciphers, W. H. Freeman & Co., 1989,ISBN 0-7167-1987-8, Chapter 4. A non-technical overview; reprint of the 1976Scientific American article.
• Polly Shulman,"Infinity Plus One, and Other Surreal Numbers",Discover, December 1995.
• A detailed treatment of surreal numbers: Norman L. Alling,Foundations of Analysis over Surreal Number Fields, 1987,ISBN 0-444-70226-1.
• A treatment of surreals based on the sign-expansion realization: Harry Gonshor,An Introduction to the Theory of Surreal Numbers, 1986,ISBN 0-521-31205-1.
• A detailed philosophical development of the concept of surreal numbers as a most general concept of number:Alain Badiou,Number and Numbers, New York: Polity Press, 2008,ISBN 0-7456-3879-1 (paperback),ISBN 0-7456-3878-3 (hardcover).
• The Univalent Foundations Program (2013).Homotopy Type Theory: Univalent Foundations of Mathematics. Princeton, NJ:Institute for Advanced Study.MR 3204653. The surreal numbers are studied in the context ofhomotopy type theory in section 11.6.

Wikiversity discussessurreal numbers

Wikibooks has a book aboutsurreal numbers

• 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

• Combinatorial game theory
• Mathematical logic
• Infinity
• Real closed field
• John Horton Conway
• Nonstandard analysis
• Numbers

• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from October 2025