Surreal number

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In mathematics, the surreal numbers are a fieldIn the original formulation, the surreals form a proper class, rather than a set, so the term field is not precisely correct. This can be overcome by limiting the construction to a Grothendieck universe, yielding a set with the cardinality of some strongly inaccessible cardinal. containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers.

The definition and construction of the surreals is due to John Horton Conway, and exemplifies Conway's characteristic notational cleverness and originality. They were introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had simply called numbers originally. Conway liked the new name, and later adopted it himself. Conway then described the surreal numbers and used them for analyzing games in his 1976 book On Numbers and Games.

Constructing surreal numbers

The basic idea behind the construction of surreal numbers is similar to Dedekind cuts. We construct new numbers by representing them with two sets of numbers, L and R, that approximate the new number; the set L contains a set of numbers below the new number and the set R contains a set of numbers above the new number. We will write such an approximation as . We will pose no restrictions upon L and R except that each of the numbers in L should be smaller than any number in R. For example, | } is a valid construction of a certain number between 2 and 5. (Which number exactly and why will be explained later on.) The sets are explicitly allowed to be empty. The informal interpretation of a pair } will be "a number higher than any number in L", and of | R } "a number lower than any number in R". This leads to the rule:

Construction Rule

If L and R are two sets of surreal numbers and no member of R is less than or equal to any member of L then is a surreal number.

Given a surreal number x = the sets X_L and X_R are called the left set of x and right set of x respectively. To avoid lots of brackets we will write | } simply as and | } as and | } as .Lower-case characters in this notation refer to individual numbers or games, while upper-case characters refer to sets of numbers or games.

The Construction rule relies on a "less than or equal to" relation (here written as ≤) between surreal numbers. This is supplied by the rule:

Comparison Rule

For a surreal number x = and y = it holds that x ≤ y if and only if y is less than or equal to no member of X_L, and no member of Y_R is less than or equal to x.

The Comparison rule is recursive, so we need some form of mathematical induction to make it well-defined. An obvious candidate would be finite induction, which would allow us to generate all numbers that can be constructed by applying the construction rule a finite number of times. More flexibility is achieved using transfinite inductionTransfinite induction requires that there be no infinite sequence x₁, x₂, x₃, ... such that x_i+1 is an option of x_i for all i ≥ 0., as will be seen below.

If we want the generated numbers to represent numbers then the ordering that is defined upon them should be a total order. However, the relation ≤ defines only a total preorder, i.e., it is not antisymmetric. To remedy this we define the binary relation == over the generated surreal numbers such that

x == y if and only if x ≤ y and y ≤ x.

Since this defines an equivalence relation the ordering on the equivalence classes implied by ≤ will be a total order. The interpretation of this will be that if x and y are in the same equivalence class then they actually represent the same number. The equivalence classes to which x and y belong are denoted as [x] and [y] respectively. So if x and y belong to the same equivalence class then [x] = [y].

Let us now consider some examples and see how they behave under the ordering. The most simple example is of course

ie: | }

which can be constructed without any induction at all. We will call this number 0 and the equivalence class [0] will be written as 0. By applying the construction rule we can consider the following three numbers

, and

The last number is however not a valid surreal number because 0 ≤ 0. If we now consider the ordering of the valid surreal numbers we will see that

< 0 <

where x < y denotes that x ≤ y but not y ≤ x. We will refer to and as −1 and 1 respectively, and the corresponding equivalence classes as simply −1 and 1, respectively. Since every equivalence class contains only one element that has so far been defined, we can replace in statements about ordering the surreal numbers with their equivalence classes without the risk of ambiguity. For example, the statement above could also have been written as:

< 0 <

or even

−1 < 0 < 1.

If we apply the construction rule once more we obtain the following ordered set:

== == == <

== −1 <

== <

== == == 0 <

== <

== 1 <

== == ==

We can now make three observations:

1. We have found four new equivalence classes: [], [], [], and [].
2. All equivalence classes now contain more than one element.
3. The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set.

The first observation raises the question of the interpretation of these new equivalence classes. Since the informal interpretation of is "the number just before −1" we will call it number −2 and denote its equivalence class as −2. For a similar reason we will call number 2 and its equivalence class 2. The number is a number between −1 and 0 and we will call it −1/2 and its equivalence class −1/2. Finally we will call the number 1/2 and its equivalence class 1/2. More justification for these names will be given once we have defined addition and multiplication.

The second observation raises the question if we can still replace the surreal numbers with their equivalence classes. Fortunately the answer is yes because it can be shown that

if [X_L] = [Y_L] and [X_R] = [Y_R] then [] = []

where [X] denotes . So the description of the ordered set that was found above can be rewritten to:

== == == <

== −1 <

== <

== == == 0 <

== <

== 1 <

== == ==

which in turn can be rewritten as

−2 < −1 < −1/2 < 0 < 1/2 < 1 < 2.

The third observation extends to all surreal numbers with finite left and right sets. For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element. The number | } therefore is equivalent to , which will be exactly calculated later.

Computing with surreal numbers

The addition and multiplication of surreal numbers are defined by the following three rules:

Addition

x + y =

where X + y = and x + Y = .

Negation

−x =

where −X =

Multiplication

xy =

where XY = , Xy = X and xY = Y.

These operations can be shown to be well-defined for surreal numbers, i.e., if they are applied to well-defined surreal numbers then the result will again be a well-defined surreal number, i.e., the left set of the result will be "smaller" than the right set.

With these rules we can now verify that the chosen names of the numbers we found so far are appropriate. It holds for instance that 0 + 0 = 0, 1 + 1 = 2, −(1) = −1 and 1/2 + 1/2 == 1. (Note the use of equality = and equivalence ==!)

The operations as defined above are defined for surreal numbers but we would like to generalize them for the equivalence classes we defined on them. This can be done without ambiguity because it holds that

if [x] = [x' ] and [y]=[y' ] then [x + y] = [x' + y' ] and [−x] = [−x' ] and [xy] = [x'y' ]

Finally it can be shown that the generalized operations on the equivalence classes have the desired algebraic properties, i.e., the equivalence classes plus their ordering and the algebraic operations constitute an ordered field, with the caveat that they do not form a set but a proper class, see below. In fact, it is a very special ordered field: the biggest one. Every other ordered field can be embedded in the surreals. (See also the definition of rational numbers and real numbers.)

From now on we don't distinguish a surreal number from its equivalence class, and call the equivalence class itself a surreal number.

Generating surreal numbers using finite induction

Until now we have not really looked at what numbers we can and cannot create by applying the construction rule. We will first start with the numbers that can be created by applying the rule a finite number of times. We do this by inductively defining S_n with n a natural number as follows:

• S₀ =
• S_{i + 1} is S_i plus the set of all surreal numbers that are generated by the construction rule from subsets of S_i.

The set of all surreal numbers that are generated in some S_i is denoted as S_ω. The first sets of equivalence classes we will find are as follows:

S₀ =

S₁ =

S₂ =

S₃ =

S₄ = ...

This leads to the following observations:

1. In every step the maximum (minimum) is increased (decreased) by 1.
2. In every step we find the numbers that are in the middle of two consecutive numbers from the previous step.

As a consequence all generated numbers are dyadic fractions, i.e., can be written as an irreducible fraction

: a / 2^b

where a and b are integers and b ≥ 0. This means that fractions such as 1/3, 2/3, 4/3, 1/5, 5/3, 1/6 et cetera, will not be generated. Note that we can generate numbers that are arbitrarily close to them, but the numbers themselves are never generated.

\"To Infinity and Beyond\"

The next step consists of taking S_ω and continuing to apply the construction rule to it and thus constructing S_ω+1, S_ω+2 et cetera. Note that the left sets and right sets may now become infinite.

In fact, we can define a set S_a for any ordinal number a by transfinite induction. The first time a given surreal number appears in this process is called its birthday. Every surreal number has an ordinal number as its birthday. For example, the birthday of 0 is 0, and the birthday of 1/2 is 2. A number is equivalent to the simplest number between L and R, i.e., the number between L and R with the smallest ordinal as its birthday. | }, therefore, is equivalent to 3, because the birthday of 3 is less than the birthday of any other number between 2 and 5.

Already in S_ω+1 will we find the fractions that were missing in S_ω. For example, the fraction 1/3 can be defined as

1/3 = | }.

The correctness of this definition follows from the fact that

3(1 / 3) == 1.

The birthday of 1/3 is ω+1.

Not only do all the rest of the rational numbers appear in S_ω+1; the remaining finite real numbers do too. For example

π = .

Another number that is already constructed in S_ω+1 is

ε = .

It is easy to see that this number is larger than zero but less than all positive fractions, and therefore an infinitesimal number. The name for its equivalence class is therefore ε. It is not the only positive infinitesimal because it holds for instance that

2ε = and

ε / 2 = .

Note that these numbers are not yet generated in S_ω+1.

Next to infinitely small numbers also infinitely big numbers are generated such as

ω = .

Its value is clearly bigger than any number in S_ω and its equivalence class is therefore called ω. This number is equivalent with the ordinal number with the same name. We also have the equality

ω = []

In fact, all ordinal numbers can be expressed as surreal numbers. Since addition and subtraction is defined for all surreal numbers we can use ω like any other number and show for example that

ω + 1 = and

ω − 1 = .

We can also do this for bigger numbers

ω + 2 = ,

ω + 3 = ,

ω − 2 = and

ω − 3 =

and even ω itself

ω + ω =

where x + Y = . Just as 2ω is bigger than ω it can also be shown that ω/2 is smaller than ω because

ω/2 =

where x − Y = . Finally, it can be shown that there is a close relationship between ω and ε because it holds that

1 / ε = ω

The usual addition and multiplication of ordinals differs from the addition of their surreal representations. The sum 1 + ω equals ω as ordinals, but as surreals 1 + ω = ω + 1 > ω. The surreal addition and multiplication of ordinals is the same as the natural sum and natural product of ordinals.

Since every surreal number is constructed from surreal numbers "older" than itself, we can prove many theorems about surreals using transfinite induction: We show that a theorem holds for 0, and then show that it holds for x = if it holds for all elements of X_L and X_R.

Lots of numbers can be generated this way; in fact so many that no set can hold them all. The surreal numbers, like the ordinal numbers, form a proper class.

Powers of ω

To classify the "orders" of infinite surreal numbers, also known as archimedean classes, Conway associated to each surreal number x the surreal number

• ω^x = ,

where r and s range over the positive real numbers. If 0 ≤ x < y then ω^y is "infinitely greater" than ω^x, in that it is greater than r ω^x for all real numbers r. Powers of ω also satisfy the conditions

• ω^x ω^y = ω^x+y,
• ω^−x = 1/ω^x,

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

Each power of ω also has the redeeming feature of being the simplest surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal number x there will always exist some positive real number r and some surreal number y so that x − r ω^y is "infinitely smaller" than x. This gets extended by transfinite induction so that every surreal number x has a "normal form" analogous to the Cantor normal form for ordinal numbers. Every surreal number may be uniquely written as

• x = r₀ ω^y₀ + r₁ ω^y₁ + …,

where every r_α is a nonzero real number and the y_αs form a strictly decreasing sequence of surreal numbers. This "sum", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number.

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.

Games

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

Construction Rule

If L and R are two sets of games then is a game.

Addition, negation, multiplication, and comparison are all defined the same way for both surreal numbers and games.

Every surreal number is a game, but not all games are surreal numbers, e.g. the game is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a field, but the class of games does not. The surreals have a total order: 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 (incomparable with zero, such as ).

A move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a fuzzy game for the first player to move.

If x, y, and z are surreals, and x=y, then x z=y z. However, if x, y, and z are games, and x=y, then it is not always true that x z=y z. Note that "=" here means equality, not identity.

Surreal numbers and

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 (named Left and Right)
• Deterministic (no dice or shuffled cards)
• No hidden information (such as cards or tiles that a player hides)
• Players alternate taking turns
• Every game must end in a finite number of moves, even when the players don't alternate, and one player can move multiple times in a row
• 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 where that player has a clear advantage. For analyzing games, it is useful to associate a Game with every board position. The value of a given position will be the Game , where L is the set of values of all the positions that can be reached in a single move by Left. Similarly, R is the set of values of all the positions that can be reached in a single move by Right. This simple way to associate Games with games yields a very interesting result. Suppose two perfect players play a game starting with a given position whose associated Game is x. The winner of the game is determined:

• If x > 0 then Left will win.
• If x < 0 then Right will win.
• If x = 0 then the player who goes second will win.
• If x || 0 then the player who goes first will win.

The notation G || H means that G and H are incomparable. G || H is equivalent to G−H || 0. 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, you might have two subgames where whoever moves first wins, but when they are combined into one big game, it's no longer the first player who wins. Fortunately, there is a way to do this analysis. Just use the following remarkable theorem:

If a big game decomposes into two smaller games, and the small games have associated Games of x and y, then the big game will have an associated Game of x+y.

A game composed of smaller games is called the disjunctive sum 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 realization

Definitions

In an alternative realization, called the sign-expansion or sign-sequence of a surreal number, a surreal number is a function whose domain is an ordinal and range is a subset of .

Define the binary predicate "simpler than" on numbers by x is simpler than y if x is a proper subset of y, i.e. if dom(x) < dom(y) and x(α) = y(α) for all α < dom(x).

For surreal numbers define the binary relation < to be lexicographic order (with the convention that "undefined values" are greater than −1 and less than 1). So x < y if one of the following holds:

• x is simpler than y and y(dom(x)) = + 1;
• y is simpler than x and x(dom(y)) = − 1;
• there exists a number z such that z is simpler than x, z is simpler than y, x(dom(z)) = − 1 and y(dom(z)) = + 1.

Equivalently, let δ(x,y) = min( ∪ ), so that x = y if and only if δ(x,y) = dom(x) = dom(y). Then, for numbers x and y, x < y if and only if one of the following holds:

• δ(x,y) = dom(x) ∧ δ(x,y) < dom(y) ∧ y(δ(x,y)) = + 1;
• δ(x,y) < dom(x) ∧ δ(x,y) = dom(y) ∧ x(δ(x,y)) = − 1;
• δ(x,y) < dom(x) ∧ δ(x,y) < dom(y) ∧ x(δ(x,y)) = − 1 ∧ y(δ(x,y)) = + 1.

For numbers x and y, x ≤ y if and only if x < y ∨ x = y, x > y if and only if y < x, and x ≥ y if and only if y ≤ x.

< is transitive, and for all numbers x and y, exactly one of x < y, x = y, x > y, holds (law of trichotomy). This means that < is a linear order (except that < is a proper class).

For sets of numbers, L and R such that ∀x ∈ L ∀y ∈ R (x < y), there exists a unique number z such that

• ∀x ∈ L (x < z) ∧ ∀y ∈ R (z < y),
• For any number w such that ∀x ∈ L (x < w) ∧ ∀y ∈ R (w < y), w = z or z is simpler than w.

Furthermore, z is constructible from L and R by transfinite induction. z is the simplest number between L and R. Let the unique number z be denoted by σ(L,R).

For a number x, define its left set L(x) and right set R(x) by

• L(x) = ;
• R(x) = ,

then σ(L(x),R(x)) = x.

One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals.

However, similar definitions can be made that obviate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule

• ∀g ∈ dom f (∀h ∈ dom g (h ∈ dom f ))

and whose range is (a subset of) . "Simpler than" is very simply defined now—x is simpler than y if x ∈ dom y. The total ordering is defined by considering x and y as sets of ordered pairs (as a function is normally defined): Either x = y, or else the surreal number z = x ∩ y is in the domain of x or the domain of y (or both, but in this case the signs must disagree). We then have x < y if x(z) = − or y(z) = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom f in order of simplicity (i.e., inclusion), and then write down the signs that f assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is (a subset of) .

Addition and Multiplication

The sum x + y of two numbers, x and y, is defined by induction on dom(x) and dom(y) by x + y = σ(L,R), where

• L = ∪,
• R = ∪.

The additive identity is given by the number 0 = , i.e. the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number x is the number − x, given by dom(− x) = dom(x), and, for α < dom(x), (− x)(α) = − 1 if x(α) = + 1, and (− x)(α) = + 1 if x(α) = − 1.

It follows that a number x is positive if and only if 0 < dom(x) and x(0) = + 1, and x is negative if and only if 0 < dom(x) and x(0) = − 1.

The product xy of two numbers, x and y, is defined by induction on dom(x) and dom(y) by xy = σ(L,R), where

• L = ∪ ,
• R = ∪ .

The multiplicative identity is given by the number 1 = , i.e. the number 1 has domain equal to the ordinal 1, and 1(0) = + 1.

Correspondence between realizations

The map from Conway's realization to the alternative realization is given by f() = σ(M,S), where M = and S = .

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

Notes

Further reading

• Donald Knuth's original exposition: Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. 1974, ISBN 0201038129. More information can be found at [the book's official homepage]
• An update of the classic 1976 book defining the surreal numbers, and exploring their connections to games: On Numbers And Games, 2nd ed., John Conway, 2001, ISBN 1568811276.
• An update of the first part of the 1981 book that presented surreal numbers and the analysis of games to a broader audience: Winning Ways for Your Mathematical Plays, vol. 1, 2nd ed., Berlekamp, Conway, and Guy, 2001, ISBN 1568811306.
• Martin Gardner Penrose Tiles to Trapdoor Ciphers chapter 4 — not especially technical overview; reprints the 1976 Scientific American article
• Polly Shulman, "[Infinity Plus One, and Other Surreal Numbers]". Discover, December 1995. Discussed online at the "Ask Dr. Math" [forum].

See also

• Non-standard analysis
• Dehn plane

External links

• [A gentle yet thorough introduction by Claus Tøndering]
• [Surreal number] on PlanetMath

 

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