Surreal number

{ | -1 }

{ | -1, 0 }

{ | -1, 1 }

{ | -1, 0, 1 } <
: { |0, 1 }

-1 <
: { -1 | 0 }

{ -1| 0, 1 } <
: { -1 | }

{ | 1 }

{ -1 | 1 }

0 <
: { 0 | 1 }

{ -1, 0 | 1 } <
: { -1, 0 | }

1 <
: { 1 | }

{ 0, 1 | }

{ -1, 1 | } == { -1, 0, 1 | }

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

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 = { 0 }
: S₁ = { -1 < 0 < 1 }
: S₂ = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2}
: S₃ = { -3 < -2 < -1 1/2 < -1 < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 }
: S₄ = ...

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.

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 = { { a / 2^b in S_ω | 3a < 2^b } | { a / 2^b in S_ω | 3a > 2^b } }.
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
: &pi = {3, 25/8, 201/64, ... | ..., 101/32, 51/16, 13/4, 7/2, 4}.

Another number that is already constructed in S_ω+1 is
: ε = { 0 | ..., 1/16, 1/8, 1/4, 1/2, 1 }.
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 = { ε | ..., ε + 1/16, ε + 1/8, ε + 1/4, ε + 1/2, ε + 1 } and
: ε / 2 = { 0 | ε }.
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 = { S_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 = [{ 1, 2, 3, 4, ... | }]

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 = { ω | } and
: ω - 1 = { S_ω | ω }.
We can also do this for bigger numbers
: ω + 2 = { ω + 1 | },
: ω + 3 = { ω + 2 | },
: ω - 2 = { S_ω | ω - 1 } and
: ω - 3 = { S_ω | ω - 2 }
and even ω itself
: ω + ω = { ω + S_ω | }
where x + Y = { x + y | y in Y }. Just as 2ω is bigger than ω it can also be shown that ω/2 is smaller than ω because
: ω/2 = { S_ω | ω - S_ω }
where x - Y = { x - y | y in Y }. Finally, it can be shown that there is a close relationship between ω and ε because it holds that
: 1 / ε = ω

Note that addition of ordinals differs from addition of their surreal representations. The sum 1 + ω equals ω as ordinals, but as surreals 1 + ω = ω + 1 > ω.

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 = { X_L | X_R } 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.

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 { L | R } 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 { 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 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 {1|-1}).

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 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 {L|R}, and the lowercase game for recreational games like Chess or Go.

We consider games with these properties:

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 {L|R}, 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:

Sometimes when a game nears the end, it will decompose into several smaller games that do not interact. 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.

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. 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.

Further reading

External links