Surreal number
The surreal numbers are a class of numbers which includes all of the real numbers, and additional "infinite" numbers which are larger or smaller than any real number. They also include "infinitesimal" numbers that are closer to zero than any real number, and each real number is surrounded by surreals that are closer to it than any real number. In this, the surreals are similar to the hyperreal numbers, but their construction is very different and the class of surreals is larger and contains the hyperreals as a subset. Mathematicians have praised the surreal numbers for being simpler, more general, and more cleanly constructed than the more common real number system.
Surreal numbers were first proposed by John Conway and later detailed by Donald Knuth in his 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. This book is actually a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea has been 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.
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 { L | R }. 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, { {1, 2} | {5, 8} } 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 { L | {} } 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 following construction 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 { L | R } is a surreal number.
Given a surreal number x = { X_{L} | X_{R} } 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 { {a, b, ... } | { x, y, ... } } simply as { a, b, ... | x, y, ... } and { {a} | {} } as { a | } and { {} | {a} } as { | a }.
In order for the generated numbers to actually qualify as numbers there has to be a "less than or equal to" relation (here written as ≤) defined on them. This is supplied by the following rule:
;Comparison Rule: For a surreal number x = { X_{L} | X_{R} } and y = { Y_{L} | Y_{R} } 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 two rules are recursive, so we need some form of induction to put them to work. An obvious candidiate would be finite induction, i.e., generate all numbers that can be constructed by applying the construction rule a finite number of times, but, as will be explained later on, things get really interesting if we also allow transfinite induction, i.e., apply the rule more often than that. 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 iff 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
- { 0 | }, { | 0 } and { 0 | 0 }
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 } < 0 < { 0 | }
where
x <
y denotes that
not(
y ≤
x). We will refer to
{ |
0 } and {
0 | } as
-1 and
1 respectively, and the
corresponding equivalence classes as simply -1 and 1, respectively. Since
every equivalence class contains only one element 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 } < 0 < { 0 | }
or even
- -1 < 0 < 1.
If we apply the construction rule once more we obtain the following ordered
set:
- { | -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 | }
We can now make three observations:
- We have found four new equivalence classes, viz., [{ | -1 }], [{ -1 | 0 }], [{ 0 | 1 }] and [{ 1 | }].
- All equivalence classes now contain more than one element.
- 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 { | -1 } 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 { 1 | } number
2 and its equivalence class 2. The number { -1 | 0 } is a
number between -1 and 0 and we will call it -1/2 and its
equivalence class -1/2. Finally we will call { 0 | 1 } 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 [{ X_{L} | X_{R} }] = [{ Y_{L} | Y_{R} }]
where [X] denotes { [x] | x in X }. So the description of the ordered set that was found above can be rewritten to:
- { | -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 | }
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 set, this is valid in an altered form, since infinite sets may not contain a maximal element. The number { {1, 2} | {5, 8} } therefore is equivalent to { 2 | 5 }, which will be exactly calculated later.
The addition and multiplication of surreal numbers are defined by the following three rules:
;Addition: x + y = { X_{L} + y ∪ x + Y_{L} | X_{R} + y ∪ x + Y_{R} }
where X + y = { x + y | x in X } and x + Y = { x + y | y in Y }.
;Negation: -x = { -X_{R} | -X_{L} }
where -X = { -x | x in X }
;Multiplication: xy = { (X_{L}y + xY_{L} - X_{L}Y_{L}) ∪ (X_{R}y + xY_{R} - X_{R}Y_{R}) | (X_{L}y + xY_{R} - X_{L}Y_{R}) ∪ (X_{R}y + xY_{L} - X_{R}Y_{L}) }
where XY = { xy | x in X and y in Y }, Xy = X{y} and xY = {x}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.
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 assumption that we only generate those 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:
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:
- S_{0} = { 0 }
- S_{1} = { -1 < 0 < 1 }
- S_{2} = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2}
- S_{3} = { -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 }
- S_{4} = ...
This leads to the following observations:
- In every step the maximum (minimum) is increased (decreased) by 1.
- 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.
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 = {0, 1, 2, 3, 25/8, 201/64, ... | 4, 7/2, 13/4, 51/16, 101/32, ...}.
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 infintesimal 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.
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.
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:
- 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 {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:
- 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 is fuzzy then the player who goes first will win
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.
In other words, gluing together several non-interacting games is equivalent to simply
adding their Games!
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. It is amazing that all this came out of the study of Go!
- A gentle yet thorough introduction by Claus Tøndering can be found at http://www.tondering.dk/claus/surreal.html
- 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 http://www-cs-faculty.stanford.edu/~knuth/sn.html
- 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