Surreal numbers

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The surreal numbers are a class of numbers which includes all of the real numbers, and additional "infinite" numbers which are larger 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 perhaps the only time a new mathematical idea has been first presented in a work of fiction. The book takes the form of a dialogue, similar to Douglas Hofstadter's Goedel Escher Bach. (More information can be found at the book's official homepage.) Conway himself also covered the idea of surreal numbers 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 { 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 = { XL | XR } the sets XL and XR 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 = { XL | XR } and y = { YL | YR } it holds that x <= y if and only if y is less than or equal to no member of XL, and no member of YR 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 two observations:

  1. We have found four new equivalence classes, viz., [{ | -1 }], [{ -1 | 0 }], [{ 0 | 1 }] and [{ 1 | }].
  2. All equivalence classes now contain more than one element.

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 [XL] = [YL] and [XR] = [YR] then [{ XL | XR }] = [{ YL | YR }]

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.

Computing with Surreal Numbers

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

Addition
x + y = { XL + y U x + YL | XR + y U x + YR }

where X + y = { x + y | x in X } and x + Y = { x + y | y in Y }.

Negation
-x = { -XR | -XL }

where -X = { -x | x in X }

Multiplication
xy = { (XLy + xYL - XLYL) U (XRy + xYR - XRYR) | (XLy + xYR - XLYR) U (XRy + xYL - XRYL) }

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 then the right set.

With these rules we can now verify that the chosen names of the numbers we found sofar are appropriate. It holds for instance that 0 + 0 = 0, 1 + 1 = 2 and 1/2 + 1/2 == 1.

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.)


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 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 Sn with n a natural number as follows:

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

The set of all surreal numbers that are generated in some Si is denoted as Sω. The first sets of equivalence classes we will find are 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 = ...

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 / 2b

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 Sω and continuing to apply the construction rule to it and thus constructing Sω+1, Sω+2 et cetera. Note that the left sets and right sets may now become infinite.

In fact, we can define a set Sa 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.

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

1/3 = { { a / 2b in Sω | 3a < 2b } | { a / 2b in Sω | 3a > 2b } }.

The correctness of this definition follows from the fact that

3(1 / 3) == 1.

The birthday of 1/3 is ω+1.

Another number that is already constructed in Sω+1 is

e = { 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 Sω+1.

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

w = { Sω | }.

Its value is clearly bigger than any number in Sω and its equivalence class is therefore called ω. This number is equivalent with the ordinal number with the same name. 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 = { Sω | ω }.

We can also do this for bigger numbers

ω + 2 = { ω + 1 | },
ω + 3 = { ω + 2 | },
ω - 2 = { Sω | ω - 1 } and
ω - 3 = { Sω | ω - 2 }

and even ω itself

ω + ω = { ω + Sω | }

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 = { Sω | ω - Sω }

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


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.

Surreal numbers and Game theory

yet to be written

Further reading

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