Surreal Number - Overview

Overview

The surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers a and b either a ≤ b or b ≤ a. (Both may hold, in which case a and b are equivalent and denote the same number.) Numbers are formed by pairing subsets of numbers already constructed: given subsets L and R of numbers such that all the members of L are strictly less than all the members of R, then the pair { L | R } represents a number intermediate in value between all the members of L and all the members of R.

Different subsets may end up defining the same number: { L | R } and { L′ | R′ } may define the same number even if L ≠ L′ and R ≠ R′. (A similar phenomenon occurs when rational numbers are defined as quotients of integers: 1/2 and 2/4 are different representations of the same rational number.) So strictly speaking, the surreal numbers are equivalence classes of representations of form { L | R } that designate the same number.

In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: { | }. This representation, where L and R are both empty, is called 0. Subsequent stages yield forms like:

{ 0 | } = 1

{ 1 | } = 2

{ 2 | } = 3

and

{ | 0 } = −1

{ | −1 } = −2

{ | −2 } = −3

The integers are thus contained within the surreal numbers. Similarly, representations arise like:

{ 0 | 1 } = 1/2

{ 0 | 1/2 } = 1/4

{ 1/2 | 1 } = 3/4

so that the dyadic 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 any real number a can be represented by { L_a | R_a }, where L_a is the set of all dyadic rationals less than a and R_a is the set of all dyadic rationals greater than a (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.

But there are also representations like

{ 0, 1, 2, 3, … | } = ω

{ 0 | 1, 1/2, 1/4, 1/8, … } = ε

where ω is a transfinite number greater than all integers and ε 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 about 2ω or ω − 1 and so forth.