Games - Surreal number

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	Games - Surreal number

In mathematics, the surreal numbers are a field 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 { 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 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 }.

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 = { 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 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 induction, 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 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 and respectively. So if x and y belong to the same equivalence class then = .

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 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

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