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Surreal Numbers
Path: CGTPath   · Prev: TreeRepresentation   · Next: NegativeOfAGame
    Keywords: EndGame

Surreal numbers are a generalization of numbers introduced by John Horton Conway in his book On Numbers and Games. In it, he also discusses positions in two-player combinatorial games. The surreal numbers are described as game positions (he calls them just 'games') of a particular form. The term 'surreal number' was coined by Donald Knuth, in the eponymous book Surreal Numbers.

A game is a set with left- and right-membership, i.e., something of the form { L | R }, where L and R are sets of games. Thanks to the empty list, this definition is not circular but recursive. { | } is a game by the definition, and it is not defined in terms of anything else.

A surreal number is a game in which all games in L and R are also surreal numbers, and in which each member of L is less than all members of R.

I've cleaned up the relationship between numbers and games a bit, but without the definition of comparison it's still inadequate. --Matthew Woodcraft

In the game-theoretical interpretation of surreal numbers as (positions of) games, L is the list of options of the "left" player and R is the list of options of the "right" player. See CGT.

On the zeroth day?
Meet { | }=0 and discover why it's zero.
On the first day?
Meet {0| }=1, { |0}=-1 and {0|0}=* and why they have the values they have.
On the second day?
Meet {1|-1} and see why {*|*}={-1|1}=0.
Meet {*| }, {*|0}, {-1|*}, {1|0}, {1|*}, {1|1} and their negatives. See why {1| }=2 and {-1| }=0.
Meet strange beasts like {*,0|*,-1}.
(Is the first number born on the Second Day supposed to be {-1|1}? -Tyler)
You could say that these are all born simultaneously. - Migeru
Sorry, looking at the question I didn't ask what I had intended to ask. Does {1|-1} simplify to *? If so, is the first term on day two a typo? -Tyler
{0|0} = *, which has a temperature/miai value of 0. {1|-1} has a temperature of 1, but it chills to *. - Migeru


Path: CGTPath   · Prev: TreeRepresentation   · Next: NegativeOfAGame
This is a copy of the living page "Surreal Numbers" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.