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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.
Details may be found for example at 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 dayMeet {0| } = 1, { |0} = -1 and {0|0} = *; and why they have the values they have. On the second dayMeet {1|-1} and see why {*|*} = {-1|1} = 0.
Sorry, I can't seem to get the diagram right... RafaelCaetano
Path: CGTPath · Prev: TreeRepresentation · Next: NegativeOfAGame This is a copy of the living page "Surreal Numbers" at Sensei's Library. ![]() |