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 http://www.wikipedia.org/wiki/Surreal_number
A game is a set with left- and right-membership[1], 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 every member of L is less than every member 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[2].
Meet {0| } = 1, { |0} = -1 and {0|0} = *; and why they have the values they have[3].
Meet {1|-1} and see why {*|*} = {-1|1} = 0.
Bildstein: I get it! Black has an option to turn it into a position where he has 1 more option than White, and White has an option to turn it into a position where he has 1 more option that Black.
Bildstein: What is the exact relationship between miai values and surreal numbers?
Bill: If the number is dyadic with a denominator, when reduced to its lowest terms, of 2^n, its miai value is -1/2^n. If the number is not dyadic (such as 1/3 or 2/5), I believe that its miai value is -1/omega.
[1] Bildstein: Oh, rats! I just read TomsTechnicalIntroductionToCGT and thought I understood that games aren't actually sets, but now I'm reading that they are. Are they, or aren't they? What is left- and right-membership?
Bill: A game is a pair of sets, a left set and a right set. A game can be a member of either set.
Bildstein: So a game is not a set, it is a different beast. Thanks for clarifying that.
[2] Bildstein: I thought {|}=0 by definition. But I can't seem to "discover why". I suppose it is still the zeroth day for me.
Bill: Well, it acts like 0. 1 + 0 = 1, for instance.
In go, a ko threat can be a zero, and adding one to a ko position can alter its value.
[3] Bildstein: Jumping the gun, perhaps, but trying not to miss the boat... Why is {{|}|} = 1? I don't know. Is {{|}|} Even a surreal number? I can't seem to figure it out from the definition. Is {|} even a surreal number?
Bill: In terms of games, {{|}|} yields one move for Left, while White has no move. That's why it is +1.
Bildstein: So the number that we give a game is the number of moves left has minus the number of moves right has? But can you really talk about "the number of moves" a player has in a game? I don't know.
If {{|}|}=1, does that have any connection to a local endgame position with miai value of 1?
Bill:
This is more complex than {0 | }, but if White has sente he should resign, so it comes to the same thing.
The miai value of this play is -1, just as it is with regular go. Making a play costs a point.