Surreal Number - Games

Games

The definition of surreal numbers contained one restriction: each element of L must be strictly less than each element of R. If this restriction is dropped we can generate a more general class known as games. All games are constructed according to this rule:

Construction Rule
If L and R are two sets of games then { L | R } is a game.

Addition, negation, and comparison are all defined the same way for both surreal numbers and games.

Every surreal number is a game, but not all games are surreal numbers, e.g. the game { 0 | 0 } is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a field, but the class of games does not. The surreals have a total order: given any two surreals, they are either equal, or one is greater than the other. The games have only a partial order: there exist pairs of games that are neither equal, greater than, nor less than each other. Each surreal number is either positive, negative, or zero. Each game is either positive, negative, zero, or fuzzy (incomparable with zero, such as {1|−1}).

A move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a fuzzy game for the first player to move.

If x, y, and z are surreals, and x=y, then x z=y z. However, if x, y, and z are games, and x=y, then it is not always true that x z=y z. Note that "=" here means equality, not identity.

Read more about this topic:  Surreal Number

Famous quotes containing the word games:

    Whatever games are played with us, we must play no games with ourselves, but deal in our privacy with the last honesty and truth.
    Ralph Waldo Emerson (1803–1882)

    Intelligence and war are games, perhaps the only meaningful games left. If any player becomes too proficient, the game is threatened with termination.
    William Burroughs (b. 1914)

    In the past, it seemed to make sense for a sportswriter on sabbatical from the playpen to attend the quadrennial hawgkilling when Presidential candidates are chosen, to observe and report upon politicians at play. After all, national conventions are games of a sort, and sports offers few spectacles richer in low comedy.
    Walter Wellesley (Red)