fyi:
http://scienceblogs.com/goodmath/2007/04/post_3.php
Today we're going to take our first baby-step into the land of surreal games.
A surreal number is a pair of sets {L|R} where
every value in L is less than every value in R.
If we follow the rules of surreal construction,
so that the members of L sets are always strictly
less than members of R sets, we end up with a
totally ordered field (almost) - it gives us
something essentially equivalent to a superset of
the real numbers. (The reason for the almost is
that technically, the surreals form a class not a
set, and a field must be based on a set. But for
our purposes, we can treat them as a field without much trouble.)
But what happens if we take away the restriction
about the < relationship between the L and R
sets? What we get is a set of things called
games. A game is a pair of sets L and R, where
each member of L and R is also a game. It should
be obvious that every surreal number is also a
game - but there are many more games than there
are surreal numbers, and most games are not surreal numbers.
Games lose some of the nice properties of the
surreal numbers. They are not a field. They are
not totally ordered. In fact, they're not even
all positive or negative. They're very strange things.
So why would we want to break the restriction on
the surreals that gives us games? Naturally,
because games have useful applications in
modeling many things - in particular, games (in
the non-mathematical sense - games like Go, Chess, Checkers, Poker, etc).
Let's take a bit more of a detailed look at games, and how they interact.
Game arithmetic is exactly the same as surreal
arithmetic: addition, subtraction,
multiplication, negation - even division (which
we haven't looked at yet) are all defined in the
same way of surreal numbers and games.
But: while surreal numbers are always either
positive, negative, or zero, games can also be
fuzzy. Remember, games are not fully ordered.
That means that there are pairs of games (a,b)
where ¬a b and ¬b a - that is, where the two
games cannot meaningfully be compared. Fuzzy
games are games that can't be compared to zero.
What does a fuzzy game look like? The simplest
example is: {1|-1}. Try to use the definition of
" " on that game with zero - it doesn't work.
Games also have some strange behaviors with
respect to multiplication. If a, b, and c are
games, then (as you would expect for numbers), if
x×z=y×z then x=y. But, with games, x=y doesn't
mean that x×z=y×z. Nasty, that, eh?
So what are these beasts useful for? Part of
Conway's motivation was trying to analyze the
game of Go (aka Wei-Chi). Go is one of the oldest
strategic games in the world; it's been played
for thousands of years in China, Japan, and
Korea. Go is the Japanese name, which is
generally used here in the US; Wei-Chi is the
chinese name for it. It's a thoroughly fascinating game.
Go is a two-player game where the players have a
17x17 grid. Each move, a player puts a piece of
their own color on one of the intersections on
the grid. The goal of the game is to surround
territory using your pieces. Whoever has the most
territory at the end wins. Mechanically, it's
about as simple as a game can get. Strategically,
it's unbelievably deep and complex. It's
frequently compared to Chess in terms of depth
and strategy. It's a wonderful game. ...
---
vice-chair http://ocjug.org/
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