David Wilson asks:
Anyway, I was wondering, does 0.99999... = 1 in the surreal numbers?
That depends on how you map infinite decimals into surreals. For sets of numbers A and B with a<b whenever a in A and b in B, surreal "A | B" is the "simplest" number between A and B. The surreals that have finite sets of reals for A and B are dyadic rationals (including integers). If .9999... is mapped to {.9, .99, .999, .9999, ...} | {} then it is 1 in the surreals. But with this mapping, .3333... becomes {.3, .33, .333, .3333, ...} | {} which is 1/2 in the surreals. A better (IMO) mapping of infinite decimals to surreals is .3333... = {.3, .33, .333, .3333, ...} | {.3+.1, .33+.01, .333+.001, ...} which is 1/3 in the surreals. This mapping gives .9999... = {.9, .99, .999, .9999, ...} | (1,1,1,1,...} which is 1 - 1/omega in the surreals. In general, for numeral D an infinite decimal converging to real r, this interpretation in surreals yields D = r - 1/omega if r is a dyadic rational D = r if r is not a dyadic rational