Joel Uckelman on Wed, 28 Jul 2010 11:00:01 -0700 (MST) |
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Re: [game-lang] Another quick tangential comment about integers and reals |
Thus spake Simon McGregor: > On Wed, Jul 28, 2010 at 2:21 PM, Joel Uckelman <uckelman@xxxxxxxxx> wrote: > > Thus spake Simon McGregor: > >> > >> In regard to rationals and reals, imagine the following: a game is > >> played by moving points on a board in R2, up to 1 unit's distance at a > >> time (with the Euclidean metric). We can model this using the > >> rationals, because rational arithmetic is adequate to determine > >> whether two points with rational coordinates are within a rational > >> distance of one another. > >> > >> However, suppose that the provably optimal move in some situation is > >> to move your point as far as possible towards another point. Then (in > >> general), no move in the rational model is optimal, although there's > >> an optimal move in the continuous model. Uh-oh! > >> > > > > I don't follow this last bit of reasoning. It seems to imply that there's > > some real r which is nearer to a target t than any rational q is. > > Yup. Some real (r_1, r_2) which is nearer to a rational target (t_1, > t_2) than any rational (q_1, q_2), given the constraints imposed by > the movement range rule - that the Euclidean distances of (r_1, r_2) > and (q_1, q_2) from a prior rational point (p_1, p_2) are a specified > rational distance (1) or less. The need for irrationals comes from the > square root sign in the Euclidean metric. I see what you're saying now---I was not considering that the target could be an irrational distance away. -- J. _______________________________________________ game-lang mailing list game-lang@xxxxxxxxx http://lists.ellipsis.cx/mailman/listinfo/game-lang