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.

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