

Re: inexactness vs. exactness

On Tue, 26 Jul 2005, Aubrey Jaffer wrote:

> So, does R5RS permit an inexact number which only one mathematical
> number rounds to?

R5RS certainly permits it, but it is difficult to imagine a practical
implementation in which it were true.

One possible such implementation (albeit not very practical) would be
a system that represented numbers as promises;  with the numbers as
written in the source code at the leaves of an expression tree, there
would be an inexact result for every exact number;  but the exact numbers
would look like formulas, and the inexact numbers could be calculated
(to an arbitrary desired precision) from them on demand.

One reason such a system wouldn't be very practical is that you'd have
to define = in such a way as to ignore differences below some epsilon;
otherwise a comparison of two exact equal numbers could be nonterminating
because otherwise = would keep asking for approximations with more and
more precision to compare as long as it hadn't yet found a difference.

Another is that the phyiscal memory of the machine would still place a
limit on the available precision.