Jim Choate writes:
"If every segment is to have a numerical measure as its length, then a new domain of numbers is needed, an extension to the domain of rational numbers. This new domain can no longer be constructed, as in the previous case, by number pairs. But hints for its construction are provided by a theoretical analysis of the measuring process for segments."
A mathematicaly rigorous defintion of the class of numbers called 'Real' is that which equates the members of that set to the possible lengths of an arbitrary line segment.
This seems a tad circular, as the real number line, from which line segments are constructed, is a copy of the set of real numbers. One can construct the reals from the rationals quite easily using any of several well-known methods, such as equivalence classes of Cauchy sequences, or Dedikind cuts. -- Mike Duvos $ PGP 2.6 Public Key available $ mpd@netcom.com $ via Finger. $