I was actually thinking, during the previous thread involving Complex numbers
It may not have any practical use, but if one wanted to define an ordering for
complex numbers that was deterministic and relatively unbiased, a way to do this
would be based on what I'll call for now the "spiral distance".
Conceptually, you take an infinite length spiral line that starts at and is
centered on the origin, where for each turn the current spot on the spiral
increases an infinitesimal radius from the origin, or a distance approaching
zero, in the calculus sense. Complex numbers closer to the origin on the spiral
will be ordered earlier than those further from the spiral.
Actually calculating this is a simple comparison of the radius and angle
components of the two complex numbers in the polar coordinate system. If the
radius value is different, then the one with the smaller radius is ordered
before the one with the larger; if the two radius values are the same, then the
one with the smaller angle is ordered first; if both are the same, then the two
complex numbers are equal.
The math is just as simple as a naive comparison that just compares the real
component and then imaginary component in a cartesian coordinate system, but the
result is much more reasonable I think.
This whole principle of "distance from origin" method of ordering does also, I
suspect, scale to any number of dimensions; the one-dimensional version is
simply comparing first the absolute value of the two numbers, and then saying
that either the positive or negative version orders first.
-- Darren Duncan