small changes

[Alan Bawden <ALAN@MC.LCS.MIT.EDU>]

    Date: Sun, 23 Mar 86 18:10:53 EST
    From: Jonathan A Rees <JAR at MC>
    - Define GCD and LCM to work on Gaussian integers.  (Complex numbers
    are inessential anyhow, so you won't be obliged to implement this - but
    I'll tell you how if you care.)

    - Define new procedures NUMERATOR and DENOMINATOR on nonzero Gaussian
    rationals which return canonical numerator and denominator (denominator
    will always be a positive integer).

Super.  I wanted these two in Common Lisp, but they were overlooked in the
rush apparently.  But you have to be careful about how you state the
contract.  What you have here is incorrect.

GCD on integers always returns a non-negative result.  On Gausian rationals
you have to define some region of the complex plane that should contain the
answer.  You want it to include the non-negative integers for compatibility
with the non-Gausian case.  I would recommend 
{ a+bi | a>0 & b>=0 or a=0 & b=0 }, which consists of the "first quadrant"
and the non-negative real axis.

LCM is constrained similarly for similar reasons.

DENOMINATOR should also be defined to return a Gausian integer in the same
region.  It would be -incorrect- to always return a positive integer!  For
example, given (3/2)-(3/2)i, DENOMINATOR should return 1+i, since in lowest
terms (3/2)-(3/2)i is 3/(1+i).