Gaussian rationals

[Alan Bawden <ALAN @ MIT-MC>]

The point is that -1 and -1+0*i are the same number in any clear way of
thinking about the situation.  To muddy the essential mathematics of the
situation (that the real rationals are a sub-field of the Gaussian
rationals) because of a user-interface issue (what users expect of certain
transcendental functions), is downright wrong.

I note that only a few transcendental functions have the potential
user-interface problem alluded to.  By the nature of a field, no rational
function can escape from the reals.  The GCD and LCM functions behave the
same way when restricted to the ring of integers as they do when their
domain is taken to be the ring of Gaussian integers.  The NUMERATOR and
DENOMINATOR functions can consequentially be naturally extended to Gaussian
rationals.  Various other miscellaneous functions like ABS and SIGNUM also
naturally extend to the complex case.  Only a handful of transcendental
functions that have been traditionally expected to err when given certain
real arguments, could now be expected to yield a complex result.

The functions in question are: LOG, SQRT, ASIN, ACOS, ACOSH, and ATANH.
If anyone can find any more, they should speak up.

Not coincidentally, these are all functions that require the specification
of branch cuts.  These are also all functions that deal in floating point
values.  (Because floating point is the usual representation chosen for
transcendental numbers.  Another unfortunate tradition, but not one we are
currently in a position to challenge.)

Not being a great fan of floating point, I don't like being in a position
where I have to suggest anything about these functions.  However, since I
seem to be the one on the crusade to clean up the Gaussian rationals, I
feel I must make some proposal about what to do about these six functions.
I'll offer 4 possibilities:

1.	Surprise everyone.  In Common Lisp, these six functions are
	complex-valued.  People who expect (SQRT -1) to err are wrong.  (I
	would favor this myself, but I appreciate the arguments against it.)

2.	These six functions are real-values only.  Introduce six more
	functions that have the full complex range.  Nobody is surprised.
	People who want complexes say (COMPLEX-SQRT -1).  (Actually more
	than six functions would need to be introduced; I presume we would
	want a COMPLEX-ASINH to match COMPLEX-ACOSH etc.)

3.	These six functions are complex-valued as in 1., and we additionally
	introduce six new functions that are real-valued.  People who want
	(SQRT -1) to be an error, have to say (REAL-SQRT -1) instead.
	(Again, more than six are really necessary.)

4.	Since in all cases the answer is floating point, and since -1.0 and
	#C(-1.0 0.0) really ARE different, define these functions to
	return a complex result (in the questionable part of their domains)
	ONLY if given a floating point complex argument.  Thus (SQRT -1),
	and (SQRT -1.0) are both errors, but (SQRT #C(-1.0 0.0)) is not.
	To get the effect of the COMPLEX-SQRT function, you would write
	something like (SQRT (COMPLEX X 0.0)).  If we were to decide to do
	this, then the next issue we get to haggle is what (SQRT #C(1 1))
	does (since you can hardly regard 1+i a a "questionable" part of
	the domain of SQRT!).