Flavors of zero, flavors of infinity Re: On writing negative zero - with or without sign

Ok, but there are two such numbers. Minus zero and plus zero are
both allowed as results of that operation.

Why not three? Minus zero, plus zero, and blank zero?

Which of those is "exact" zero?

...

As far as I am concerned, there is only one exact zero, not two, and
not three, and there should be no sign associated with an exact zero.

Part of the problem here is that people "think" that zero should be
simple. But it is NOT simple. The same issue arises with infinity,
but in that case no one is surprised, after all, infinity should be
difficult.


Consider the complex numbers. In cartesian representation (i.e.,
real and imaginary), zero is (apparently) simple. There is only
one "zero". Now try to define infinity, and you need infinitely
many infinities.

But reconsider the complex numbers in polar representation (i.e.,
magnitude and angle), and clearly "zero" is not so simple any more.
There are infinitely many representations for complex zero, all
with zero magnitude but with arbitrary angle, just as there are
infinitely many representations for infinity.

Finally, consider the complex numbers geometrically, as the
complex plane which can be mapped to a unit sphere uniquely,
with all points on the plane having a unique point on the sphere
to correspond, (the standard mapping is to have the unit circle
on the plane be the equator on the unit sphere, then rays from
the "north pole" of the sphere downward intersect the plane in but
one point and the sphere in but one point, establishing a 1-1
mapping). In this representation, a unique point at infinity
can be defined to be the one point on the sphere which is not
mapped to the plane, i.e., the "north pole". There is a SINGLE
unique point called "zero" and a single unique point called
"infinity".

Which of these three representations for complex numbers is
right? (Trick question) ALL THREE ARE.

So is complex infinity singular or infinite in representation? Answer,
depends upon the representation chosen.

So is complex zero singular or infinite in representation? Answer,
depends upon the representation chosen.

Then we add machine representation issues to the theoretical representation
issues and add yet another layer of complexity needing interpretation.

NO, ZERO IS NOT A SIMPLE CONCEPT. AND REPRESENTATIONS OF ZERO CANNOT
BE PRESEUMED TO BE SIMPLE, EVEN THOUGH SOME REPRESENTATIONS MANAGE TO
ACHIEVE A CERTAIN DECEPTIVE SIMPLICITY.
.

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