We can make `inf - inf' be any real number we like, say,
a, just by claiming that we added a to the first
infinity but not to the second. This is just as true for complex
values of a, so nan can stand for a complex number.
(And, similarly, uinf can stand for an infinity that points
in any direction in the complex plane, such as `(0, 1) inf').
In fact, we can multiply the first inf by two. Surely
`2 inf - inf = inf', but also `2 inf - inf = inf - inf = nan'.
So nan can even stand for infinity. Obviously it's just
as easy to make it stand for minus infinity as for plus infinity.
The moral of this story is that "infinity" is a slippery fish
indeed, and Calc tries to handle it by having a very simple model
for infinities (only the direction counts, not the "size"); but
Calc is careful to write nan any time this simple model is
unable to tell what the true answer is.