In some (albeit rare) scenarios, you might
want to do some operation with a value that
is extremely close to, but not equal, to a
number. An example of this would be
expressing the maximum value of a line as it
approaches an asymptote, without reverting
to complicated inequalities.

The
constant would have a value of
0.0000000000...1, and would simply be used
by adding to or subtracting from another
number.

I actually have automated my startup to set a system variable to 0.0000000001 at work. I would like "almost zero" as zero is unacceptable for that particular system variable.
+ to you.

well, the next time I have to... ...express the maximum value of a line as it approaches an asymptote, without reverting to complicated inequalities...

I remember having versions of
POV Ray rendering incorrectly if you
looked exactly along an axis. I got into
the habit of making sure that none of
the camera parameters were exactly
zero by putting a number in the 5th or
6th significant digit.

There are situations where you need to
perturb variables, but I don't think a
single value is the answer. Different
applications will need different levels of
accuracy. Plus, any perturbations should
appear random. You wouldn't want to
eliminate the effect when you subtract
one variable from another.

Vaguely remembering that there are different levels of infinity, would this number be 1/infinity? It would be slicky if I could type that infinity character but I have had no luck getting funky characters to show up in the HB.

This sounds very useful for calculus proofs (of limits, derivatives, etc.). These proofs often require wording such as "suppose x equals y + epsilon", and end with numerical computations leading to "... Choose delta = 13/743 epsilon."

To get rid of the fraction part of that proof, you propose a number AlmostZero, which is smaller than itself! Presumably, it is only smaller than itself if you compare it to itself a finite number of times. Then you can use AlmostZero instead of Epsilon in the above proof and the proof is much simpler.