On 01/12/08 08:39, Bill McCarthy wrote:
[...]
> My C docs say a negative number produces a domain error. A
> zero produces a range error. Assuming these Open Watcom
> docs are correct, I've specified the argument must be a
> positive NUMBER or FLOAT, hence my notation (0,inf] - you
> correctly assumed my "(" means open and "]" means closed for
> ranges. Of course it may be better to always consider "inf"
> as open.
>
> Please let me know if you spot any errors or a need for
> clarification.
>
Not sure if it was an error, a need for clarification, or both, or neither.
":echo log(0)" answers -inf
":echo log(log(0))" answers nan
":echo log(log(log(0)))" still answers nan
":echo log( 1. / 0. ) gives three Vim errors (the first of which is
"E15: Invalid expression: / 0. )"
IOW, specific floating-point errors are not handled the same way as
errors on operators which were primarily designed for integers (even if,
like "/", they were later extended to floats). I suppose generating INF
and NaN results, and no exceptions, is correct from the point of view of
the IEEE floating-point standards.
<digression>
And about the interval, I didn't spot it, or my old latin-math high
school teachings would have kicked in, where, indeed, infinity used to
be supposed to be always an open bound. Strictly speaking, log(0) is
undefined and the log function has a minus-infinity limit when its
argument tends towards zero by positive values. In common language, one
could say (except, maybe, at a math exam) that the logarithm of zero
"is" minus infinity, and that's obviously what the IEEE standard (or at
least my gcc version's implementation of it) is following.
However, I learnt algebra (and calculus) on number sets the fullest of
which is the field of complex numbers, which has no infinity element; in
projective geometry I "sort of used" lines with one infinity point
(actually, "infinity" had no meaning from a projective POV but the
"completed affine plane" used to map the projective plane onto the
affine has one line at infinity -- one of my math teachers used to say:
"take a projective plane, remove one line, and what you get is an affine
plane. If instead of removing it you paint it red, what you get is the
completed affine plane". When using analytical geometry, real or complex
numbers (still with no "infinity" number) were still used, the trick
being that instead of using N cartesian coordinates (where N is the
dimension, 2 for a plane, later 3 for 3-dimensional "space") one would
use N+1 real (or complex) coordinates, not all zero and given (how does
one say in English?) "à un facteur près" (modulo an arbitrary common
factor? I don't like using "modulo" in this sense because "modulo", in
math, is usually the arithmetic remainder operation).
</digression>
So a number set with two infinity elements is, I guess, as far as one
could go from the math I learnt and still remain [somewhat] understandable.
I suppose the previous digression is out of place here: if you like math
you probably already knew all of it, and if you don't you probably
skipped it (and rightly so). Ah, well... maybe I just didn't teach math
long enough to lose my passion for analytical geometry (as taught in
upper high school and lower college years, on second-degree curves and
surfaces using, mostly, real polynomials on complex number sets) -- but
i'm starting to digress again. Let's stop before I lose my current
audience. ;-)
Best regards,
Tony.
--
It is something to be able to paint a particular picture, or to carve a
statue, and so to make a few objects beautiful; but it is far more
glorious to carve and paint the very atmosphere and medium through
which we look, which morally we can do. To affect the quality of the
day, that is the highest of arts.
-- Henry David Thoreau, "Where I Live"
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