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October 8, 2002
SH10116
Interpretation Number: 1
Topic: Division by Zero
Relevant Clause: Subclause 7.2
In IEEE Std 754-1985, subclause 7.2- Division by Zero, it says:
"If the divisor is zero and the dividend is a finite nonzero number,
then the division by zero shall be signaled. The result, when no trap occurs,
shall be a correctly signed (infinity symbol)(6.3)."
While this is apparently the convention decided on by the committee, it is mathematically incorrect and it seems as if it should have been designated as Not-a-Number, since division by zero is mathematically undefined and implies that 0*infinity=1, which is patently absurd.
Why was this convention chosen instead of NaN, since it leads to a further
degradation of our children's math abilities, given that the IEEE floating-point
standard would be considered to be authoritative on this subject, yet produces
an erroneous results.
When zero is divided by zero, no such extrapolation can be made. If it is
caused by an attempt to take the limit of
the ratio of two numbers when both become two small to be represented, then
the limit cannot be determined. If it
is caused by some mistake in the programming, then no limit exists. Thus, this
case is thought to be invalid and a NaN
of appropriate sign is returned. (The sign is the only bit of information that
can be determined.)
While counter examples to the mathematical interpretation of both of these
results can be constructed they tend to be either the result of extreme scaling
or an attempt to evaluate a non-analytic function. The former can be resolved
by
rescaling. But, as the latter involve functions that cannot (formally) be evaluated
on a computer (without extreme effort
anyway) in the region of their non-analyticity, usually no good solution exists.
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