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zeros and infinities
On 2007-06-03 14:56:07 +1000, Peter Henderson wrote:
I don't know where the idea originated, or who adheres to it, but the
view that +/-infty is a positive integer is a bad one and contrary to
the philosophy of the standard.
"philosophy" is the right term here, and I agree.
In the standard, f(infty) is the limit as
the real argument tends to infinity, NOT the limit as the floating point
numbers tend to infinity if we allow the exponent to grow indefinitely.
Such descriptions in the standard are indeed just "philosophy", i.e. they
document the thinking that went into various specific definitions, but
they are NOT the definition itself.
The actual definitions try to accomodate multiple points of view, and tend
to conflate mutually incompatible views into a smaller number of specific
concepts. Zeros, NaNs and Infinities are all overloaded in this sense, so
it should not be surprising that one can construct logical arguments from
the same (incompletely specified) premisses to derive "inconsistencies".
As has been pointed out before, by many, there are too few bits in the short
representations to accomodate the missing distinctions. Furthermore, upon
more detailed analysis, it often turns out that the "inconsistencies" are
just pushed a bit further out. This becomes apparent when one elaborates on
the idea of incorporating an explicit "Inexact" bit in the representation,
e.g. to distinguish between the various types of zeros. It seems to work
for one operation, but then one is faced with the result of subtracting two
inexact signed zeros, for example: does the result have a sign? As the
distinctions are refined, we face a combinatorial explosion with diminishing
returns, never reaching a conclusion. So the decision was made 20 years ago
to support only the crudest distinctions and to stop at that point.
Michel.
Sent: 2007-06-03 15:43:19 UTC