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Re: Zeroes and infinities
Thorsten Siebenborn <7_born@xxxxxx> wrote:
What exactly yields the example of one case 0^0 == 1 in regard of
the question what NaN^0 should be ? And 0^0 == 1 is not just a
"feeling", it fulfills the valuable binominal theorem in case of 0.
Er, what exactly do you mean?
But I am not talking about Infinity. I am talking about the result
of the *number* one divided by *number* zero. It seems to me that
you have substituted automatically "infinity" for 1/0 when I haven't
talked about the number "infinity" at all (I have said that 1/0 isn't
an infinity).
That is correct, even in the extended reals.
Should you ever take a course in advanced theory of
integration (usually called measure theory), among the first things
defined is the extended real numbers, which adds +infty and -infty,
together with appropriate the rules of arithmetic. In this case
+/-infty*0 = 0, as the area of a rectangle with a side of length zero is
zero, no matter how long the other side. infty-infty is still
undefined. Similarly, in complex analysis, an unsigned projective
infinity is always introduced when dealing with algebraic functions.
Well, I haven't taken a course in measure theory yet, probably
because I am too dumb for such complicated things, but you
can certainly answer all my humble inquiries if I don't understand
something...
Well, I did, and I have taught it. I am afraid that Peter Henderson
is wrong. Measure theory has damn all to do with the standard
compactification of the real line, and a great deal to do with Borel
sets and Lebesgue measure (which, in turn, are about the limits of
countable sequences). While what he says about rectangles is true
in Lebesgue measure, it is EXTREMELY misleading.
For example, the measure of every rectangle with corners at (x,0) and
(x,1) is 0, and there are an infinite number of such rectangles for
x in the range (0,1), but the measure of the union of those rectangles
is 1. The contradiction is because that is a non-countable infinite
number, not Aleph null.
However, there IS a standard compactification of the real line,
which adds infinities, and another one that adds both infinities
and infinitesimals.
One thing -- in school I learned that numbers are
something like "fields", containing an inverse element for
multiplication and therefore the operation of division. Are these new
extended real numbers fields, too ? Is therefore
the operation "/" comparable with the operation division in a field ?
No. The standard compactifications are not fields. They are
topological, not algebraic, constructions. Unless I am REALLY
misremembering what I did so long ago ....
While FP numbers are crude approximations of the fields R and C
(not obeying the distributive law), they are intended to model them,
right ?
Traditionally, yes. But I believe not in IEEE 754.
Of course I assume you are a standards committee member ? That's fine,
because if someone tells me that he is a compiler writer, a FP hardware
designer or a library routine author indicating that I am not one, I
will gladly ask him all my questions. He is an expert and he will never
ever tell me stupid errors and totally wrong answers because that would
be really embarassing for a pro, wouldn't it ? Especially (just in case)
if the amateur realizes it and points it out.
I, in contrast, as an amateur may tell as many errors as I like because
I have much to learn yet.
Heck. Do you think that I don't? I have only been a professional
programmer for 40 years, on and off, and an 'expert' for 35 :-)
And, no, I am not a compiler writer or FP hardware designer ....
Regards,
Nick Maclaren,
University of Cambridge Computing Service,
New Museums Site, Pembroke Street, Cambridge CB2 3QH, England.
Email: nmm1@xxxxxxxxx
Tel.: +44 1223 334761 Fax: +44 1223 334679