## Positive / negative zero

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• If there is a positive infinity and a negative infinity, isn t there a positive zero and a negative zero, the reciprocals of the infinities? -- Barubary
Message 1 of 3 , Nov 30, 2001
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If there is a positive infinity and a negative infinity, isn't there a
positive zero and a negative zero, the reciprocals of the infinities?

-- Barubary
• ... Given the two assumtions: If there is a positive and negative infinity which are distinct _AND_ Both the positive and negative infinities are in the set
Message 2 of 3 , Dec 1, 2001
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On Fri, 30 November 2001, "Barubary" wrote:
> If there is a positive infinity and a negative infinity, isn't there a
> positive zero and a negative zero, the reciprocals of the infinities?

Given the two assumtions:
If there is a positive and negative infinity which are distinct
_AND_
Both the positive and negative infinities are in the set with which you are permitted to perform divisions
_then maybe_ you can come up with a model that includes both a positive and a negative zero that's useful. However, the set you end up with will be sadly lacking in many of the sensible properties that we are familiar with with the reals.

So _don't_ make those two assumtions unless you're prepared for the consequences.

IEEE754 and 854 are both working models for arithmetic which contains the concept of +/-0 and +/-inf. However, they are _lousy_ as a substitute for the real reals.

Phil

Don't be fooled, CRC Press are _not_ the good guys.
They've taken Wolfram's money - _don't_ give them yours.
http://mathworld.wolfram.com/erics_commentary.html

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• ... Ah, another question which has my favourite answer: it depends. The IEEE floating point standard has an encoding where the two infinities are distinguished
Message 3 of 3 , Dec 1, 2001
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> If there is a positive infinity and a negative infinity, isn't there a
> positive zero and a negative zero, the reciprocals of the infinities?

Ah, another question which has my favourite answer: it depends.

The IEEE floating point standard has an encoding where the two
infinities are distinguished by having a one-bit difference between
them: the sign bit. In this mode, there are two encodings of zero
which are distinguished in the same way. The obvious identities are
then preserved when doing arithmetic on these quantities. However, -0
compares equal to +0 and an interesting philosophical question arises as
to how they differ when they are equal!

In mathematics, we often come across limits which are taken as some
quantity tends to zero from above or below. These limiting values could
reasonably be taken as definitions of +0 and -0. Although the limiting
values themselves are equal, the limiting values of the expression need
not be the same or, for that matter, equal to the value of the
expression at zero. For an example, consider the sign(x) function which
is -1 for negative x, 0 for x=0 and +1 for positive x.

Paul
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