- On Fri, 30 November 2001, "Barubary" wrote:
> If there is a positive infinity and a negative infinity, isn't there a

Given the two assumtions:

> positive zero and a negative zero, the reciprocals of the infinities?

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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http://www.shopping.altavista.com > If there is a positive infinity and a negative infinity, isn't there a

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

> positive zero and a negative zero, the reciprocals of the infinities?

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