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Re: [PrimeNumbers] Positive / negative zero

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  • Phil Carmody
    ... 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 1 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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    • Paul Leyland
      ... 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 2 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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