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Positive / negative zero

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  • Barubary
    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
    • 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 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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      • 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 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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