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ZERO has NO SIGNS

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  • Archie Medrano
    Are there any other mathematicians on this list? The number 0 does not use any signs. It is neither positive nor negative. The year 2002 is not an interval.
    Message 1 of 4 , May 2 12:45 AM
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      Are there any other mathematicians on this list?

      The number 0 does not use any signs. It is neither positive nor negative.

      The year "2002" is not an interval. It is one number.


      Archie
    • Karl Ove Hufthammer
      Archie Medrano wrote in ... I m a mathematics student, FWIW. ... This is correct, cf. . But
      Message 2 of 4 , May 2 7:58 AM
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        Archie Medrano <amedrano@...> wrote in
        news:3CD0EE89.3010404@...:

        > Are there any other mathematicians on this list?

        I'm a mathematics student, FWIW.

        > The number 0 does not use any signs. It is neither positive
        > nor negative.

        This is correct, cf. <URL: http://mathworld.wolfram.com/Zero.html >.

        But sometimes negative and positive zeroes are used in computers
        and programming languages. E.g., IEEE 754 defines a (widely used)
        floating point format, where we can have both positive and
        negative zeroes. These are (naturally!) considered equal, though.

        --
        Karl Ove Hufthammer
      • bam
        1. You asked for the opinion of a mathematician . I suppose I qualify (see below). 2. What you say below is essentially correct. 3. The number zero is
        Message 3 of 4 , May 2 9:36 AM
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          1. You asked for the opinion of a "mathematician".
          I suppose I qualify (see below).


          2. What you say below is essentially correct.


          3. The number zero is neither negative nor positive-definite.
          A plus or minus sign in front of it is THEORETICALLY irrelevant.
          However, IN PRACTICE, experimenters and others sometimes use a
          plus or minus zero to indicate results that are uncertain but
          close to zero and are known to lie on one side of zero (if not
          actually zero). However, this convention does not imply that
          the number zero itself has two signed states!

          (By the way, it has long been noted that: IN THEORY, there is no
          difference between theory and practice, but IN PRACTICE there is. ;^>


          4. When years (or months) are numbered, the same whole number applies
          to the entire year, regardless of when the year begins and regardless
          of how far forward from the beginning (or backward from the end) the
          date in question is. We are not in the habit of referring to the year
          for today's date as 2002.456789 +/- .06 (or whatever -- I just made
          that up, without resort to a calculator!). Nor do we refer to this
          month as month number 5.04 (or "May plus 1/31"). This year is 2002
          (not 2002. not 2002-point-something) all year. This month is May or
          month number 5, today, yesterday, and until Memorial Day (US) an a
          few days afterward; then, the month number becomes 6 (a whole number)
          for every one of the next 30 days!


          5. It is true that some measurements (such as temperature) are made
          on a scale that runs to both sides (positive and negative) from the
          origin value of zero (unsigned). However, the natural numbers that
          we use as ordinals, to designate discrete items -- such as years --
          are whole numbers, which may have a sign but which may not have a
          fractional value between the integer values that are allowed.


          6. In most programming languages, the natural or whole numbers are
          generally represented by a data type such as "int" or "INTEGER" and
          another type such as "float" or "REAL" is used to indicate fractional
          values approximately (!) (by using one of the infinitesmally-small set
          of rational values that are representable on the hhardware available).
          (However, there is not much point in discussing the fractional or "float"
          numbers herein, since they are NOT conventionally used to indicate
          either year numbers or month numbers -- as discussed above, in point 4.)


          7. The implementation of mathematics on a digital computer is a
          somewhat different matter, however, and this may be where some of
          the confusion comes from.

          In the past, computer circuitry for implementing integer (and float)
          arithmetic came in (at least) three flavors. Some hardware used
          "twos-complement" arithmetic, some used "one-complement", some
          used "sign-magnitude" representation for the values and corresponding
          circuitry for the arithmetic.

          In a twos-complement machine (such as the IBM 7040, my second computer),
          negative numbers are indicated by complementing (0 becomes 1; 1 becomes 0)
          every bit of the corresponding positive value. Thus, in a six-bit word
          (or field), the bits representing ten and twelve would be 001010 and 001100
          respectively, and bits representing -10 an -12 would be: 110101 and 110011
          respectively. Unfortuantely, this leaves two bit configurations "between"
          those for the signed numbers +1 and -1, i.e. 000000 and 111111. These
          degenerate representations for the same value were were sometimes written
          as +0 and -0 (and systems sometimes used -0 as a flag for uninitialized
          data). Nevertheless, the mathematical/arithmetic value represented by
          000000 and 111111 is exactly the same with two-acomplement arithmetic.
          (As an aside, I might note that even without twos-complement, there are
          many, many degenerate representations possible in unnormalized floating-
          point notation. However, I shall try to resist the temptation to digress.)

          I'll skip sign-magnitude, except to say that one bit was used to indicate
          the sign (1 for minus; 0 for plus) and the remaining bits were treated as
          an unsigned value for the magnitude of the number being represented.

          Ones-complement arithmetic (which I first saw on my third machine,
          the CDC-6600) eliminates the degenerate representation between that for
          zero and that for -1. In ones-complement hardware, the six-bit field
          containing 111111 would represent -1 (not 0) and the numbers -10 and -12
          would be 110110 and 110100 respectively.

          To change the sign of a ones-complement number, the operation would be:
          "complement and increment" (or CIA on a PDP-8 machine), i.e. first
          complent all of the bits, then add one. (Note that, for 000000,
          complementing gives 111111 and adding one gives 000000.)

          Twos-complement and sign-magnitude representation schemes have the
          advantages that the hardware to change sign is simpler and the range
          of possible values is symmetrical (e.g. 15-bits can express values
          from -32767 to +32768).

          (Incidentally, my first computer was a IBM-650, which represented
          numbers in a "bi-quinary" scheme, using a 5-state device and a
          two-state device, with a drum for memory and vacuum-tube based
          hardware -- but you don't really want to hear about that!)
          (But if you do, I still have the manuals!!)


          8. Anyhow, most computers today use ones-complement hardware,
          and the -0 is not an issue. (There is a sort-of minus zero
          for floating-point numbers, when the sign is negative but the
          magnitude is too small to normalize. But I said I'd avoid
          the subject of floating-point approximations.)

          So, I think some of the confusion is that the relic of -0
          lingers on in the standards for some programming languages.
          (On X3J3, we eliminated it for Fortran 90.)

          Also, some confusion comes from the use of linear measurement
          scales, such as temperature or distance. However, a measurement
          scale is NOT appropriate for assigning ordinal numbers or COUNTING.
          When you count things, a distinct whole number is given to each
          discrete item being counted, e.g. months, states of dice,
          population census, chips in a pot, pennies in a piggy bank,
          number of college courses passed, etc.

          Further confusion arises when the identifying numbers assigned to
          the distinct items happen to be negative. However, the use of -1
          or -999 to identify a specific year (or a specific person or a
          specific student in a classroom) does not require either fractions
          or the use of degenerate -0 values.


          9. Finally, consider this: People are queued up in a straight line,
          to wait for a movie, etc. While it would be possible to count them
          by assigning 1 to the first one, 2 to the next, etc., I ould just as
          easily assign 1 to the person ahead of me, to the one ahead of him,
          and so forth. Now, if I also wish to see how many people follow me,
          I could assign the number -1 to the one directly behind me, -2 to the
          next, etc. I chose zero for myself (but one could just as easily
          pick someone else on the line as the arbitrary zero item).

          Note that I may begin this assignment prcess even without being
          able to see the beginning of the line and/or the end of the line
          (which might be growing as I count!) (And the line might already
          be entering the theatre, too!) Still, it is easy to calculate how
          many seats will be taken by the portion of the line from the lady
          in the red hat to the man with the popcorn, provided I know their
          assigned (positive or negative) numbers.

          If I really wanted to calculate the length of the line in millimeters -- from halfway
          thru him to 3/4 of the way thru her, this could be easily done
          using real numbers. However, that measurement calculation does
          not in any way invalidate the assignment of discrete numbers to
          the individual peopl in the line.

          Numbering the people ahead or behind an arbitraryly chosen discrete
          item as a "starting point" does not require the establishment of
          a degenerate minus zero -- because we are not slicing the people
          up into fractional parts, nor are we ignoring the boundary between
          each whole person. The same is true for the numbering of years in
          a sequence (with no certain end or precise beginning).

          Bruce A. Martin
          B.S. Mathematics, Polytechnic University, 1964
          Ph.D. candidate in Physics, Polytechnic University, 1965-1970
          Staff member in the Applied Mathematics, Brookhaven National Laboratory, 1966-1980
          Adjunct Professor of Computer Science (at four different colleges)




          Archie Medrano wrote:
          >
          > Are there any other mathematicians on this list?
          >
          > The number 0 does not use any signs. It is neither positive nor negative.
          >
          > The year "2002" is not an interval. It is one number.
          >
          > Archie
          >
          >
          >
          >
          > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
        • Ed Davies
          ... I think there s some confusion here. Ones complement (as used on the CDC machines, IIRC) is to simply invert the bits (giving two representations of zero
          Message 4 of 4 , May 3 2:12 AM
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            bam wrote:

            > ....
            > In a twos-complement machine (such as the IBM 7040, my second computer),
            > negative numbers are indicated by complementing (0 becomes 1; 1 becomes 0)
            > every bit of the corresponding positive value. Thus, in a six-bit word
            > (or field), the bits representing ten and twelve would be 001010 and 001100
            > respectively, and bits representing -10 an -12 would be: 110101 and 110011
            > respectively. Unfortuantely, this leaves two bit configurations "between"
            > those for the signed numbers +1 and -1, i.e. 000000 and 111111.
            > ....
            > Ones-complement arithmetic (which I first saw on my third machine,
            > the CDC-6600) eliminates the degenerate representation between that for
            > zero and that for -1. In ones-complement hardware, the six-bit field
            > containing 111111 would represent -1 (not 0) and the numbers -10 and -12
            > would be 110110 and 110100 respectively.

            I think there's some confusion here.

            Ones complement (as used on the CDC machines, IIRC) is to simply invert
            the bits (giving two representations of zero (000000 and 111111)) whereas
            twos complement is to invert the bits then add one (000000 => 111111 =>
            000000, discarding the carry) so there is only one representation of
            zero. Most machines these days (e.g., those based on the Intel '86 family
            or the Motorola 68000 (I think - at least the 6800 did)) use twos complement
            for integer arithmetic though not necessarily for floating point.

            I've never used a 7040 but the first computer I used was its little brother,
            the 1130 which used twos complement arithmetic on 16 bit words. The FORTRAN
            IV manual for this machine had a good explanation of twos complement using
            a way of looking at the matter I have not seen described since. Instead of
            considering rather magic bit manipulations as above it pointed out that the
            the effect is that the weight of the top bit is -32768.

            To follow the examples above the weights on a six bit twos complement
            value would be:

            -32, 16, 8, 4, 2, 1

            so +5 = 000101. We get -5 by inverting to get 111010 then add one to
            get 111011 which is -32 + 16 + 8 + 2 + 1 = -32 + 27 = -5.

            Another way of looking at twos complement is that the top half of the number
            range is picked up and moved down below zero. So 111111 (-1) is followed by
            00000 (0) which seems reasonable.

            I don't want to get drawn into this debate about +/- year zero but couldn't
            let this misinformation on one and twos complement stand, however irrelevant.

            Ed.

            P.S., just to make sure it wasn't me spreading misinformation I double
            checked this by having a look in "The Art Of Electronics" by Paul
            Horowitz and Winfield Hill. On pages 476 and 477 (of my second
            edition) they describe two other representations we needn't go into
            here then under "2's complement representation" they say:

            > ....a negative number is simply represented as the binary number you
            > add to a positive number of the same magnitude to get zero. To form a
            > negative number, first complement each of the bits of the positive
            > number (i.e., write 1 for 0, and vice versa; this is called the "1's
            > complement"), then add 1 (that's the "2's complement").
            > ....
            > There's only one zero, conveniently represented by all bits 0....
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