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Re: [PrimeNumbers] Re: Is 2 a prime

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  • Jack Brennen
    ... I would suggest that one can define divisibility by zero without needing a definition for division by zero. Say that X is divisible by A if there exists
    Message 1 of 15 , Aug 5, 2006
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      Dr. Michael Paridon wrote:
      > I actually think division by zero is not defined.
      >

      I would suggest that one can define divisibility by zero without
      needing a definition for division by zero. Say that X is divisible by A
      if there exists any element B such that X = AB.


      Thus you do not need to define exactly which B represents X/A, only
      that some such B exists. By this definition, zero is divisible by zero.
      Non-zero is not divisible by zero.
    • Dr. Michael Paridon
      Sorry, but I do not agree. Due to correction: a) I think divisibility is defined for natural numbers only. b) You suggested Say that X is divisible by A if
      Message 2 of 15 , Aug 7, 2006
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        Sorry, but I do not agree.

        Due to correction:

        a) I think divisibility is defined for natural numbers only.

        b) You suggested "Say that X is divisible by A if there exists any element B such that X = AB."

        I think it is "...there exists one and only one distinct element B such that X = AB." Which of course leads to non-definition of division by zero in all cases. As a) does, too.

        Best regards

        Michael Paridon

        -------- Original-Nachricht --------
        Datum: Sat, 05 Aug 2006 09:05:57 -0700
        Von: Jack Brennen <jb@...>
        An: primenumbers@yahoogroups.com
        Betreff: Re: [PrimeNumbers] Re: Is 2 a prime

        > Dr. Michael Paridon wrote:
        > > I actually think division by zero is not defined.
        > >
        >
        > I would suggest that one can define divisibility by zero without
        > needing a definition for division by zero. Say that X is divisible by A
        > if there exists any element B such that X = AB.
        >
        >
        > Thus you do not need to define exactly which B represents X/A, only
        > that some such B exists. By this definition, zero is divisible by zero.
        > Non-zero is not divisible by zero.
        >

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      • Phil Carmody
        ... With what? Please don t top-post, m kay? ... But Jack provides a definition which works for N / { 0 }. Yes, there exist a handful of simple and convenient
        Message 3 of 15 , Aug 7, 2006
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          --- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
          > Sorry, but I do not agree.

          With what?

          Please don't top-post, m'kay?

          Fixing:

          > Von: Jack Brennen <jb@...>
          > > Dr. Michael Paridon wrote:
          > > > I actually think division by zero is not defined.
          > >
          > > I would suggest that one can define divisibility by zero without
          > > needing a definition for division by zero. Say that X is divisible by A
          > > if there exists any element B such that X = AB.
          > >
          > >
          > > Thus you do not need to define exactly which B represents X/A, only
          > > that some such B exists. By this definition, zero is divisible by zero.
          > > Non-zero is not divisible by zero.

          > Due to correction:
          >
          > a) I think divisibility is defined for natural numbers only.

          But Jack provides a definition which works for N \/ { 0 }.
          Yes, there exist a handful of simple and convenient definitions which only work
          for natural numbers, but Jack's wording was pedantically correct - one can
          provide a definition which does what Jack says it does.

          > b) You suggested "Say that X is divisible by A if there exists any element B
          > such that X = AB."
          >
          > I think it is "...there exists one and only one distinct element B such that
          > X = AB." Which of course leads to non-definition of division by zero in all
          > cases. As a) does, too.

          That's one possible definition, yes. If Jack were to rely on his definition of
          divisibility in a paper, I feel sure that he would include that definition if
          there was any chance of ambiguity.

          To be deliberately contrary (shock horror!) I would propose that the simplest
          definition of divisibility is one which doesn't mention division at all, it
          simply refers to properties of ideals. a|b := (b) \subset (a).

          You might enjoy working out the divisibility properties of 0 using this
          definition.

          Phil

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        • Dr. Michael Paridon
          ... Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT) Von: Phil Carmody An: primenumbers@yahoogroups.com Betreff: Re: [PrimeNumbers] Re: Is
          Message 4 of 15 , Aug 7, 2006
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            -------- Original-Nachricht --------
            Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT)
            Von: Phil Carmody <thefatphil@...>
            An: primenumbers@yahoogroups.com
            Betreff: Re: [PrimeNumbers] Re: Is 2 a prime

            > --- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
            > > Sorry, but I do not agree.
            >
            > With what?
            >
            > Please don't top-post, m'kay?
            >
            > Fixing:
            >
            > > Von: Jack Brennen <jb@...>
            > > > Dr. Michael Paridon wrote:
            > > > > I actually think division by zero is not defined.
            > > >
            > > > I would suggest that one can define divisibility by zero without
            > > > needing a definition for division by zero. Say that X is divisible by
            > A
            > > > if there exists any element B such that X = AB.
            > > >
            > > >
            > > > Thus you do not need to define exactly which B represents X/A, only
            > > > that some such B exists. By this definition, zero is divisible by
            > zero.
            > > > Non-zero is not divisible by zero.
            >
            > > Due to correction:
            > >
            > > a) I think divisibility is defined for natural numbers only.
            >
            > But Jack provides a definition which works for N \/ { 0 }.
            > Yes, there exist a handful of simple and convenient definitions which only
            > work
            > for natural numbers, but Jack's wording was pedantically correct - one can
            > provide a definition which does what Jack says it does.
            >
            > > b) You suggested "Say that X is divisible by A if there exists any
            > element B
            > > such that X = AB."
            > >
            > > I think it is "...there exists one and only one distinct element B such
            > that
            > > X = AB." Which of course leads to non-definition of division by zero in
            > all
            > > cases. As a) does, too.
            >
            > That's one possible definition, yes. If Jack were to rely on his
            > definition of
            > divisibility in a paper, I feel sure that he would include that definition
            > if
            > there was any chance of ambiguity.
            >
            > To be deliberately contrary (shock horror!) I would propose that the
            > simplest
            > definition of divisibility is one which doesn't mention division at all,
            > it
            > simply refers to properties of ideals. a|b := (b) \subset (a).
            >
            > You might enjoy working out the divisibility properties of 0 using this
            > definition.
            >
            > Phil
            >
            > () ASCII ribbon campaign () Hopeless ribbon campaign
            > /\ against HTML mail /\ against gratuitous bloodshed
            >
            > [stolen with permission from Daniel B. Cristofani]


            Sorry, if I did not post right. I was not about to cause any inconvenience.

            I did not say Jack's definition is not possible. I think is not the definition mostly used, as usally division by zero is not defined. As well as 0^0, if I remember right. Jack's definition leads to an agreeable result, I admit.

            Best regards

            Michael Paridon



            --


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          • Peter Kosinar
            ... It depends on the area people work in. For a combinatorian, the equality 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number of
            Message 5 of 15 , Aug 7, 2006
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              > Sorry, if I did not post right. I was not about to cause any
              > inconvenience.
              >
              > I did not say Jack's definition is not possible. I think is not the
              > definition mostly used, as usally division by zero is not defined. As
              > well as 0^0, if I remember right. Jack's definition leads to an
              > agreeable result, I admit.

              It depends on the area people work in. For a combinatorian, the equality
              0^0 = 1 can work perfectly well; as the left-hand-side denotes the number
              of functions from empty set to empty set [*]. Moreover, things like
              binomial theorem also work nice with this extension; it allows you to
              evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or
              simply 0^n (ok, I admit, this is just a contrived academic example).

              Likewise, if you work in the area of foundations of mathematics, defining
              divisibility using the operation of division is a bit more complicated
              than using the straight existential-quantifer with multiplication (just
              like Jack did); for the division is only a derived operation in e.g. Peano
              Arithmetics and one needs to prove its well-definedness (and possibly some
              other properties) first.

              On the other hand, an analyst would probably bop you over the head
              if he saw 0^0 :-)

              Peter

              [*] This works even in the much more general framework of set-theory --
              If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined
              to be the cardinality of the set A^B which is the set of all functions
              from B to A. If the sets A and B are finite, the cardinal exponentation
              agrees with the usual exponentation of natural numbers.

              --
              [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
            • Dr. Michael Paridon
              So I thank everybody very much for explanation. Mathematics has more hidden beauties I will ever learn, and as I am not in a professional way dealing with
              Message 6 of 15 , Aug 7, 2006
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                So I thank everybody very much for explanation. Mathematics has more hidden
                beauties I will ever learn, and as I am not in a professional way dealing
                with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

                Best regrads

                Michael


                ----- Original Message -----
                From: "Peter Kosinar" <goober@...>
                To: "Dr. Michael Paridon" <dr.m.paridon@...>
                Cc: <primenumbers@yahoogroups.com>
                Sent: Monday, August 07, 2006 4:19 PM
                Subject: Re: [PrimeNumbers] Re: Is 2 a prime


                > > Sorry, if I did not post right. I was not about to cause any
                > > inconvenience.
                > >
                > > I did not say Jack's definition is not possible. I think is not the
                > > definition mostly used, as usally division by zero is not defined. As
                > > well as 0^0, if I remember right. Jack's definition leads to an
                > > agreeable result, I admit.
                >
                > It depends on the area people work in. For a combinatorian, the equality
                > 0^0 = 1 can work perfectly well; as the left-hand-side denotes the number
                > of functions from empty set to empty set [*]. Moreover, things like
                > binomial theorem also work nice with this extension; it allows you to
                > evaluate the sum [k=0,n,(-1)^k*(n choose k)] as being equal to (1-1)^n, or
                > simply 0^n (ok, I admit, this is just a contrived academic example).
                >
                > Likewise, if you work in the area of foundations of mathematics, defining
                > divisibility using the operation of division is a bit more complicated
                > than using the straight existential-quantifer with multiplication (just
                > like Jack did); for the division is only a derived operation in e.g. Peano
                > Arithmetics and one needs to prove its well-definedness (and possibly some
                > other properties) first.
                >
                > On the other hand, an analyst would probably bop you over the head
                > if he saw 0^0 :-)
                >
                > Peter
                >
                > [*] This works even in the much more general framework of set-theory --
                > If A and B are sets with cardinalities |A| resp. |B|, |A|^|B| is defined
                > to be the cardinality of the set A^B which is the set of all functions
                > from B to A. If the sets A and B are finite, the cardinal exponentation
                > agrees with the usual exponentation of natural numbers.
                >
                > --
                > [Name] Peter Kosinar [Quote] 2B | ~2B = exp(i*PI) [ICQ] 134813278
                >
                >
                > __________ NOD32 1.1694 (20060805) Information __________
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                >
              • Alan McFarlane
                ... Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction. ... [snip] -- Alan
                Message 7 of 15 , Aug 7, 2006
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                  Dr. Michael Paridon wrote:
                  > So I thank everybody very much for explanation. Mathematics has more hidden
                  > beauties I will ever learn, and as I am not in a professional way dealing
                  > with Mathematics, I hope I will find out what, e.g., Peano Arithmetics, is.

                  Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction.

                  > Best regrads
                  >
                  > Michael
                  >
                  >
                  > ----- Original Message -----
                  [snip]


                  --
                  Alan
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