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RE: factor distinguished from divisor ; was [PrimeNumbers] Prime formula

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  • Jon Perry
    The point is we have two words which mean exactly the same thing. I m not going to force anyway to accept my usage, but I can t see any harm in the splitting
    Message 1 of 16 , Nov 1, 2001
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      The point is we have two words which mean exactly the same thing. I'm not
      going to force anyway to accept my usage, but I can't see any harm in the
      splitting of the definitions. Helps me out anyway.

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry
      BrainBench MVP for HTML and JavaScript
      http://www.brainbench.com


      -----Original Message-----
      From: Jud McCranie [mailto:jud.mccranie@...]
      Sent: 31 October 2001 22:43
      To: Jon Perry
      Cc: primenumbers@yahoogroups.com
      Subject: RE: factor distinguished from divisor ; was [PrimeNumbers]
      Prime formula


      At 07:52 PM 10/31/2001 +0000, Jon Perry wrote:
      >As they are vague, I think I'll stick with my factor is a prime, and
      divisor
      >is the product of one or more primes.


      This website says they are the same.

      http://www.utm.edu/research/primes/glossary/Divisor.html

      In fact the entry for "factor" links to "divisor".


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      | Programming Achieved with Structure, Clarity, And Logic |
      +---------------------------------------------------------+





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    • Walter Nissen
      Thanks for your responses. I apologize for not quoting all of your relevant points below. ... Wow, and all along I thought the subject to be number theory,
      Message 2 of 16 , Nov 3, 2001
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        Thanks for your responses. I apologize for not quoting all of your
        relevant points below.


        > "Humpty Dumpty"

        Wow, and all along I thought the subject to be number theory, not the
        politics of personal destruction. 8->


        Factorization is a process which produces factors. Any user would be
        less than satisfied by a program which produces too many "factors" or
        too few. A claim to have factors is a claim to have "soup for
        Goldilocks", just the right number. Even so, unless the product is
        important, such a claim may be somewhat superfluous. In which case,
        "divisor" will do just fine, just as it does when you don't have the
        "right" number.


        The rest of this message is at serious risk of being "much ado about
        nothing much more".


        Whenever a factor or divisor appears, there is always a product about;
        if not in the foreground, then lurking in the background. Any set of
        naturals can form a product. Simply multiply them all together. Then
        the elements of the set are the factors of the product. In determining
        the perfectness of 28, that product is 784. But it is perhaps merely a
        distraction. Also, every division implies a product. If d | D, then
        that product is D = d * q .


        > However, in that context, the set of factors I view more as a
        > /decomposition/

        Exactly. As you know, when applied to naturals, this process is usually
        called factorization.

        Every product is associated with at least one set of factors. This
        begins with the closure axiom itself. Every element of such a set is a
        factor.


        > Are 2 and 14 factors of 28?
        > Are 4 and 7 factors of 28?
        > Are 1, 4 and 7 factors of 28?

        Yes. Yes. Yes.

        > Are 14 and 28 divisors of 784, or are they factors of 784?

        Yes; no, not by themselves.


        > a factor can be equal to zero

        Again, as you know, not in Z+.

        In Z+, you eliminate the associates problem and reduce the units problem
        to the multiplicative identity. It's a pretty nice system. Especially
        with the Fundamental Theorem, i.e., unique factorization.


        > The word "reasonable" doesn't seem to make sense in this context.

        I notice you don't suggest an improvement. The word "reasonable"
        could easily be the weakest in my earlier message. By using
        "reasonable", rather than, say, "specific" or "uniquely determined", I'm
        leaning over backwards to avoid correcting an author who may be able to
        cite some tenuous reference or connection to the product.


        It's best not to be mesmerized by parallels like this:

        If a * b = c , then a and b are factors of c.

        If a * b = c , then a and b are divisors of c.

        These theorems suggest the close relationship, but don't extinguish the
        distinction.


        If you think this subject hasn't been beaten to death already, and wish
        to reply, I would ask that you please not bloat hundreds of mailboxes by
        quoting this entire message. Anyone who wants to see it can readily
        look in the message archive.


        Cheers.


        Walter Nissen
      • Milton Brown
        Better get used to it in this group. Its worse than being a politician. ... From: Walter Nissen To: Cc: my
        Message 3 of 16 , Nov 3, 2001
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          Better get used to it in this group. Its
          worse than being a politician.


          ----- Original Message -----
          From: "Walter Nissen" <wnissen@...>
          To: <primenumbers@yahoogroups.com>
          Cc: "my e-mail address is" <wnissen@...>
          Sent: Saturday, November 03, 2001 3:22 PM
          Subject: [PrimeNumbers] Re: factor distinguished from divisor


          > Thanks for your responses. I apologize for not quoting all of your
          > relevant points below.
          >
          >
          > > "Humpty Dumpty"
          >
          > Wow, and all along I thought the subject to be number theory, not the
          > politics of personal destruction. 8->
          >
          >
          > Factorization is a process which produces factors. Any user would be
          > less than satisfied by a program which produces too many "factors" or
          > too few. A claim to have factors is a claim to have "soup for
          > Goldilocks", just the right number. Even so, unless the product is
          > important, such a claim may be somewhat superfluous. In which case,
          > "divisor" will do just fine, just as it does when you don't have the
          > "right" number.
          >
          >
          > The rest of this message is at serious risk of being "much ado about
          > nothing much more".
          >
          >
          > Whenever a factor or divisor appears, there is always a product about;
          > if not in the foreground, then lurking in the background. Any set of
          > naturals can form a product. Simply multiply them all together. Then
          > the elements of the set are the factors of the product. In determining
          > the perfectness of 28, that product is 784. But it is perhaps merely a
          > distraction. Also, every division implies a product. If d | D, then
          > that product is D = d * q .
          >
          >
          > > However, in that context, the set of factors I view more as a
          > > /decomposition/
          >
          > Exactly. As you know, when applied to naturals, this process is usually
          > called factorization.
          >
          > Every product is associated with at least one set of factors. This
          > begins with the closure axiom itself. Every element of such a set is a
          > factor.
          >
          >
          > > Are 2 and 14 factors of 28?
          > > Are 4 and 7 factors of 28?
          > > Are 1, 4 and 7 factors of 28?
          >
          > Yes. Yes. Yes.
          >
          > > Are 14 and 28 divisors of 784, or are they factors of 784?
          >
          > Yes; no, not by themselves.
          >
          >
          > > a factor can be equal to zero
          >
          > Again, as you know, not in Z+.
          >
          > In Z+, you eliminate the associates problem and reduce the units problem
          > to the multiplicative identity. It's a pretty nice system. Especially
          > with the Fundamental Theorem, i.e., unique factorization.
          >
          >
          > > The word "reasonable" doesn't seem to make sense in this context.
          >
          > I notice you don't suggest an improvement. The word "reasonable"
          > could easily be the weakest in my earlier message. By using
          > "reasonable", rather than, say, "specific" or "uniquely determined", I'm
          > leaning over backwards to avoid correcting an author who may be able to
          > cite some tenuous reference or connection to the product.
          >
          >
          > It's best not to be mesmerized by parallels like this:
          >
          > If a * b = c , then a and b are factors of c.
          >
          > If a * b = c , then a and b are divisors of c.
          >
          > These theorems suggest the close relationship, but don't extinguish the
          > distinction.
          >
          >
          > If you think this subject hasn't been beaten to death already, and wish
          > to reply, I would ask that you please not bloat hundreds of mailboxes by
          > quoting this entire message. Anyone who wants to see it can readily
          > look in the message archive.
          >
          >
          > Cheers.
          >
          >
          > Walter Nissen
          >
          >
          > Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
          > The Prime Pages : http://www.primepages.org
          >
          >
          >
          > Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
          >
          >
        • Michael Bell
          ... Surely this is a stupid definition, you say 4 is not a factor of 36, but 4 and 9 are factors of 36. Surely you can see why that isn t reasonable? Michael.
          Message 4 of 16 , Nov 3, 2001
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            >
            > > Are 2 and 14 factors of 28?
            > > Are 4 and 7 factors of 28?
            > > Are 1, 4 and 7 factors of 28?
            >
            > Yes. Yes. Yes.
            >
            > > Are 14 and 28 divisors of 784, or are they factors of 784?
            >
            > Yes; no, not by themselves.
            >

            Surely this is a stupid definition, you say 4 is not a factor of 36, but 4
            and 9 are factors of 36. Surely you can see why that isn't reasonable?

            Michael.
          • Jud McCranie
            ... By themselves ? Hmmm. Doesn t the fact that 28 is a divisor/factor of 784 imply that 784/28 is also a divisor/factor?
            Message 5 of 16 , Nov 3, 2001
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              At 06:22 PM 11/3/2001 -0500, Walter Nissen wrote:
              > Are 14 and 28 divisors of 784, or are they factors of 784?

              >Yes; no, not by themselves.


              "By themselves"? Hmmm. Doesn't the fact that 28 is a divisor/factor of
              784 imply that 784/28 is also a divisor/factor?


              +---------------------------------------------------------+
              | Jud McCranie |
              | |
              | Programming Achieved with Structure, Clarity, And Logic |
              +---------------------------------------------------------+
            • Paul Leyland
              ... the ... Hmmm. Too subtle perhaps. I suggest that you consult the seminal works of that great 19th century mathematician Charles Dodgson for further
              Message 6 of 16 , Nov 4, 2001
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                > From: Milton Brown [mailto:miltbrown@...]
                > Better get used to it in this group. Its
                > worse than being a politician.
                ...
                > From: "Walter Nissen" <wnissen@...>
                ...
                > > Thanks for your responses. I apologize for not quoting all of your
                > > relevant points below.
                > >
                > >
                > > > "Humpty Dumpty"
                > >
                > > Wow, and all along I thought the subject to be number theory, not
                the
                > > politics of personal destruction. 8->

                Hmmm. Too subtle perhaps. I suggest that you consult the seminal works
                of that great 19th century mathematician Charles Dodgson for further
                insights into what I meant by those words.

                Paul

                P.S. Don't take it too personally 8-)
              • d.broadhurst@open.ac.uk
                ... Here s the passage that Paul had in mind: There s glory for you! I don t know what you mean by glory , Alice said. I meant, there s a nice
                Message 7 of 16 , Nov 4, 2001
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                  Paul Leyland wrote:

                  > I suggest that you consult the seminal works
                  > of that great 19th century mathematician Charles Dodgson
                  > for further insights into what I meant by those words.

                  Here's the passage that Paul had in mind:

                  "There's glory for you!"

                  "I don't know what you mean by 'glory'," Alice said.

                  "I meant, 'there's a nice knock-down argument for you!'"

                  "But 'glory' doesn't mean 'a nice knock-down argument',"
                  Alice objected.

                  "When I use a word,"
                  Humpty Dumpty said in a rather scornful tone,
                  "it means just what I choose it to mean -- neither more nor less."

                  Through the Looking-Glass (1872) ch. 6
                • Walter Nissen
                  I m not sure why certain people have trouble with this concept. I really don t have an explanation for that. The only thing I can think of is ring theory,
                  Message 8 of 16 , Nov 11, 2001
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                    I'm not sure why certain people have trouble with this concept. I
                    really don't have an explanation for that. The only thing I can think
                    of is ring theory, ideal theory.


                    Here is an analogy.

                    Take a typical flawed definition of perfect:
                    A perfect is the sum of all its factors.

                    Actually this isn't too typical.
                    Authors often account for one of these factors 8->
                    positive
                    proper

                    Let's account for both:
                    A perfect is the sum of its positive factors, except itself.

                    Take 28 as an example.
                    The "factors" are 1, 2, 4, 7 and 14.
                    Since factors imply a product, this product would be 784.
                    It's really hard to imagine how to obtain 28 as a product from that set
                    (though I confess 29 would be even more of a challenge 8-> ).
                    What would the rationale for selection be?

                    Now, by analogy, would you apply the same standard to divisor that you
                    apply to factor?

                    Suppose the set { 31 , 32 } is under discussion. Would you say the
                    elements of this set are integers? Would you say the elements of this
                    set are divisors? Of course, it _is_ a fact that they are divisors, of
                    say, 4063232. So what? How is that relevant to anything worth saying
                    here? Unless something more is happening, this is a set of integers.

                    And, I would suggest, in the context above, { 1 , 2 , 4 , 7 , 14 } is a
                    set of divisors.


                    Simple (albeit incomplete) operational process for authors:

                    If all numbers in a set are real, call them real.
                    If all are algebraic, call them algebraic.
                    If all rational, call them rational.
                    If all integer, call them integer.
                    If all divisors of some number under consideration, call them
                    divisors.
                    If all are factors of some number under consideration, call them
                    factors.
                    If all are prime factors of some number under consideration, call them
                    prime factors.

                    You can stop short in this process; you won't be incorrect. But you
                    also may not be making the best use of your communication. But going
                    too far in this process is wrong, misleading, confusing.

                    These distinctions are valuable and they should be preserved.

                    It's a simple matter of clarity.


                    > Doesn't the fact that 28 is a divisor/factor of
                    > 784 imply that 784/28 is also a divisor/factor?

                    Surely, yes, even 784\28. 8->


                    > > > a factor can be equal to zero

                    > > Again, as you know, not in Z+.

                    > Who did decide this restriction?

                    My remark, two messages ago, was so restricted. By me. I didn't want
                    to wander off into ring theory, and in a mailing list named
                    primenumbers, the naturals seemed like a natural 8-> set. Others are
                    welcome to take up Z, Q, etc.


                    > Of course, 'to be a factor' and 'to be a divisor' are not intrinsic
                    > properties of a number, they obviously depend on the context.
                    > Personally, I regard these words as the will of their author to give
                    > me more information (even fuzzy) about the origin, the destination,
                    > the possible use, whatever, concerning a number in a given context.

                    Yes, good.


                    > the meaning of a word is neither definitive nor the same for
                    > everybody.

                    Not all the money you receive is genuine. That doesn't mean the
                    counterfeit is less bad.

                    If we can agree on the meaning of terms such as "greater", "product",
                    "distinct", "odd", "set", "positive" (as distinguished from
                    "non-negative"), "integer", "monotonic", "prime", "factor", "twin
                    primes", "circle", "phi", then we can communicate more clearly and more
                    easily.

                    > Trying to legislate on that is not far from a pure waste of time.

                    Short of legislation is a process called standardization. It can be
                    formal or de facto. It's more of a good thing than a bad thing.


                    Cheers.

                    Walter Nissen
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