## Re: [PrimeNumbers] Re: Is 2 a prime

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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
Message 1 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@...>
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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• ... 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 2 of 15 , Aug 7, 2006
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--- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
> Sorry, but I do not agree.

With what?

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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• ... Datum: Mon, 7 Aug 2006 05:30:17 -0700 (PDT) Von: Phil Carmody An: primenumbers@yahoogroups.com Betreff: Re: [PrimeNumbers] Re: Is
Message 3 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@...>
Betreff: Re: [PrimeNumbers] Re: Is 2 a prime

> --- "Dr. Michael Paridon" <dr.m.paridon@...> wrote:
> > Sorry, but I do not agree.
>
> With what?
>
>
> 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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• ... 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 4 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

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
• 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 5 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.

Michael

----- Original Message -----
From: "Peter Kosinar" <goober@...>
To: "Dr. Michael Paridon" <dr.m.paridon@...>
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
>
>
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> Diese E-Mail wurde vom NOD32 antivirus system geprüft
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>
>
• ... Try: http://en.wikipedia.org/wiki/Peano_arithmetic for a nice introduction. ... [snip] -- Alan
Message 6 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.