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

> > 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]

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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"Feel free" mit GMX DSL: http://www.gmx.net/de/go/dsl > Sorry, if I did not post right. I was not about to cause any

It depends on the area people work in. For a combinatorian, the equality

> 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.

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 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

>

>

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>

> - Dr. Michael Paridon wrote:
> So I thank everybody very much for explanation. Mathematics has more hidden

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

> 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

[snip]

>

> Michael

>

>

> ----- Original Message -----

--

Alan