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In article <IINNj.2620$ko5.41@news-server.bigpond.net.au>,
 "Don H" <donlhumphries@bigpond.com> wrote:

> 1^0 = 1 ; 2^0 = 1; 3^0 = 1 ; etc.
> Is this axiomatic, or is there a proof?
> ==============================

It follows from the rule a^x/a^y = a^(x-y), for a^y =/= 0.

On Thu, 17 Apr 2008 19:40:24 GMT, "Don H" <donlhumphries@bigpond.com>
wrote:

>1^0 = 1 ; 2^0 = 1; 3^0 = 1 ; etc.
>Is this axiomatic, or is there a proof?
>==============================
>
 It isn't a math fact --- it's a definition --- made so as to be
consistant and for convenience when writing infinite series etc.

<Pfsszxt@aol.com> wrote in message
news:jlah04drbo3nucjn5n7n438gqbng4auf0r@4ax.com...
> On Thu, 17 Apr 2008 19:40:24 GMT, "Don H" <donlhumphries@bigpond.com>
> wrote:
>
> >1^0 = 1 ; 2^0 = 1; 3^0 = 1 ; etc.
> >Is this axiomatic, or is there a proof?
> >==============================
> >
>  It isn't a math fact --- it's a definition --- made so as to be
> consistant and for convenience when writing infinite series etc.

# And so is an arbitrary definition?
   What is a number n to the power x?
   The x indicates how many n must be multiplied together.
   Thus,
   n^x = n by n (x times)
   2^2 = (2, twice) = 4
   2^1 = (2, once) = 2
   2^0 = (2, no times) = 1 ?
     - but if you multiply 2 by itself, no times, you've still 2 ?
Whereas, if you multiple 0 by 0, two times, you've still 0 ?

In article <QIMOj.3408$ko5.2468@news-server.bigpond.net.au>,
 "Don H" <donlhumphries@bigpond.com> wrote:

> <Pfsszxt@aol.com> wrote in message
> news:jlah04drbo3nucjn5n7n438gqbng4auf0r@4ax.com...
> > On Thu, 17 Apr 2008 19:40:24 GMT, "Don H" <donlhumphries@bigpond.com>
> > wrote:
> >
> > >1^0 = 1 ; 2^0 = 1; 3^0 = 1 ; etc.
> > >Is this axiomatic, or is there a proof?
> > >==============================
> > >
> >  It isn't a math fact --- it's a definition --- made so as to be
> > consistant and for convenience when writing infinite series etc.
> 
> # And so is an arbitrary definition?
>    What is a number n to the power x?
>    The x indicates how many n must be multiplied together.
>    Thus,
>    n^x = n by n (x times)
>    2^2 = (2, twice) = 4
>    2^1 = (2, once) = 2
>    2^0 = (2, no times) = 1 ?
>      - but if you multiply 2 by itself, no times, you've still 2 ?
> Whereas, if you multiple 0 by 0, two times, you've still 0 ?

The standard inductive (or, if  you prefer, recursive) definition of 
powers with positive integer exponents is:
   For any real/complex number x, x^1 = x  and x^(n+1) = x^n * x.

"Virgil" <Virgil@gmale.com> wrote in message
news:Virgil-6FD0F3.13355320042008@comcast.dca.giganews.com...
> In article <QIMOj.3408$ko5.2468@news-server.bigpond.net.au>,
>  "Don H" <donlhumphries@bigpond.com> wrote:
>
> > <Pfsszxt@aol.com> wrote in message
> > news:jlah04drbo3nucjn5n7n438gqbng4auf0r@4ax.com...
> > > On Thu, 17 Apr 2008 19:40:24 GMT, "Don H" <donlhumphries@bigpond.com>
> > > wrote:
> > >
> > > >1^0 = 1 ; 2^0 = 1; 3^0 = 1 ; etc.
> > > >Is this axiomatic, or is there a proof?
> > > >==============================
> > > >
> > >  It isn't a math fact --- it's a definition --- made so as to be
> > > consistant and for convenience when writing infinite series etc.
> >
> > # And so is an arbitrary definition?
> >    What is a number n to the power x?
> >    The x indicates how many n must be multiplied together.
> >    Thus,
> >    n^x = n by n (x times)
> >    2^2 = (2, twice) = 4
> >    2^1 = (2, once) = 2
> >    2^0 = (2, no times) = 1 ?
> >      - but if you multiply 2 by itself, no times, you've still 2 ?
> > Whereas, if you multiple 0 by 0, two times, you've still 0 ?
>
> The standard inductive (or, if  you prefer, recursive) definition of
> powers with positive integer exponents is:
>    For any real/complex number x, x^1 = x  and x^(n+1) = x^n * x.

# Thanks.
   However, consider two rabbits, say, Rob and Ruby, living in a Platonic
relationship (2^0), who then decide to copulate, initially without result
(2^1), but finally generating two kids (Richard, Rachel) (2^2), who all then
continue to propagate, exponentially = 2 x 2 x 2 x 2...
   Likewise, two non-entities, Zero and Zilch, who, no matter how many times
they try, produce nothing - or nothing evident to our eyes.
   Does it depend on what you start out with, as to what the result will be?

In article <vz4Pj.3714$ko5.1622@news-server.bigpond.net.au>,
 "Don H" <donlhumphries@bigpond.com> wrote:

> "Virgil" <Virgil@gmale.com> wrote in message
> news:Virgil-6FD0F3.13355320042008@comcast.dca.giganews.com...
> > In article <QIMOj.3408$ko5.2468@news-server.bigpond.net.au>,
> >  "Don H" <donlhumphries@bigpond.com> wrote:
> >
> > > <Pfsszxt@aol.com> wrote in message
> > > news:jlah04drbo3nucjn5n7n438gqbng4auf0r@4ax.com...
> > > > On Thu, 17 Apr 2008 19:40:24 GMT, "Don H" <donlhumphries@bigpond.com>
> > > > wrote:
> > > >
> > > > >1^0 = 1 ; 2^0 = 1; 3^0 = 1 ; etc.
> > > > >Is this axiomatic, or is there a proof?
> > > > >==============================
> > > > >
> > > >  It isn't a math fact --- it's a definition --- made so as to be
> > > > consistant and for convenience when writing infinite series etc.
> > >
> > > # And so is an arbitrary definition?
> > >    What is a number n to the power x?
> > >    The x indicates how many n must be multiplied together.
> > >    Thus,
> > >    n^x = n by n (x times)
> > >    2^2 = (2, twice) = 4
> > >    2^1 = (2, once) = 2
> > >    2^0 = (2, no times) = 1 ?
> > >      - but if you multiply 2 by itself, no times, you've still 2 ?
> > > Whereas, if you multiple 0 by 0, two times, you've still 0 ?
> >
> > The standard inductive (or, if  you prefer, recursive) definition of
> > powers with positive integer exponents is:
> >    For any real/complex number x, x^1 = x  and x^(n+1) = x^n * x.
> 
> # Thanks.
>    However, consider two rabbits, say, Rob and Ruby, living in a Platonic
> relationship (2^0), who then decide to copulate, initially without result
> (2^1), but finally generating two kids (Richard, Rachel) (2^2), who all then
> continue to propagate, exponentially = 2 x 2 x 2 x 2...
>    Likewise, two non-entities, Zero and Zilch, who, no matter how many times
> they try, produce nothing - or nothing evident to our eyes.
>    Does it depend on what you start out with, as to what the result will be?

Of course! Otherwise 2^2 would be the same as 3^2.