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Topic: 0^0=1
Replies: 145   Last Post: Jun 5, 2012 1:10 PM

 Messages: [ Previous | Next ]
 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: 0^0=1
Posted: Apr 25, 2012 10:53 PM

On Apr 25, 4:06 pm, Jussi Piitulainen <jpiit...@ling.helsinki.fi>
wrote:
> Dan Christensen writes:
> > I have always seen exponentiation defined recursively for integer or
> > natural number exponents n as:

>
> > x^1 = x
> > x^(n+1)= x^n * x.

>
> > For integer n, and non-zero x, we have x^n=x^(n+1)/x. Thus x^0 = 1
> > for non-zero x and x^0 is left undefined for x=0.

>
> ...
>

> > > Division by zero is quite different. Zero cannot have a
> > > multiplicative inverse, call it w, because then we would have both
> > > 0*w = 0 (zero does that to any number) and 0*w = 1 (the
> > > multiplicative inverse does that to the number), which is a
> > > contradiction. I'm taking for granted that (a = b and a = c)
> > > implies a = c, and not 0 = 1.

>
> > Good point. x/y is is undefined for y=0. As a direct result, x^0 is
> > undefined for x=0 (see above).

>
> I make no such point.

Didn't you make the point x/y is undefined for y=0? If not, I
apologize.

> The failure to define x^0 by a rule that is not
> valid when x=0 is entirely yours, and only you are impressed by it.
>

As I recall, many sources leave 0^0 undefined.

> > > > (x+y)^n = x^n if y=0 and not x = n = 0
> > > >         = y^n if x=0 and not y = n = 0
> > > >         = 0 if x+y = 0 and n > 0
> > > >         = sum for k = 0, ..., n of C(n,k) x^k y^(n - k) if x =/= 0 and
> > > > y =/= 0 and not x+y = n = 0

>
> > > > (Have I covered all the cases?)
>
> ...
>

> > > My source says the theorem is "too important to be arbitrarily
> > > restricted", which is what happens when one leaves 0^0 undefined.

>
> > Too big to fail? I have argued here that BT does not require 0^0 to
> > be defined.

>
> No, "too important to be arbitrarily restricted".

[snip]

Do you think it is impossible to prove the binomial theorem with 0^0
being undefined?

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Also see video demo

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