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Topic: The ambiguity of 0^0 on N
Replies: 106   Last Post: Sep 29, 2013 10:06 AM

 Messages: [ Previous | Next ]
 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: The ambiguity of 0^0 on N
Posted: Sep 19, 2013 1:23 AM

On Wednesday, September 18, 2013 7:11:53 PM UTC-4, Rotwang wrote:
> On 18/09/2013 15:53, Dan Christensen wrote:
>

> > [...]
>
> >
>
> > Thus, if the Product of Powers Rule is to hold on N, 0^0 will be ambiguous -- being either 0 or 1. Unless one of these alternatives can be formally proven
>
>
>
> Obviously if you start of with a "definition" of ^ which leaves 0^0
>
> undefined,

Actually, I start with a definition which leaves out any explicit mention at all of exponents of 0 or 1. I define only exponents greater than 1. Starting with this definition, I prove that x^0 = 1 and x^1=x for x=/=0. I also prove that if you want to extend the Product of Powers Rule to all of N, then we must have 0^1=0 and either 0^0=0 or 0^0=1. Until we are able to prove or disprove one of these alternatives, we should probably leave 0^0 undefined.

> then nothing can be formally proven about 0^0.

It seems you can only prove something about 0^0 if PPR is extended over all of N.

> There's
>
> nothing particularly special about ^ in this regard; if I defined x + y
>
> to be equal to its usual value when y > 0 but left x + 0 undefined then
>
> nothing could be proved about x + 0 either. That doesn't mean that
>
> partially defining addition in this way would be a sensible thing to do.
>
>
>
>
>

> > or shown to give rise to a contradiction,
>
>
>
> Definitions cannot give rise to contradictions.
>

They most certainly can. If you define 0^0=0 AND 0^0=1, you get a contradiction because 0=/=1.

>
>
>
>

> > the prudent course
>
>
>
> You keep saying this. Why?

Because I believe it is true.

> What negative consequences do you imagine
>
> will follow from people defining exponentiation in the usual way?

Whatever consequences may arise from a calculation that results in a value of 1 when it should be 0. The result could be catastrophic.

> Why do
>
> you imagine these consequences are more serious, or more likely, than
>
> the obvious negative consequences of changing the behaviour of C, Java,
>
> Javascript, Python, Perl, Ruby, Lisp, Haskell and probably most other
>
> programming languages, or of changing a mathematical definition in such
>
> a way as to render many well-known theorems false?

The most well known theorem that makes use of 0^0 is probably the Binomial Theorem. And it can easily be restated to avoid its use, e.g. by stating at the outset that (x+0)^n = x^n, etc.

Dan

Date Subject Author
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Dan Christensen
9/18/13 Peter Percival
9/18/13 Virgil
9/18/13 Dan Christensen
9/18/13 Rotwang
9/18/13 Rock Brentwood
9/18/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/19/13 Virgil
9/19/13 Virgil
9/19/13 Rotwang
9/18/13 Virgil
9/18/13 fom
9/18/13 Rotwang
9/28/13 Shmuel (Seymour J.) Metz
9/29/13 Marshall
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Michael F. Stemper
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/19/13 fom
9/19/13 fom
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 Helmut Richter
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 fom
9/19/13 fom
9/19/13 JT
9/19/13 JT
9/19/13 Michael F. Stemper
9/19/13 JT
9/19/13 JT
9/19/13 JT
9/19/13 Helmut Richter
9/28/13 Shmuel (Seymour J.) Metz
9/19/13 fom
9/19/13 Peter Percival
9/19/13 Dan Christensen
9/19/13 Peter Percival
9/19/13 Karl-Olav Nyberg
9/19/13 fom
9/19/13 fom
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/25/13 Rotwang
9/26/13 Dan Christensen
9/27/13 Brian Q. Hutchings
9/19/13 fom
9/18/13 Rock Brentwood
9/19/13 Dan Christensen
9/19/13 Dan Christensen
9/19/13 Rotwang
9/19/13 Dan Christensen
9/19/13 fom
9/20/13 Dan Christensen
9/20/13 fom
9/20/13 Dan Christensen
9/20/13 Peter Percival
9/20/13 Peter Percival
9/20/13 Dan Christensen
9/20/13 Virgil
9/20/13 Peter Percival
9/20/13 fom
9/20/13 Michael F. Stemper
9/20/13 LudovicoVan
9/21/13 Michael F. Stemper
9/21/13 LudovicoVan
9/21/13 Richard Tobin
9/20/13 Peter Percival
9/20/13 Peter Percival
9/21/13 Dan Christensen
9/19/13 Karl-Olav Nyberg