Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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 11:33 AM

On Thursday, September 19, 2013 9:31:44 AM UTC-4, Rotwang wrote:
> On 19/09/2013 06:23, Dan Christensen wrote:
>

> > 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.
>
>
>
> But your definition leaves 0^0 undefined. So nothing can be proved about
>
> it without an additional assumption.
>

Yes. That additional assumption is the extension of the Product of Power Rule to all of N. But even then, you are left with two possibilities: 0^0=1 or 0^0=0.

>
>
>
>

> > I also proven 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.
>
>
>
> Yes, I know. "If you want to extend the Product of Powers Rule to all of
>
> N" is an additional assumption. But what I'd like to know is, what's so
>
> special about the Product of Powers Rule? I mean, your "definition"
>
> implies that x^0 = 1 for x != 0 - let's call this the Power of Zero
>
> Rule. Why do you want to extend the Product of Powers Rule to all of N,
>
> but not the Power of Zero rule?
>

You may be onto something here. If you don't want to make this extension -- the most conservative option, I suppose -- you also could not assign a value to 0^1.

>
>
>
>

> > Until we are able to prove or disprove one of these alternatives, we should probably leave 0^0 undefined.
>
>
>
> Why? Note that I'm not asking you to repeat your argument that if one
>
> defines ^ a certain way one finds that there are two different functions
>
>
> why anyone should define ^ the way you do in your OP, rather than the
>
> much simpler and more common way that it's usually defined, which has
>
> the additional benefit that there is a unique function that satisfies
>
> it, and that PPR can be derived from it, rather than just assumed.
>

It was much simpler to allow unrestricted comprehension for sets. But it eventually led to the now well-known contradictions (e.g Russell's Paradox). So, simpler is not always better.

>
>
>
>

> > It seems you can only prove something about 0^0 if PPR is extended over all of N.
>
>
>
> Or if you use a sensible definition of ^.
>
>
>
>
>

> >>> the prudent course
>
> >>
>
> >> You keep saying this. Why?
>
> >
>
> > Because I believe it is true.
>
>
>
> Why?
>

See above.

>
>
>
>

> >> 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.
>
>
>
> But the consequences from a calculation that results in "undefined" when
>
> it should be 1 could be similarly catastrophic.

[snip]

No more catastrophic than division by zero which is similarly undefined, and is treated by computers as an error condition.

Again, the notion of 0^0 being undefined is not some radical notion. Many standard textbooks make this assumption. It is probably more mainstream than assuming 0^0=1. I'm sure that a poll of all math instructors would confirm this.

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