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

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 Dan Christensen Posts: 8,156 Registered: 7/9/08
Re: The ambiguity of 0^0 on N
Posted: Sep 19, 2013 2:26 PM
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On Thursday, September 19, 2013 1:02:30 PM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
>
>
>

> >
>
> > I have shown that we need only explicitly define ^ in N for exponents
>
> > of 2 or more.
>
>
>
> Actually you haven't, you have shown that in that case 0^0 might be
>
> either of 0 and 1.

That is not part of the definition. That is a derived result assuming PPR on N.

> So your "definition" leaves x^y undefined at one
>
> point; therefore it is no definition.

I disagree.

> The purpose of a definition is to
>
> give the thing defined a precise meaning, you haven't given 0^0 a
>
> precise meaning.
>

In group theory, we define a set of binary functions with certain properties on an arbitrary set. I guess you could say that, here, I am defining a set of binary functions with certain properties on N -- the set of exponential functions. It turns out to have only 2 elements that satisfy PPR.

>
>

> > Apart from the base of 0,
>
>
>
> Why apart from? Why are you leaving it out?

We can't divide by 0. Unless you want to assign a value to 0/0 as well.

> What is even odder, having
>
> left it out you seem to think that 0^0 being undefined is a good thing.
>
> No it isn't, it's a job half done.
>
>
>

> > we can derive values for
>
> > exponents 0 and 1. In that sense, the usual definition has redundant
>
> > elements.
>
>
>

> >>
>
> >
>
> > Many standard textbooks leave 0^0 undefined. This is not the realm of
>
> > cranks or radicals. On the contrary, it is the most conservative
>
> > option.
>
>
>
> Are they talking about x^y on the reals?
>

Usually real numbers, but real or natural numbers -- it shouldn't matter.

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com

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

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