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Topic: Leaving 0^0 undefined -- A number-theoretic rationale
Replies: 48   Last Post: Sep 15, 2013 1:06 PM

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 David C. Ullrich Posts: 3,555 Registered: 12/13/04
Re: Leaving 0^0 undefined -- A number-theoretic rationale
Posted: Sep 12, 2013 10:17 AM

On Wed, 11 Sep 2013 11:25:54 -0700 (PDT), Dan Christensen
<Dan_Christensen@sympatico.ca> wrote:

>Is there a more divisive is issue in all of mathematics?

Calling this a divisive issue in mathematics is utterly
silly. It's not even an issue, much less divisive.

In any given context we use the definition that we
want to use in that context. No problem.

IF one thinks that there must be One True Definition
for any bit of notation, then there would be a problem.
But that's just silly.

It's exactly as silly as saying that the meaning of the greek
lettter pi is an "issue", just because sometimes it means
3.14..., sometimes the prime counting function,
sometimes various other things.

>It seems to me to be the single most popular topic in just about every open online math forum.

so?

>At the "Ask A Mathematician" forum, a single, ongoing thread (since December 2010) on this topic has perhaps hundreds of postings in it!
>
>Following is a number-theoretic rationale for leaving 0^0 undefined. Unlike most if not all arguments for this approach presented elsewhere, it makes no use of the concept of a limit or the set of real numbers.
>
>The Rationale:
>
>There are precisely two binary functions on the set of natural numbers that satisfy the usual Laws of Exponents. One has 0^0 = 1, the other has 0^0 = 0. At all other points, they agree.
>
>More precisely...
>
>If e and e' are binary functions on N (including 0) such that
>
>(1) for all x in N, if x=/=0 then e(x,0)=1 and e'(x,0)=1
>
>(2) for all x,y in N, e(x,y+1)=e(x,y)*x and e'(x,y+1)=e'(x,y)*x
>
>then it can be shown that, irrespective of the values of e(0,0) and e'(0,0), we have
>
>(1) for all x in N, if x=/0 then e'(x,0)=e(x,0) (trivial)
>
>(2) for all x,y in N, if y>0 then e'(x,y)=e(x,y)
>
>Therefore, every exponential function on the set of natural numbers is identical except for the value assigned to 0^0. But there are only two such possibilities for 0^0.
>
>If the Product of Powers Rule is to hold, then we must have
>
>0^0 = 0^(0+0) = 0^0 * 0^0
>
>or
>
>0^0 = 0^0 * 0^0
>
>Therefore, 0^0 = 0 or 1. To my knowledge, there is no purely number-theoretic justification for eliminating either possibility.
>
>Given this ambiguity, the prudent course is to leave 0^0 undefined (like division by zero), especially in general purpose programming languages. Currently, most programming languages seem to have 0^0 = 1.
>
>Dan