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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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 Dan Christensen Posts: 8,219 Registered: 7/9/08
Re: Leaving 0^0 undefined -- A number-theoretic rationale
Posted: Sep 11, 2013 5:04 PM

On Wednesday, September 11, 2013 3:26:05 PM UTC-4, Peter Percival wrote:
> Dan Christensen wrote:
>
>
>

> >
>
> > 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.
>
>
>
> I'm sure this
>
> https://en.wikipedia.org/wiki/0%5E0#Zero_to_the_power_of_zero has been
>
> pointed out to you (either the page or the facts on the page), so your
>
> claim seems to be wrong.
>

I see there no purely number-theoretic justification (no limits or real numbers) that would eliminate 0^0 = 0 as a possibility. Note: Citing applications (like BT) where 0^0 = 1 is convenient an 0^0 = 0 is inconvenient is not enough.

>
>
> If the fact 0^0 = 0^0 * 0^0 doesn't settle 0^0 being 0 or 1, then more
>
> data is needed. You can't just stop at
>
> 0^0 = 0^0 * 0^0 -> (0^0 = 0 or 1)
>
> and say 0^0 is or should be undefined.
>

Show me a contradiction that arises from 0^0 = 0.

Example: A contradiction arises from 0^0 = 2 by substituting into 0^0 = 0^0 * 0^0. 2=/=4.

>
>

> > 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.
>
>
>
> In your arguments you seem to switch between mathematics and
>
> programming.

In the purely formal world of programming, the issue is somehow brought into sharper focus. In programming, there is no special "context" for arithmetic operators.

> In mathematics 0^0 = 1, in programming 0^0 should be what
>
> the specification says it should be.
>

Tell it to the judge.

Dan