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Topic: Considering exponentiation when both 0 and 1 are omitted
Replies: 6   Last Post: Nov 7, 2013 11:31 PM

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fom

Posts: 1,969
Registered: 12/4/12
Re: Considering exponentiation when both 0 and 1 are omitted
Posted: Nov 7, 2013 9:08 AM
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On 11/7/2013 2:29 AM, William Elliot wrote:
> On Wed, 6 Nov 2013, fom wrote:
>

>> Given all of the posts about exponentiation, I thought
>> anyone interested in something more mathematical
>> on the subject, might find it interesting. It is the
>> paper mentioned at Wikipedia for the link on Tarski's
>> high school algebra problem.

>
> What posts? Those about 0^0?
> For n in N, a^n = prod(j=1,n) a and na = sum(j=1,n) a.
> When n = 0, the expression defaults to the identity of * or +.
> This allows statements like for all a in R, n,m in N
> . . a^n a^m = a^(n + m) and (a^n)^m = a^nm.
>
> That is discrete math. For continuous math, ie calculus, it's
> required that x^y be continuous. That is not possible for 0^0.
> Thus for x^y to be continuous, it's defined avoiding (0.0). Also
> x < 0 is avoided for x^y can't defined continuous over x < 0, for
> every y, such as y = 1/2.
>
> Though there's more sophisticated examples, here's a simple reason why
> it has to be defined for 0^0.
>
> 0 = lim(n->oo) 0^(1/n) = 0^0
> 1 = lim(n->oo) (1/n)^0 = 0^0
>
> For analysis, the approach is yet defferent.
> First exp is defined as the solution to y' = y, exp 0 = e^0 = 1.
> The log = exp^-1 is defined and finally
> . . a^x = e^(x.log a) for all a > 0.
>
> since the image of exp is the positive numbers
> requiring the domain of log to be the positive numbers.
>
> So about what the big Much ado about Nothing flap that's been
> ongoing? Smart alexs' hot air pretense of smarts?
>


This has nothing to do with the pretense in those
threads. Nor do I have any expectation that it would
resolve the arguments of the disputants since the
paper is specifically about when 0 and 1 are omitted.

The paper contains some good mathematics on exponentiation.

Because of the threads, I thought some people
might find it of interest.

That is all.




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