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.