On Thursday, October 17, 2013 6:28:33 AM UTC-4, Bart Goddard wrote: > Dan Christensen <Dan_Christensen@sympatico.ca> wrote in > > news:email@example.com: > > > > > To follow-up on my previous postings at sci.math > > > > ...because you think garbage will evolve into real > > mathematics if you just keep repeating it...? > > > > Your assertion is that 0^0 can be defined to be > > any value and yet have all the usual rules of > > exponents apply.
As I show in Theorem 4, if we avoid base values of 0, we have x^y * x^z = x^(y+z). Likewise for the other Laws of Exponents (Theorems 5 and 7). In practice, this no more onerous a restriction than disallowing zero denominators.
While we can't use Theorem 4 to prove that 0^2 * 0^3 = 0^(2+3), we can still prove by other means that 0^2 * 0^3 = 1 and 0^(2+3) = 1.
> You've been given examples > > that show this is a false statement and have > > declined to respond to those examples. >
I don't recall any such examples.
> > Note that each of your "theorems" about the > > rules of exponents specifically exclude using > > 0 as a base.
> Using "logic" like this, you could > > define 3^2 to be anything and just word your > > theorems "If a is not 3, then these rules work." >
Using my definition of ^ of N, if you assume any value for 3^2 other than 9, you will get a contradiction.
> > > So you're asserting that 0^0 can be defined any > > way one wants, and it won't matter, _provided that > > one doesn't actually look at 0^0.
It works because you will still have the same values at every other point. It just formalizes what has been the practice by mathematicians for nearly two centuries -- leaving 0^0 undefined. (Yes, many still cling to 0^0=1.)