Hi Robert, > > It's not true because it's neither true nor false. It's a not well > > formulated statement. (Mathematically). > > I disagree with this, we certainly agree that 0.0 ** negative value > is undefined, i.e. that this is outside the domain of the ** function, > and I think normally in mathematics one would say the same thing, > and simply say that 0**0 is outside the domain of the function. > > However, we indeed extend domains for convenience. After all > typically on computers 1.0/0.0 yielding infinity, and that > certainly does not correspond to the behavior of the division > operator over the reals in mathematics, but it is convenient :-)
the problem with 1.0/0.0 is not so much the domain but the range: it is the range which needs to be extended to contain an infinite value (representing both + and - infinity), at which point the definition 1.0/0.0 = infinity is an example of the standard notion of "extension by continuity". The problem with x^y is that the range of limits as (x,y) converges to zero (through x>0,y>0) is the entire interval of real numbers between 0 and 1 inclusive. Attempts to extend by continuity are doomed in this case (the fact that the limiting values also arise as values of x^y for x,y>0 means that attempts to "fix up" the range bork the usual function meaning). Ciao, Duncan.
