Real Number Operations Caveat

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As an "aside", ruveyn, I would suggest that "1/2" is the best definition of the value of "0^0", as it is halfway between the two values to most easily "prove correct".

Alternatively, one could define "0^0" to be "15+sqrt(2)*i", as the limit of some suitably grotesque functions "f(x)/g(x)".



(I am not being serious.)

I think the problem is there's no notational distinction between the discrete definition of x^n when n is only allowed to be an integer and the case of a real valued exponents x^p.

In the case of real exponents x^p is usually defined as exp(log(x) * p). Because log(0) is undefined, the real power operation x^p should really be undefined when x is zero. However, since lim(x->0) x^p = 0 for all positive p, it's a convenient shorthand to define 0^p as 0 for positive p, making x^p continuous w.r.t. x on the interval 0 <= x < infinity.

I'd take x^n with n being strictly an integer as a totally separate operation. In combinatorics it makes a lot of sense to define x^n as...

1 when n is zero
x * x * ... * x (n terms) when n is positive
1 / (x * x * ... * x) (-n terms) when x is nonzero and n is negative

So my opinion is that when dealing with discrete integer exponents, it's perfectly sensible to define 0^0 as 1. When dealing with real valued exponents 0^0 should be considered undefined. I just consider x^n and x^p as different operations using the same exponent notation because they are identical on a restricted subset.