(The real number 0) ^ (The real number 0) I think is best regarded as undefined, since x^y has an essential singularity at (0,0). But (The cardinal 0) ^ (The cardinal 0) is clearly 1. X^Y is the number of functions from a set of size Y to a set of size X, and there is exactly one function with range and domain the null set. Andy On Sat, Jun 13, 2020 at 7:27 AM Andres Valloud <ten@smallinteger.com> wrote:
I hear 0^0 should be defined as 1 for convenience, but perhaps there are proofs this has to be so (the *value* zero raised to the *value* 0, not the indeterminate limits of the form 0^0). What's your favorite proof that 0^0 = 1? Here's one.
0^0 = (1 - 1)^0 = \sum_{k=0}^0 \binom{0}{k} 1^{0-k} (-1)^k = \binom{0}{0} * 1 * 1 = \frac{0!}{0! \cdot 0!} * 1 = 1 * 1 = 1
Andres.
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