I'm happier starting with the natural numbers, and just defining m^n as m multiplied by itself n times. Everyone gets that this is the point of exponentiation o

I'm happier starting with the natural numbers, and just defining m^n as m multiplied by itself n times. Everyone gets that this is the point of exponentiation of natural numbers, and it has the obvious recursive definition: m^0 = 1 m^n = m^(n-1). And now 0^0 = 1 follows immediately, just as it does for 0+0 = 0 and the recursive definition of addition in terms of successor. After that, I consider the question of what x^y means for non naturals as motivated largely by considerations of algebraic and analytic extension. We get negative exponents simply by extending the operation with the requirement that m^0 = 1 m^(n+p) = m^n*m^p. We get rational exponents by extending the operation with the requirement that (m^n)^p = m^(n*p). We get real exponents by extending the operation to the limits where they exist. We get complex exponents by extending the operation to the largest analytic extension. The tension then is that the real and complex extensions don't "confirm" the original reasonable setting of 0^0=1, but I personally find it more objectionable poking a hole in the function, when it was so clear in the case of natural numbers that 0^0 should be 1 (and see the set theoretic and category theoretic accounts for more reasons why this is clear).