Question on evaluating complex exponents...

Question

anonymous · Answer

The complex numbers are a superset of the real numbers if I understand correctly, and any real number can be written as a complex number. If you take the expression:$$0^{2}$$, this is equal to 0. However, if you rewrite the exponent as a complex number...$$0^{2+0i} = 0^{2}0^{0i}$$ You run into a problem with 0^0i being indeterminate. How would you evaluate this? I've tried using $$0^{2}e^{\ln(0^{0i})}  = 0^{2}e^{0i \ln(0)} = \cos(0\ln(0)) + i \sin(0\ln(0)) $$ But ln(0) is undefined.

unklerhaukus · Answer

zero as a complex number can be written $$0=0+0i$$

anonymous · Answer

But I'm not trying to rewrite 0 as a complex number, I'm rewriting 2, the exponent, as a complex number and evaluating the expression, which seems to become indeterminate.

unklerhaukus · Answer

why are you trying to do that?

anonymous · Answer

Because it should work... why would 0^2 be defined but not 0^(2+0i)?

unklerhaukus · Answer

zero is a strange base

anonymous · Answer

My calculus 2 teacher couldn't answer it...