Prove that every complex number with modulus 1 can be expressed as the principal value $(−1)^a$ for some real number a.

Question

"Prove that every complex number with modulus 1 can be expressed as the principal value $(−1)^a$ for some real number a."

I understand that a modulus of 1 results in a radius of 1 as well, so then z = i$\theta$. But then that would be a principle value of (i$\theta$)$^c$, where c is some real number. How am I supposed to get rid of the $\theta$, or am I doing something else wrong?

Answer 1

Every number with modulus 1 can be written $e^{i\theta}=e^{i\pi \theta/\pi}=(e^{i\pi})^{\theta/\pi}=(-1)^a $

Answer 2

It's probably $e^{i\theta} $ you intended to write instead of $(i\theta)^c$.

Hint: $e^{i\pi} = -1$.