Possible size = 1 complex numbers that when divided by their conjugate, produce a size = 1 number

Question

Let’s assume we have a complex number $z$ that $|z| =1$.

How many different complex numbers like $a$ exist, when $|a| = 1$ and $$z = \frac{a}{\bar{a}}$$? How could we prove that?

Answer 1

We can use exponential notation: $$z=e^{i\xi}\\a=e^{i\phi}$$ Then $$\bar a=e^{-i\phi}$$ and $$z=e^{2i\phi}=e^{i\xi}$$ The obvious solution is $\phi=\frac{\xi}2$, but you need to account for the periodicity:$$e^{2i\phi}=e^{2i\phi+i2\pi}$$ So assuming that $\xi$ is in the interval $[0,2\pi)$, then $\phi=\frac{\xi}2$ is in the interval $[0,\pi)$. Then you have the other solution $\frac{\xi}2+\pi$ in the interval $[\pi,2\pi)$