Square roots of a complex number

Question

My book says that given the complex number $z$ with modulus $r$, its square roots are $±√re^{iΘ/2}$ where $Θ$ is the principal value of $\arg z$. My question is that why must we consider the principal value of its argument?

Answer 1

We consider the principal value by convention. But also if we take the other value $\frac{\theta}{2}+\pi$ we find the same two roots because $e^{i(\frac{\theta}{2}+\pi)}=-e^{i(\frac{\theta}{2})}$

Answer 2

You must not consider the principal value of the argument ! If $z=re^{i \phi}$ with $r \ge 0$ and $ \phi \in \mathbb R$, then put

$w_1:= \sqrt{r} e^{i \frac{\phi}{2}}$ and $w_2:=-w_1$.

Then we have

$$w_1^2=z=w_2^2.$$