Why: $\sqrt{z} => φ_\sqrt {z}=\frac {1}{2}φ_z$

Question

Let's consider a complex number z=a+bi, with angle φ Then why is $φ_\sqrt {z} = \frac {1}{2}φ_z$ ?

So if a complex number z has for example an angle of 90°, then $\sqrt {z}$ has an angle of 45°.

Answer 1

The simple explanation is this: If you take a (non-zero) complex number with argument $45^\circ$ and square it, what is the argument of the result?