Why square r when duplicating an angle?

Question

I am wondering if this is just a feature of complex numbers or a general geometric/vector requirement.

But even if it is that way, cam you give a rationale and explain the logic behind it?

Answer 1

Recall that by De Moivre's formula $$z=r(\cos \theta + i \sin \theta)\implies z^2=r^2(\cos 2\theta + i \sin 2\theta)$$

and more in general

$$z=r(\cos \theta + i \sin \theta)\implies z^n=r^n(\cos n\theta + i \sin n\theta)$$

Answer 2

write the complex number in polar form.

$$z = re^{i\theta}$$

then we have

$$z^2 = r^2e^{i(2\theta)}$$

Hence the angle is doubled.

Answer 3

When interpreting a complex number $w=a+ib$ as a geometric transformation of the (complex) plane, namely as the map $\Bbb C\to \Bbb C$, $z\mapsto w\cdot z$, it turns out that it is a combination of a rotation and a scaling with $0$ (of course) as fix point. If we rotate by an angle $\theta$ twice, we rotate by a total of $2\theta$. If we scale by a factor $r$ twice, we scale by $r^2$.