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?

Please remember that you can choose an answer among the given if the OP is solved, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work — user, Apr 19 '18 at 19:20
So we are considering a right triangle with sides $1,2$ and $\sqrt 5$ and we are doubling an angle? Sorry but It wasn’t completely clear that the OP was about it. I’m happy to revise but you should better specify what you mean with angle doubling and in particular how sides change. — user, Apr 20 '18 at 07:27
yes it chenge a lot, you need to define exactly what kind of doubling angle you are considering, if you refer to a triangle you must specify what side can change with the angle — user, Apr 20 '18 at 08:20

score 0 · Answer 1 · answered Apr 15 '18 at 06:52

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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

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

answered Apr 15 '18 at 06:52

user

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@user157860 Yes it is a property valid for complex number and not for vectors. – user Apr 15 '18 at 06:56
@user157860 If we multiply $z$ by the complex number $\frac{z}{|z|}$ we obtain a rotation without scaling. – user Apr 15 '18 at 10:44

score 0 · Answer 2 · answered Apr 15 '18 at 06:52

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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.

answered Apr 15 '18 at 06:52

Siong Thye Goh

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You might like to read about polar form. It is a property of complex numbers. what does it mean by squaring a vector? some relevant reading – Siong Thye Goh Apr 15 '18 at 07:03
oops, I must have made a typo in my url, reading – Siong Thye Goh Apr 15 '18 at 07:42

score 0 · Answer 3 · answered Apr 15 '18 at 07:49

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 $r^2$.

Why square r when duplicating an angle?

3 Answers3