1. ## Finding polar form

This makes no sense to me. I definitely think I'm doing it right.

Convert 3sqrt(3) - 9i into polar form.

Every time I do it, I get 6sqrt(3)(1/2 - .....) The second part doesn't work.

What am I doing wrong?

EDIT: Bonus Question:
Prove there is no field F such that R is not a subset of F which is not a subset of C. (R is the real numbers and C is the complex numbers, sorry I don't know how to do the math writing on here well yet.)

2. Hello, Janu42!

You never did write it in polar form . . .

Convert $3\sqrt{3} - 9i$ into polar form.
The triangle looks like this:
Code:
            _
3√3
* - - - - *
* θ     |
*     |9
r  *   |
* |
*

$r \:=\:\sqrt{(3\sqrt{3})^2 + 9^2} \;=\;\sqrt{108} \:=\:6\sqrt{3}$

$\cos\theta \:=\:\frac{3\sqrt{3}}{6\sqrt{3}} \:=\:\frac{1}{2} \quad\Rightarrow\quad \theta \:=\:\frac{\pi}{3}$

On the graph, we see that the angle is $-\frac{\pi}{3}$

Therefore, the polar form is:

. . $6\sqrt{3}\bigg[\cos\left(\text{-}\tfrac{\pi}{3}\right) + i\sin\left(\text{-}\tfrac{\pi}{3}\right)\bigg] \;=\;6\sqrt{3}\left(\cos\tfrac{\pi}{3} - i\sin\tfrac{\pi}{3}\right)$