Ok so basically I'm supposed to understand complex numbers (so that we can use them for eigenvalues and eigenvectors), but then someone dropped the ball and we got almost no teaching on the subject and now pretty much everyone in my class who hasn't worked with complex numbers before (which includes me) are having trouble understanding.

Ok, so I'm doing an example question and I just can't seem to get it to work:

$\displaystyle z^4 = -64$ and I have to find z. Immediately my response was to find the fourth root of 64 ($\displaystyle 2\sqrt{2}$) and multiply it by some form of $\displaystyle i$ whose fourth power would be equal to negative 1. This turned out to be $\displaystyle \sqrt{i}$ and I'm sure that there's various terms that could be made positive or negative (or some duplicate terms) yielding four roots. Great. I still don't understand it, that was essentially guess and check.

So my teacher did show us a method for finding the nth root of a complex number (I'll cut out most of the algebraic steps as I'm sure most of the people who can help me will be well aware of this method):

So if you have

$\displaystyle z^n = w$, where w is a complex number, you convert w into polar form and then set

$\displaystyle z = te^{(i\phi)}$ so that you get

$\displaystyle (te^(i\phi))^n = re^{(i\theta)}$

After some algebraic manipulation, you get

$\displaystyle z = \sqrt[n]{r} e^{(i(\frac{\theta + m2\pi}{n}))}$ where m is an integer.

So I tried that with $\displaystyle z^4 = -64 = -64 + 0i$. To convert, for the modulus I got $\displaystyle r = \sqrt{(-64)^2 + (0i)^2} = 64$ and for the argument I got $\displaystyle \theta = arctan(\frac{0}{-64}) = 0$. So...

$\displaystyle z = \sqrt[4]{64} e^{(i(\frac{0 + m2\pi}{4}))}

= 2\sqrt{2} e^{(i(\frac{m\pi}{2}))}$ and when you change that back into algebraic form (the form the answer is required in) you get(for m=0) $\displaystyle z = 2\sqrt{2}(\cos0 + i\sin0) = 2\sqrt{2}$ which is clearly not right. What am I doing wrong? The problem I see is that when I convert a pure real number into a complex number in polar form, the polar form is the same regardless of whether the number is positive or negative. Any help with solving problems of this type is greatly appreciated.