I am reading Spivak (Chapter 25 - Complex Numbers) and I need some assistance in understanding the following: In developing "Theorem 2", which explains the method for finding the n-th roots of a complex number, he explains:
A complex number $\ z=r(\cos\theta+i \sin\theta)$ satisfies $\ z^n=w$ if and only if
$\cos (n\theta) + i \sin (n\theta) = \cos \phi + i \sin \phi$
From the first equation it follows that $\sqrt[n]{s}$ (which denotes the positive real n-th root of s
From the second equation, it follows that for some integer $k$ we have
$\theta=\theta_k=\frac{\phi}{n}+\frac{2k\phi}{n}$
Could anyone paraphrase (explain a bit) the $\theta=\theta_k=\frac{\phi}{n}+\frac{2k\phi}{n}$ part, please?
Thank you very much!
You need to remember that complex numbers (other than $0$) have $n$ $n^{th}$ roots and not just one as you may expect from real numbers. Of course, even there, for even $n$ there are two roots. For the even case, picking the positive one as the principal root generally behaves well.
With the complex numbers, it is not possible to pick a principal root that behaves so well. It is usually necessary to remember all of the possibilities. The argument $\frac{\phi}{n}$ gives one root. Adding $\frac{2 k \pi}{n}$ for $k$ from $1$ to $n-1$ gives you the other roots. Note that this assumes the correction suggested by EuklidAlexandria.