Powers and Roots of Complex Numbers

If a complex number $z$ is written in exponential form: $$z=re^{i\theta},$$ then the $n$th power of $z$ is: \begin{eqnarray*} z^n&=&r^n\, (e^{i\theta})^n\\ &=&r^n\, e^{in\theta} \end{eqnarray*} and we see that the distance of the point $z$ from the origin in the complex plane has been raised to the $n$th power, but the angle has been multiplied by $n$. Similarly, an $n$th root of $z$ is: \begin{eqnarray*} z^{\frac{1}{n}}&=&r^{\frac{1}{n}}e^{i\frac{\theta}{n}}. \end{eqnarray*} For example, a square root of $-4=4\exp(i\pi)$ is given by: \begin{eqnarray*} z^{\frac{1}{2}}&=&(4e^{i\pi})^{\frac{1}{2}}\\ &=&2e^{\frac{i\pi}{2}}\\ &=&2i \end{eqnarray*} This is one of the square roots of $-4$; what about the other root?

It turns out that we can find the other root by including in our original expression for $z$ the multiplicity of angles, all of which give the same point in the complex plane, i.e. \begin{equation} z=r\, e^{i\pi + 2\pi im} \end{equation} where $m$ is any positive or negative integer. Now, when we take the root, we get an infinite number of different factors of the form $exp(i\frac{2\pi m}{n})$. How many of these correspond to different geometric angles in the complex plane? For $m=\{0, 1, \dots, n-1\}$, we will get different angles in the complex plane, but as soon as $m=n$ the angles will repeat. Therefore, we find $n$ distinct $n$th roots of $z$. If $z$ is real and positive, then one of these roots will be the positive, real $n$th root that you learned about in high school.

FIXME add an example of cube roots with a picture. Show that the roots are equally spaced around a circle in the complex plane with radius $r^{1/n}$.

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