## Section2.7Roots of Complex Numbers

If a complex number $z$ is written in exponential form:

\begin{equation} z=re^{i\theta}\text{,}\tag{2.7.1} \end{equation}

then the $n$th power of $z$ is:

\begin{align} z^n\amp = r^n\, (e^{i\theta})^n\notag\\ \amp = r^n\, e^{in\theta}\tag{2.7.2} \end{align}

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 $n\text{.}$ Similarly, an $n$th root of $z$ is:

\begin{align} z^{\frac{1}{n}}\amp = r^{\frac{1}{n}}e^{i\frac{\theta}{n}}\text{.}\tag{2.7.3} \end{align}

For example, a square root of $-4=4\exp(i\pi)$ is given by:

\begin{align} z^{\frac{1}{2}}\amp = (4e^{i\pi})^{\frac{1}{2}}\notag\\ \amp = 2e^{\frac{i\pi}{2}}\notag\\ \amp = 2i\tag{2.7.4} \end{align}

This is one of the square roots of $-4\text{;}$ 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}\tag{2.7.5} \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 $exp(i\frac{2\pi m}{n})\text{.}$ How many of these correspond to different geometric angles in the complex plane? For $m=\{0, 1, \dots, n-1\}\text{,}$ 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 $z\text{.}$ 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 $r^{1/n}\text{.}$