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Section 2.8 Roots of Complex Numbers

If a complex number \(z\) is written in exponential form:

\begin{equation} z=re^{i\theta}\text{,}\tag{2.8.1} \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{2.8.2} \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{2.8.3} \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{2.8.4} \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{2.8.5} \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 \(r^{1/n}\text{.}\)