# Roots of Complex Numbers

Recall that if $z=x+iy$ is a nonzero complex number, then it can be written in polar form as $z=r(\cos \theta +i \sin \theta)$ where $r=\sqrt{x^2+y^2}$ and $\theta$ is the angle, in radians, from the positive $x$-axis to the ray connecting the origin to the point $z.$

Now, de Moivre's formula establishes that if $z=r(\cos \theta +i\sin \theta)$ and $n$ is a positive integer, then \begin{eqnarray*} z^n=r^n(\cos n\theta+i\sin n\theta). \end{eqnarray*}

Let $w$ be a complex number. Using de Moivre's formula will help us to solve the equation $$z^n=w$$ for $z$ when $w$ is given.

Suppose that $w=r(\cos \theta +i\sin \theta)$ and $z=\rho (\cos \psi +i\sin \psi).$ Then de Moivre's formula gives $$z^n=\rho^n(\cos n\psi+i\sin n\psi).$$ It follows that $$\rho^n=r=|w|$$ by uniqueness of the polar representation and $$n\psi = \theta +k(2\pi),$$ where $k$ is some integer. Thus

$z=\sqrt[n]{r}\left[\cos\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right) \right].$
Each value of $k=0,1,2,\ldots ,n-1$ gives a different value of $z.$ Any other value of $k$ merely repeats one of the values of $z$ corresponding to $k=0,1,2,\ldots ,n-1.$ Thus there are exactly $n$th roots of a nonzero complex number.

Using Euler's formula: $$e^{i\theta}=\cos \theta +i \sin \theta,$$ the complex number $z=r(\cos \theta +i\sin \theta)$ can also be written in exponential form as $$z=re^{i\theta} = r \,\mbox{exp}(i \theta).$$

Thus, the $n$th roots of a nonzero complex number $z\neq 0$ can also be expressed as

\begin{eqnarray}\label{expform} z=\sqrt[n]{r}\;\mbox{exp}\left[i\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right)\right] \end{eqnarray}
where $k=0, 1, 2, \ldots , n-1.$

The applet below shows a geometrical representation of the $n$th roots of a complex number, up to $n=10.$ Drag the red point around to change the value of $z$ or drag the sliders.

Code

Enter the following script in GeoGebra to explore it yourself and make your own version. The symbol # indicates comments.

#Complex number
Z = 1 + ί

#Modulus of Z
r = abs(Z)

#Angle of Z
theta = atan2(y(Z), x(Z))

#Number of roots
n = Slider(2, 10, 1, 1, 150, false, true, false, false)

#Plot n-roots
nRoots = Sequence(r^(1/n) * exp( ί * ( theta/n + 2 * pi * k/n ) ), k, 0, n-1)


Exercise: From the exponential form (\ref{expform}) of the roots, show that all the $n$th roots lie on the circle $|z|=\sqrt[n]{r}$ about the origin and are equally spaced every $2\pi/n$ radians, starting with argument $\theta/n.$

Topology of the Complex Plane