Complex Analysis

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 $$z=re^{i\theta}$$

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.

Sorry, the applet is not supported for small screens. Rotate your device to landscape. Or resize your window so it's more wide than tall.


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 $\theta/n$.

NEXT: Complex Functions