Roots of Complex Numbers

A complex number is of the form $z = a + bi$, where:

• $a$ is the real part,
• $b$ is the imaginary part, and
• $i$ is the imaginary unit such that $i^2 = -1.$

nth Roots of a Complex Number

To find the $n$-th roots of a complex number, we express the complex number in polar form and then apply De Moivre's Theorem.

Step 1: Convert the complex number to polar form

A complex number $z = a + bi$ can be written in polar form as:

$$ z = r_0(\cos \theta + i \sin \theta) = r_0e^{i\theta} $$

Where:

• $r_0 = \sqrt{a^2 + b^2}$ is the modulus (or absolute value) of $|z|$
• $\theta = \text{arg}(z) = \tan^{-1} \left( \frac{b}{a} \right)$  is the argument (or angle) of $z$.

Step 2: Apply De Moivre's Theorem

The $n$-th roots of a complex number are given by: $z_k = r_0^{1/n} \left[ \cos \left( \frac{\theta + 2k\pi}{n} \right) + i \sin \left( \frac{\theta + 2k\pi}{n} \right) \right]$

For $k = 0, 1, 2, \dots, n-1$, where:

• $r_0^{1/n}$ is the magnitude of each root,
• The angles $\theta + 2k\pi$ are equally spaced around the unit circle, with each root separated by $\frac{2\pi}{n}$

We call this root, principal root of complex number $z$

Example: Finding the Square Roots of $z = 1 + i$