Finding roots of complex numbers

How to Find the n-th Roots of Complex Numbers - Examples and Step-by-Step Guide

How to Find the \(n\)-th Roots of Complex Numbers

Finding the \(n\)-th root of a complex number involves converting the number to its polar form and using De Moivre’s Theorem. Below is a detailed, step-by-step guide on how to find both fourth roots and cube roots of complex and real numbers with examples.

Steps to Find the \(n\)-th Roots

Let’s break down the process of finding the \(n\)-th roots of a complex number into three main steps:

Step 1: Convert the Complex Number to Polar Form

A complex number \( z = x + yi \) can be written in polar form as:

\[ z = r(\cos \theta + i \sin \theta) \]

Where:

  • r is the modulus: \( r = |z| = \sqrt{x^2 + y^2} \)
  • \(\theta\) is the argument: \( \theta = \arg(z) \) found using trigonometry based on the quadrant.

Step 2: Use De Moivre's Theorem

De Moivre's Theorem gives us the formula to find the \(n\)-th roots of \( z \):

\[ z^{1/n} = r^{1/n} \left( \cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n} \right) \]

Where \( k = 0, 1, 2, \dots, n-1 \) represents the different \(n\)-th roots.

Step 3: Calculate All Roots

For each \( k \), compute the \(n\)-th root of the modulus and the argument, and substitute into the formula to find each root.

Example 1: Find the Fourth Roots of \( \frac{1}{2} - \frac{\sqrt{3}}{2} i \)

We will now find the fourth roots of the complex number \( \frac{1}{2} - \frac{\sqrt{3}}{2} i \).

Step 1: Convert to Polar Form

The modulus is \( r = 1 \) and the argument is \( \theta = 300^\circ \). Therefore, the polar form is:

\[ z = 1 \left( \cos 300^\circ + i \sin 300^\circ \right) \]

Step 2: Apply De Moivre’s Theorem

To find the fourth roots, we apply the formula:

\[ z^{1/4} = 1 \left( \cos \frac{300^\circ + 2k\pi}{4} + i \sin \frac{300^\circ + 2k\pi}{4} \right) \]

The four roots are found by substituting \( k = 0, 1, 2, 3 \). These roots are:

  • Root 1: \( \cos 75^\circ + i \sin 75^\circ \)
  • Root 2: \( \cos 165^\circ + i \sin 165^\circ \)
  • Root 3: \( \cos 255^\circ + i \sin 255^\circ \)
  • Root 4: \( \cos 345^\circ + i \sin 345^\circ \)

Example 2: Find the Cube Roots of \( z = 8 \)

We now find the cube roots of the real number \( z = 8 \).

Step 1: Convert to Polar Form

The polar form of \( z = 8 \) is:

\[ z = 8 \left( \cos 0^\circ + i \sin 0^\circ \right) \]

Step 2: Apply De Moivre’s Theorem

The cube roots are found using the formula:

\[ z^{1/3} = 2 \left( \cos \frac{0^\circ + 2k\pi}{3} + i \sin \frac{0^\circ + 2k\pi}{3} \right) \]

The three cube roots are:

  • Root 1: \( 2 \)
  • Root 2: \( -1 + i \sqrt{3} \)
  • Root 3: \( -1 - i \sqrt{3} \)

Conclusion

By following the above steps, you can find the \(n\)-th roots of any complex or real number using De Moivre’s Theorem. It’s important to convert the number into polar form, calculate the modulus and argument, and then apply the formula for the roots.