Finding roots of complex numbers

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.

2.3 Circles

Equation of a Circle: MPM2D Grade 10 Mathematics

1. Introduction to the Equation of a Circle

A circle is a set of all points in a plane that are at a fixed distance (called the radius) from a fixed point (called the center). The standard form of the equation of a circle depends on the location of its center.

1.1. Equation of a Circle with Center at the Origin

If the center of the circle is at the origin (0,0), the equation of the circle is:

\[ x^2 + y^2 = r^2 \]

Example:

Consider a circle with a radius of 5 units centered at the origin.

The equation is:

\[ x^2 + y^2 = 5^2 \]

\[ x^2 + y^2 = 25 \]