Complex numbers are a fundamental part of mathematical theory with practical applications in various fields such as engineering, physics, and computer science. They are crucial in the analysis of electrical circuits, signal processing, quantum mechanics, and in solving certain differential equations.

In this chapter, we will explore imaginary and complex numbers, understand how they go beyond the real numbers system, and learn basic operations like addition, subtraction, multiplication, and division. We will also introduce a new operation in complex conjugation, learn how to compute the norm of a complex number, and how to represent them in an Argand diagram. These are crucial for practical applications, such as in electrical engineering, to represent impedance.

This chapter also introduces polar coordinates, a valuable tool in representing and manipulating complex numbers, especially useful in fields like signal processing. We will see how this perspective simplifies many operations and understanding of complex number behaviour.

The key learning objectives for this chapter are listed below. They outline the essential skills and knowledge to be developed and provide a reference point for what is expected by the end of the chapter.

Learning objectives

The student...

1. understands complex numbers as an extension of real numbers, evaluates square roots of negative numbers, and recognises that complex numbers arise as solutions of equations of the form #x^2+c^2=0#;
2. distinguishes between their imaginary and real components, represents them in both cartesian and polar form, as well as performs basic operations between them, such as addition, subtraction, multiplication, complex conjugation, and division.