In this section, we will discuss the topic of complex numbers. A complex number, which will be discussed in detail later, is generally considered as an entity defined by real numbers a and b, expressed in the form a + ib. Here, i is called the imaginary unit and has the unusual property that i² = -1. This property is unusual because, in the world of real numbers, no number squared yields a negative result. Therefore, i is not a real number.

Complex numbers can be represented in three different forms in mathematical expressions: Cartesian, polar, and exponential. Each form offers advantages in solving specific mathematical and physical problems. In Cartesian form, complex numbers are expressed as a + ib, whereas in polar form, a number consists of two components, r (magnitude) and θ (angle), and is written as r(cos θ + i sin θ). The exponential form is expressed as r e^{i θ}, using Euler’s formula e^{i θ} = cos θ + i sin θ.

Each of these representations is especially useful for solving problems where phase differences and transformations are important.

Graphically, complex numbers are represented using the Argand diagram, which will be discussed in detail in later sections. This diagram provides a geometric interpretation of complex numbers and is used to visually express the magnitude and angle of the numbers. The Argand diagram is particularly useful for understanding vector addition and the multiplication of complex numbers.