1. Introduction

Definition

Imaginary Number

$$i = \sqrt{-1}$$

Complex Number

$$4i,\;\;\; 3 + i,\;\;\; 8-4i$$

 

Complex numbers are all real & imaginary numbers or a combination of.

$$z = a + b i, \quad a, b \in \mathbb{R}, \quad z \in \mathbb{C}$$

 

Example

$$\begin{aligned} \text{a)} \quad & (3 + 2i) + (5 - 4i) &&= 8 - 2i \\[1mm] \text{b)} \quad & (5 - 3i)(5 + 2i) &&= 25 + 10i - 15i - 6i^2 \\ & &&= 25 - 5i + 6 \\ & &&= 31 - 5i \end{aligned}$$

Notation

Let $\; \; \; \; z = a + bi \;$

then $\; Re(z) = a$

& $\; \; \; \; \; Im(z) = b$