Definition: A complex number is a pair of real number $(a,b) \in \mathbb{R}^2$ written as

$$a + b\cdot i.$$

The number $a$ is called the real part and the number $b$ is called the imagionary part. The set of all complex numbers is denoted as $\mathbb{C}$.

Definition: Given a pair of complex numbers $a + bi$, $c + d i$, we define their product as

$$(a + b i )\cdot(c + di) = (ac - db) + (ad + bc) \cdot i.$$

This is a commutative operation.

Corollary: The complex number $i$, has the property

$$i^2 = -1.$$

Hence, we view the number $i$ as a square root of -1.