Definition and Algebra
In this section we will define complex numbers as an extension of $\mathbb{R}$ as the following
\[\mathbb{C} \cong \mathbb{R}[x] / \langle x^2+1 \rangle = \{a+bx+\langle x^2+1 \rangle\ | ~ a,b \in \mathbb{R} \}\]Algebra of Complex Number
Addition of two complex numbers are done as follows
$\displaystyle a + b x + c + d x = a + c + x \left(b + d\right)$
Multiplication is done as follows
$\displaystyle \left(a + b x\right) \left(c + d x\right) = a c + a d x + b c x + b d x^{2}$
$\displaystyle a c - b d + x \left(a d + b c\right)$