A complex number is of the form $a + bi \in \comp$ where $a, b \in \real$ and $i^{2} = -1$. Additon and multiplication is performed as
\begin{align} (a + bi) + (c + di) &= (a + c) + (b + d)i\newline (a + bi) \times (c + di) &= a \times (c + di) + bi \times (c + di) = ac + adi + bci + bd \times (-1)\newline &= (ac - bd) + (ad + bc)i \end{align}
Complex arithmetic satisfies the following properties
Commutativity
$\alpha + \beta = \beta + \alpha$ and $\alpha \beta = \beta \alpha \quad \forall \; \alpha, \beta \in \comp$
Associavity
$(\alpha + \beta) + \lambda = \alpha + (\beta + \lambda)$ and $(\alpha \beta)\lambda = \alpha(\beta \lambda) \quad \forall \alpha, \beta, \lambda \in \comp$
Identity Elements
There exist identity elements $0$ and $1$ such that $\alpha + 0 = \alpha$ and $\alpha \times 1 = \alpha \quad \forall \alpha \in \comp$
Additive inverse
For every $\alpha \in \comp, \; \exists \beta \in \comp$ which is unique with $\alpha + \beta = 0$
Multiplicative Inverse
For every $\alpha \in \comp, \; \exists \beta \in \comp$ which is unique with $\alpha \beta = 1$
Distributive Property
$\lambda(\alpha + \beta) = \lambda \alpha + \lambda \beta \quad \forall \lambda, \alpha, \beta \in \comp$
Subtraction and Division are also defined for complex numbers
\begin{align} \alpha - \beta &= \alpha + (-\beta)\newline \alpha \div \beta &= \alpha * (\frac{1}{\beta}) \quad \text{where $1/\beta$ is the multiplicative inverse of $\beta$} \end{align}