The complex numbers are given by ordered pairs of real numbers: \(\mathbb C = \{(a,b)\mid a,b\in \mathbb R\}\) and endowed with two binary operations: multiplication and addition.
Addition is defined pairwise, while multiplication is defined as \((a,b)*(c,d) = (ac-bd,bc+ad)\) for all $(a,b),(c,d) \in \mathbb C$.
We often write $a+bi$ for $(a,b)$ where $i^2=-1$.