The complex numbers are given by ordered pairs of real numbers: C={(a,b)∣a,b∈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)∈C.
We often write a+bi for (a,b) where i2=−1.