MathGloss

The complex numbers are given by ordered pairs of real numbers: C={(a,b)a,bR}\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)=(acbd,bc+ad)(a,b)*(c,d) = (ac-bd,bc+ad) for all (a,b),(c,d)C(a,b),(c,d) \in \mathbb C.

We often write a+bia+bi for (a,b)(a,b) where i2=1i^2=-1.