Complex numbers/Pairs/Definition

The set ${\displaystyle {}\mathbb {R} ^{2}}$ with ${\displaystyle {}0:=(0,0)}$ and ${\displaystyle {}1:=(1,0)}$, with componentwise addition and the multiplication defined by

${\displaystyle {}(a,b)\cdot (c,d):=(ac-bd,ad+bc)\,,}$

is called the field of complex numbers. We denote it by

${\displaystyle \mathbb {C} .}$