with addition and multiplication operations defined as:
(a,b) * (c,d) = (a * c - b * d, a * d + b * c),
for any a,b,c,d belonging to R (R is Real numbers set).
let's note that addition's neutral element is (0,0).
let's note that multiplication's neutral element is (1,0).
let's note that opposite element to (a,b) is -(a,b) = (-a,-b).
let's note that inverse element to (a,b) ≠ (0,0) is:
let's define multiplication of complex number by real number as:
let's note that with this we have:
at last, identifying complex number (a,0) with real number a, and adding additional notation:
a = Rz is real part and b = Iz imaginary part of complex number.
i itself we call imaginary unit.
let's note that:
See also: Groups & Bodies.