Complex Numbers and Functions

Abstract

Definition: A complex number z is a number of the form z a ib = + , where the symbol i = −1 is called imaginary unit and a b R ,. ∈ a is called the real part and b the imaginary part of z, written a z = Re and b z = Im. With this notation, we have z z i z = + Re Im. The set of all complex numbers is denoted by C a ib a b R = + ∈ ,. If b = 0, then z a i a = + = 0 , is a real number. Also if a = 0, then z ib ib = + = 0 , is a imaginary number; in this case, z is called pure imaginary number. Let a ib + and c id + be complex numbers, with a b c d R , , ,. ∈ 1. Equality a ib c id + = + if and only if a c b d = = and. Note: In particular, we have z a ib = + = 0 if and only if a b = = 0 0 and .