Defination
Complex numbers be defined as numbers having a real part which is followed by an imaginary part. The imaginary part is either denoted by 'i' or they are denoted by 'j'. thecomplex number is simply denoted by $ i=\sqrt{-1} $. They are generally written as a+ib and are represented on a complex plane in both Cartesian(x,y) and polar form(r,theta). The Cartesian and the polar form are inter-convertible where $ r=\sqrt{x^2+y^2} $ and $ theta=\tan^{-1}{y/x} $.
general
$ i=\sqrt{-1} $
$ i^2=\ {-1} $
$ i^3=\ {-i} $
$ i=\ { 1} $
addition
z1=a+ib
z2=c+id
$ z1+z2=\ (a+c)+i(b+d) $
subtraction
z1=a+ib
z2=c+id
$ z1-z2=\ (a-c)+i(b-d) $
multiplication
z1=a+ib
z2=c+id
$ z1*z2=\ (ac-bd)+i(bc+ad) $
division
z1=a+ib
z2=c+id
$ \,\frac{z1}{z2}=\frac{(a + bi)}{(c + di)} = \frac{(ac + bd)}{(c^2 + d^2)} +i \frac{(bc - ad)}{(c^2 + d^2)}\, $
