Complex Analysis- Lecture 2- Complex numbers

1.2 complex number:-
A complex number Z can be defined as an ordered pair of real number x,y.
Z=(x,y)

Let Z_1=(x_1,y_1 ) and Z_2=(x_2,y_2 ) be any two complex numbers, then their sum is
Z_1+Z_2=(x_1,y_1 )+(x_2,y_2 )=(x_1+x_2 ,y_1+y_2 )
and their product is
Z_1 Z_2=(x_1,y_1 )(x_2,y_2 )=(x_1 x_2-y_1 y_2 ,x_1 y_2+y_1 x_2 )

Remarks:-
The order pair (0,y) is called a pure imaginary numbers.
The order pair (x,0) is called real numbers and we write it as x, so the set of real numbers R is subset of the set of complex numbers C.
x is called real part of complex number Z=(x,y) and we denote it as x=R(Z) .
y is called imaginary part of complex number Z=(x,y) and we write it as y=I(Z) , so y=I(Z) is real numbers.
We can write a complex number Z=(x,y) as algebraic (ordinary) form Z=(x,y)=(x,0)+(0,y)
=(x,0)(0,1)(y,0)
Z=x+iy
where i=(0,1) is called imaginary unit.
The algebraic form of complex numbers is a best the Cartesian form of complex numbers, especially in expression of complex number and satisfied its properties.
By using product of complex numbers, we get
? i?^2=i?i=(0,1)?(0,1)=(0-1,0?1+1?0)
=(-1,0)=-1
i^3=i^2?i=(i)i=-i
i^4=i^2?i^2=(-1)(-1)=1