introduction

Complex numbers:
In this chapter, we survey the algebraic and geometric structure of the complex numbers system, we assume various corresponding properties of real numbers to be known.
1-sums and products:
Complex numbers can be defined as order pairs (x,y) of real numbers coordinates x and y ,when real numbers x are displayed as points(x,0) on the real axis,it is clear that the set of complex numbers includes the real numbers as a subset,
Complex numbers of the form(0,y) correspond to points on the y axis are called pure imaginary numbers the y axis is then referred as the imaginary axis ,so that z=(x,y),the real numbers x and y ,moreover known as the real and imaginary parts of z,so that
Rez=x & Imz=y
Now ,two complex z1=(x1,y1)&z2=(x2,y2) are equal whenever the have the same real parts and the same imaginary parts,thus
z1=z2 ?Rez1=Rez2 & Imz1=Imz2
the sum z1+z2 and the product z1z2 of two complex numbers z1=(x1,y1),z2=(x2,y2) are defined as follows:
1-z1+z2=(x1,y1)+(x2,y2)=(x1+x2,y1+y2)
2-z1z2=(x1,y1).(x2,y2)=(x1x2-y1y2,y1x2+y2x1)