moduli and conjugate

It is natural to associate any non zero complex number z=x+iy with the direction line segment ,or vector from the origin to the point (x,y) that represent z ,in the complex plane ,in fact we often refer to z as the point z or the vector z in the fig(1) the number z=x+iy and z=-x+iy are displayed graphically as both points and radius vector
The number ?z? is the distance between the point (x,y) and the origin or the length of the vector represent z , ?z?=?(x^2+y^2 )
Note:
The inequality z1Example:
Since ?-3+2i?=?(?(-3)?^2+?(2)?^2 ) =?(9+4)=?13
And ?1+4i?=?(?(1)?^2+?(4)?^2 ) =?(1+16)=?17
The point -3+2i is closer to the origin than 1+4i
Now as
z1-z2=(x1-x2)+i(y1-y2)
then
?z1-z2?=?(?(x1-x2)?^2+?(y1-y2)?^2 )
The complex number z corresponding to the points lying on the circle with center z0 and radius R
i.e. ?z-z0?=R
example:
the equation ?z-1+3i?=2 represent the circle whose center is z0=(1,-3) and whose radius is R=2
so that as Rez=x &Imz=y
?z?2=(Rez)2+(Imz)2
Rez??Rez???z? & Imz??Imz???z?