Chapter 6

The geometrical interpretation of algebraic form of a complex number Applications in plane geometry

The ruler axiom (permits geometrical representation of the real numbers on a line)

Let d be a line and A, Bd two distinct points. There is a bijective function f:d→R through which for every Md, f(M)=xMR and which satisfies the following conditions:

1.) xA=0; xB>0;

2.) for every P, Qd, “the distance”

The function f:d→R it is called coordinates system for line d, the point A(xA=0)- its origin and xM- the abscissa(coordinate) of point Md in the considerate coordinate system. Sometimes B(xB>0) it is called point of orientation because it gives the line a direction.

This axiom permits the “identification” of a point M d with its abscissa xM R.

Let  be a plane in which we fix a system of orthogonal axes Ox, Oy, each of the axes having a proper coordinate system, with common origin- O. Like this, we can associate each point M the ordered pair of real numbers (complex number ) z=(a,b) C, where a-abscissa and b- ordinate of point M are the coordinates of the perpendicular projections of  point M on the Ox axis, respectively on Oy axis in the considerate coordinate system. Point M it is called the geometrical image of the complex number z=(a,b)=a+bi.

Like that, it is made a bijective function f:  →C, where, for every M, f(M)=zM=(xM,yM)C which permits the “identification” of a point M with an unique coordinate. In this case the plane  will be called complex plane.

Through this function, f(Ox)=R (Ox- the axis of real numbers) and f(Oy)={(0,b)| bR}- the imaginary axis(the set of complex imaginary numbers reunited with {0}).

z=4+4i; -z=-4-4i;

=4-4i; -=-4+4i;

Properties in Complex Plane:

1.)   If M(x), N(y), then the distance between M and N is

2.)   For every R*\{1}, the point M(m) for which =, MAB, m=, where A(x), B(y);

a.) if <0 then M(AB);

b.) if >1 then B(AM);

c.) if  (0,1) then A(MB).

3.)   The middle point of the segment AB is M=(x+y), =-1.

4.)    If A(a), B(b), C(c), D(d), then the lines AB, CD are parallel (AB||CD)  rR such that (a-b)=r(c-d).

5.)   ABCD is a parallelogram  (a+c)=(b+d).( A, C- opposite)

6.)   Points A(a), B(b), C(c) are collinear  rR such that (b-a)=r(c-a) in this case =rR

7.)   If Ak(ak), k= , n ≥2, is a system of n points from the complex plane , then the center of  mass G(g) of this system is: g=(g- the arithmetic media of the points ak, k=