# Geometric representation of complex numbers.

Let , with real . To this complex number we associate the point and the vector , both with coordinates .

When used to represent complex numbers, the Euclidean plane is called the Cauchy-Argand plane or Gauss plane. Biographies of Augustin Cauchy (1789-1857) and Jean Argand (1768-1822) can be found in The MacTutor History of Mathematics archive, at the following URL:

$z+z'=(a+c)+(b+d)i$

A biography

Let and be two points with respective complex coordinate'' and . Then .

Example 1.2.1 (A perpendicular bisector)   Represent in the plane the set of complex numbers such that .

Geometrically, the given equation represents the set of all the points at the equal distances from the points and , i.e. the perpendicular bisector of the segment .

Let , where are real. We have:

This is the equation of the perpendicular bisector of (see Figure 3).

Example 1.2.2 (Circles.)   If is the complex coordinate of a point in the plane, then the circle whose center is in and whose radius is equal to is defined by the equation (see Figure 2(b)).

Example 1.2.3 (A solved exercise.)   Find the geometric locus of all the points in the complex plane such that

On the one hand we have:

Thus, the required geometric locus is contained in the unit circle.

On the other hand, if, and only if, the point is on the perpendicular bisector of the segment whose endponts are the origin and the point . Therefore the coordinate of is equal to 1.

It follows that the required geometric locus is made of all the points on the unit circle whose coordinate is equal to 1; it contains exactly one point, namely ; see Figure 4.

Noah Dana-Picard 2007-12-24