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:
Let and be two points with respective ``complex coordinate'' and . Then .
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:
On the one hand we have:
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