Geometric Representation of Complex Numbers

 

If the x-axis (Ox) is chosen as the real axis and the y-axis (Oy) as the imaginary axis in the analytic plane, complex numbers can be mapped one-to-one with the points of the analytic plane. In this mapping, the point \((x,y)\) corresponds to the complex number \(x + yi\). As shown in the figure, the number \(0 + 0i\) is mapped to the origin \(O(0,0)\),

numbers of the form \(x + 0i\) are mapped to points on the real axis (Ox),

and numbers of the form \(0 + yi\) are mapped to points on the imaginary axis (Oy).

\[
z = x_1 + y_1 i
\]

A plane that is mapped one-to-one with complex numbers in this manner is called the complex plane (or Argand diagram).

 

Example

 

In the accompanying figure, the complex numbers

\[ z_1 = 1 + 2i \]
\[ z_2 = \ – \ 2 + i \]
\[ z_3 = \ – \ 1 \ – \ i \]
\[ z_4 = 2 \ – \ i \]

are represented on the complex plane.

 

Example

 

In the accompanying figure, the complex numbers

\[ z_1 = 1 + 2i \]
\[ z_2 = -2 + i \]
\[ z_3 = -1 – i \]
\[ z_4 = 2 – i \]

are represented on the complex plane.