Geometrical Representation of Complex Number

Since every complex number z = x + iy is an order pair of real numbers (x, y), it can therefore be represented by a point P(x,y) in the xy plane by taking the real part along the x-axis and the imaginary part along the y-axis.

This representation of a complex number in a plane is called the argand diagram.

Let points P and Q in the complex plane represent two complex numbers z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, respectively, as shown in figure.

Join the origin O with the points P and Q and complete the parallelogram OPRQ.

It is clear from the figure that the coordinates of R are

(x₁ + x₂, y₁ + y₂) and it represents the complex number

(x₁ + x₂) + i (y₁ + y₂), i.e., z₁ + z₂.

The absolute values of z₁, z₂ and z₁ + z₂ are geometrically given by

We know that the sum of any two sides of a triangle is greater than the third. Hence, in DORP, we have

with the equality holding only when O, P, Q are lying in a straight line. That is why this inequality for the absolute values of complex numbers is called the triangle inequality.

The triangle inequality can be extended to n complex numbers by finite induction, i.e., for any n complex numbers z₁, z₂…..., z_n, we obtain