Graphical representation of complex numbers

Complex numbers can also be represented in the classical Cartesian coordinate system. The plane where we represent complex numbers is called the plane of complex numbers or also the Gaussian plane.

Representation in the Gaussian plane

Each complex number z = x + yi is represented in the plane by a point with coordinates [x, y]. The axis x is called the real number axis in the Gaussian plane, and the axis y is called the imaginary number axis. So let us have two complex numbers, z₁ = 2 + 5i and z₂ = 4 − 3i. In the Gaussian plane, we would represent them as follows:

Opposite and complex number

We can easily plot the reciprocal and complex associated numbers on the Gaussian plane. Consider the complex number z = 3 + 5i. The reciprocal of z, which has the form −3 − 5i, is symmetric with the origin of the Gaussian plane:

For a complex-associated number, the sign of the imaginary part changes, so that the complex-associated number will be axisymmetric with the original number along the real axis (the axis of x). See the figure ($z^\prime$ denotes a complex-associated number):

Absolute value

In the real number field, the absolute value represents the positive version of a given number. In complex numbers, we calculate the absolute value in a slightly more complicated way. This is because the absolute value of a complex number represents the distance of a point in the Gaussian plane from its origin.

We can calculate the distance from the origin using the Pythagorean theorem, which tells us that |z|² = x² + y², where x and y are the real and imaginary parts of the complex number. We can then just get the absolute value of z = x + yi by square root:

$$|z|=\sqrt{x^2+y^2}$$