Geometric representation of complex numbers
With complex numbers, operations can also be represented geometrically.
With the geometric representation of the complex numbers we can recognize new connections,
which make it possible to solve further questions.
The complex plane (Gaussian plane)
Complex numbers are defined as numbers in the form \(z = a + bi\),
where \(i\) is the imaginary part and \(a\) and \(b\) are real numbers.
A complex number \(z\) is thus uniquely determined by the numbers \((a, b)\).
The geometric representation of complex numbers is defined as follows
A complex number \(z = a + bi\)is assigned the point \((a, b)\) in the complex plane.
The complex plane is similar to the Cartesian coordinate system,
it differs from that in the name of the axes.
The x-axis represents the real part of the complex number.
This axis is called real axis and is labelled as \(ℝ\) or \(Re\)
The y-axis represents the imaginary part of the complex number.
This axis is called imaginary axis and is labelled with \(iℝ\) or \(Im\).
The origin of the coordinates is called zero point.
On the complex plane, the number \(1\) is a unit to the right of the zero point on the real axis and the
Number \(i\) is a unit above the zero point on the imaginary axis.
The figure below shows the number \(4 + 3i\).