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.

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 \(4 + 3i\).

As another example, the next figure shows the complex plane with the complex numbers

\(-4 + 5i\), \(-4 – 5i\), \(5 + 3i\) and \(5 - 3i\)

To a complex number \(z\) we can build the number \(-z\) opposite to it, or the complex number konjugierte \(\overline{z}\) to it.

The next figure shows the complex numbers \(w\) and \(z\) and their opposite numbers \(-w\) and \(-z\), as well as the conjugate complex numbers \(\overline{w}\) and \(\overline{z}\).

The position of an opposite number in the Gaussian plane corresponds to a point reflection around the zero point. Following applies

\(-z=-(a+bi)=-a+(-b)i\)

The position of the conjugate complex number corresponds to an axis mirror on the real axis in the Gaussian plane. Following applies

\(\displaystyle \overline{z}=\overline{a+bi}=a-bi=a+(-b)i\)

The opposite number \(-ω\) to \(ω\), or the conjugate complex number konjugierte komplexe Zahl to \(z\) plays an important role in solving quadratic equations. With ω and \(-ω\) is a solution of\(ω2 = D\), even if the discriminant \(D\) is not real. Because it is \((-ω)2 = ω2 = D\).

If \(z\) is a non-real solution of the quadratic equation \(az^2 +bz +c = 0\) with real coefficients \(a, b, c\), then \(z\) is always a solution of this equation. This is evident from the solution formula. Non-real solutions of a quadratic equation with real coefficients are symmetric in the Gaussian plane of the real axis.

Summary

Forming the opposite number corresponds in the complex plane to a reflection around the zero point.

Forming the conjugate complex number corresponds to an axis reflection around the real axis in the complex plane.