Complex Numbers
Introduction to the basics of Complex Number Calculation
This article gives a short introduction to the basics of Complex Number Calculation. More detailed descriptions can be found in the chapter of complex numbers
Definition of a complex number
With quadratic equations, there is not always a real solution. For example, the equation
\(X^2 + 1=0\) oder eben \(X^2 = 1\)
In order to be able to count on solutions of such equations, the mathematician Leonard Euler introduced a new imaginary number and designated it with the letter \(i\).
A complex number \(z\) consists of a real part \(a\) and an imaginary part \(b\). The imaginary part is marked with the letter \(i\).
\(z=a+bi\)
The imaginary unit \(i\) has the property
\(z^2=1\)
The value of a complex number corresponds to the length of the vector \(z\) in the Argand plane.
Graphical interpretation of complex numbers
For the graphical interpretation of complex numbers the Argand plane is used. The Argand plane is a special form of a normal Cartesian coordinate system. The difference is in the name of the axles.
The real part of the complex number is displayed on the xaxis of the argand plane. The axis is called the real axis.
The imaginary part of the complex number is displayed on the yaxis of the argand plane. The axis is called the imaginary axis.
The following figure shows a graphical representation of a complex number \(3 + 4i\).
The absolute value \(z\) is \(5\).
