Berechnen des Betrags oder Absolutwert für eine komplexe Zahl

In the article on the complex plane, it was described that every complex number z can be clearly assigned a vector. The length of the vector has a special name in the complex numbers. We call it the absolute value of the complex number.

The figure below shows the graphical representation of the complex number \(3 + 4i\).

The representation with vectors always results in a right triangle, which consists of the two catheters \(a\) and \(b\) and the hypotenuse \(z\). The absolute value of a complex number corresponds to the length of the vector.

The absolute value of the complex number \(z = a + bi\) is

\(|z|=\sqrt{a^2+b^2} = \sqrt{Re^2 + Im^2}\)

Calculation of the absolute value of the complex number \(z = 3 - 4i\)

\(|z|=\sqrt{a^2+b^2} = \sqrt{3^2 + 4^2}=\sqrt{25}=5\)

Es gilt auch

\(|z|=\sqrt{z·\overline{z}}=\sqrt{(3-4i)·(3+4i)}=\sqrt{25}=5\)

Note that the absolute value at \(3 + 4i\) and \(3 – 4i\) is positive. The absolute value of complex and real numbers is always a positive value.

In most programming languages or math software, the name Abs is used for the function for determining the absolute value.