# Absolute value of a complex number

Berechnen des Betrags oder Absolutwert für eine komplexe Zahl

## The absolute value of a complex number

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.