Absolute value of a complex number

Absolute value of a complex number

The length of the vector in the Gaussian plane has a special name for the complex numbers. We call it the absolute value of the complex number

The figure below shows the graphical representation of the complex number.

When represented a complex number by vectors, the result is always a right-angled 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 a complex number \(z = a + bi\) is:

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

Example

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

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

It also applies

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

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

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