Absolute value of a complex number
Calculator and formulas for calculating the absolute value (magnitude) of a complex number
This calculator calculates the absolute value or magnitude of a complex number. For calculation enter a complex number, then click the 'Calculate' button.
The result is displayed as a real value and as a vector.

Absolute value z =
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 rightangled 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
\(z=\sqrt{z·\overline{z}}=\sqrt{(34i)·(3+4i)}=\sqrt{25}=5\)
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.
More complex functions
Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
BesselI • BesselIe • BesselJ • BesselJe • BesselK • BesselKe • BesselY • BesselYe •
