# Absolute value of a complex number

Calculator and formulas for calculating the absolute value of a complex number

## Calculating the absolute value (or modulus or magnitude)

Absolute value calculator

 Input Complex number + i Decimal places 0 1 2 3 4 6 8 10 Results Abs

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 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.