# Modulus of a complex number

This online tool calculates the modulus of a complex number.

## Modulus of a complex number

Let z be a complex number expressed in its algebraic form,

`z = a + b . i`

a - the real part of z

b - the imaginary part of z

Then, z **modulus**, denoted by |z|, is a real number is defined by,

`|z| = \sqrt(a^2+b^2)`

**Examples**

- The modulus of z = 0 is 0

- The modulus of a real number equals its absolute value `|-6| = 6`

- if `z = i` then, `|z| = sqrt(0^2 + 1^2) = 1`

- if `z = 1 + 2 * i` then, `|z| = \sqrt(1^2 + 2^2) = sqrt(5)`

- if `z = -5 + 6*i` then, `|z| = \sqrt((-5)^2 + 6^2) = sqrt(61)`

## Properties of Complex Number modulus

- The modulus of a complex number is always positive : `|z| >= 0`- if `bar z` is the conjugate of z then, `z*bar z = |z|^2`, which can be written in another way,

`|z| = \sqrt(z*bar z)`

- The modulus of the product of two complex numbers is equal to the product of their modulus.

`|z_1 * z_2| = |z_1| * |z_2|`

- The modulus of the ratio between two complex numbers is equal to the ratio of their modulus.

`|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}` with z being non-zero

- Triangle inequality

`|z_1 + z_2| <= |z_1| + |z_2|` for every two complex numbers `z_1` and `z_2`

- Modulus of a complex number and modulus of its opposite are equal.

`|-z| = |z|`

- Modulus of a complex number and modulus of its conjugate are equal.

`|bar z| = |z|`

## See also

Conjugate of a complex number

Operations on complex numbers