# Modulus of a complex number

Allowed: constants, operators and i. To multiply use a*b not ab

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|