Complex Number Modulus/Magnitude

The modulus (or magnitude) is the length (absolute value) in the complex plane, qualifying the complex number $ z = a + ib $ (with $ a $ the real part and $ b $ the imaginary part), it is denoted $ | z | $ and is equal to $ | z | = \sqrt{a ^ 2 + b ^ 2} $.

The module can be interpreted as the distance separating the point (representing the complex number) from the origin of the reference of the complex plane.

To find the module of a complex number $ z = a + ib $ carry out the computation $ |z| = \sqrt {a^2 + b^2} $

Example: $ z = 1+2i $ (of abscissa 1 and of ordinate 2 on the complex plane) then the modulus equals $ |z| = \sqrt{1^2+2^2} = \sqrt{5} $

The calculation also applies with the exponential form of the complex number.

The modulus (or magnitude) of a real number is equivalent to its absolute value.

Example: $ |-3| = 3 $

For the complex numbers $ z, z_1, z_2 $ the complex modulus has the following properties:

$$ |z_1 \cdot z_2| = |z_1| \cdot |z_2| $$

$$ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \quad z_2 \ne 0 $$

$$ |z_1+z_2| \le |z_1|+|z_2| $$

A modulus is an absolute value, therefore necessarily positive (or null):

$$ |z| \ge 0 $$

The modulus of a complex number and its conjugate are equal:

$$ |\overline z|=|z| $$