Module

Module and its properties

Given a complex number z=a+b\cdot{i}, is defined as module, or absolute value to the expression:

r=\|a+b\cdot{i}\|=\sqrt{a^2+b^2}

Given z_1, z_2, \cdots, z_n \in{\mathbb{C}} it is fulfilled that:

1. \|z_1\cdot{z_2}\cdot\text{ }\cdots\text{ }\cdot{z_n}\|=\|z_1\|\cdot\|z_2\|\cdot\text{ }\cdots\text{ }\cdot\|z_n\|
2. \|\frac{z_1}{z_2}\|=\frac{\|z_1\|}{\|z_2\|}\text{ con }\|z_2\|\not{=}0
3. \|z_1+z_2\|\leq\|z_1\|+\|z_2\|
4. \|z_1+z_2+\text{ }\cdots\text{ }+z_n\|\leq\|z_1\|+\|z_2\|+\text{ }\cdots\text{ }+\|z_n\|

Example of module

z=(-3)+4\cdot{i}

\|z\|=\|(-3)+4\cdot{i}\|=\sqrt{(-3)^2+4^2}=\sqrt{9+16}=\sqrt{25}=5

So we have to: