Modulus of a complex number

Definition

Modulus of a complex number $z$ is:

$$|z|=\sqrt{a^2+b^2}$$

Observe that:

$$z\bar{z}=(a+bi)(a-bi)=a^2-(bi{)}^2=a^2+b^2=|z{|}^2$$

Then,

$$|z|=\sqrt{z\bar{z}}$$

Only number with zero modulus is $z=0$.