What is the use of modulus in complex number?

Question

The modulus of the complex number $z = a + bi$ is denoted by $|z|$ and is given by $|z| = \sqrt{a^2+b^2}$.

What is the meaning of this? What is a modulus?

Answer 1

A complexe number $z=a+ib$ is a point $ M$ in the plane $\Bbb R^2$. the coordinates of the point $ M $ are $(a,b)$.

the modulus of $ z$ represents the Euclidian distance between the point $M(a,b)$ and the origin $O(0,0)$.

$$d(O,M)=||\vec{OM}||=\sqrt{(a-0)^2+(b-0)^2}$$ $$=\sqrt{a^2+b^2}=|z|$$