# Geometrical Interpretation of Complex Modulus

## Theorem

Let $z \in \C$ be a complex number expressed in the complex plane.

Then the modulus of $z$ can be interpreted as the distance of $z$ from the origin.

## Proof

Let $z = x + i y$.

By definition of the complex plane, it can be represented by the point $\tuple {x, y}$.

By the Distance Formula, the distance $d$ of $z$ from the origin is:

 $\ds d$ $=$ $\ds \sqrt {\paren {x - 0}^2 + \paren {y - 0}^2}$ $\ds$ $=$ $\ds \sqrt {x^2 + y^2}$

which is precisely the modulus of $z$.

$\blacksquare$