# Modulus of Complex Number equals its Distance from Origin

## Theorem

The modulus of a complex number equals its distance from the origin on the complex plane.

## Proof

Let $z = x + y i$ be a complex number and $O = 0 + 0 i$ be the origin on the complex plane.

We have its modulus:

 $\displaystyle \left\vert{z}\right\vert$ $=$ $\displaystyle \left\vert{x + y i}\right\vert$ $\displaystyle$ $=$ $\displaystyle \sqrt {x^2 + y^2}$ Definition of Modulus

and its distance from the origin on the complex plane:

 $\displaystyle d \left({z, O}\right)$ $=$ $\displaystyle d \left({\left({x, y}\right), \left({0, 0}\right)}\right)$ $\displaystyle$ $=$ $\displaystyle \sqrt{\left({x - 0}\right)^2 + \left({y - 0}\right)^2}$ Distance Formula $\displaystyle$ $=$ $\displaystyle \sqrt {x^2 + y^2}$

The two are seen to be equal.

$\blacksquare$