# Distance between Points in Complex Plane

## Theorem

Let $A$ and $B$ be points in the complex plane such that:

$A = \tuple {x_1, y_1}$
$B = \tuple {x_2, y_2}$

Then the distance between $A$ and $B$ is given by:

 $\ds \size {AB}$ $=$ $\ds \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}$ $\ds$ $=$ $\ds \cmod {z_1 - z_2}$

where $z_1$ and $z_2$ are represented by the complex numbers $z_1$ and $z_2$ respectively.

## Proof

We have:

 $\ds AB$ $=$ $\ds z_2 - z_1$ Geometrical Interpretation of Complex Subtraction $\ds$ $=$ $\ds \paren {x_2 + i y_2} - \paren {x_1 + i y_1}$ $\ds$ $=$ $\ds \paren {x_2 - x_1} + \paren {y_2 - y_1} i$ Definition of Complex Subtraction $\ds \leadsto \ \$ $\ds \size {AB}$ $=$ $\ds \cmod {\paren {x_2 - x_1} + \paren {y_2 - y_1} i}$ $\ds$ $=$ $\ds \sqrt {\paren {x_2 - x_1}^2 + \paren {y_2 - y_1}^2}$ Definition of Complex Modulus

$\blacksquare$