# Subtraction of Complex Numbers

## Theorem

Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers.

The subtraction operation on $z_1$ and $z_2$ is:

$z_1 - z_2 = \paren {a_1 - a_2} + i \paren {b_1 - b_2}$

## Proof

 $\ds z_1 - z_2$ $=$ $\ds z_1 + \paren {- z_2}$ Definition of Complex Subtraction $\ds$ $=$ $\ds a_1 + \paren {-a_2} + i \paren {b_1 + \paren {-b_2} }$ Inverse for Complex Addition $\ds$ $=$ $\ds \paren {a_1 - a_2} + i \paren {b_1 - b_2}$ Definition of Complex Subtraction

$\blacksquare$