## Theorem

Let $z = x + i y \in \C$ be a complex number.

Let $-z = -x - i y \in \C$ be the negative of $z$.

Then $-z$ is the inverse element of $z$ under the operation of complex addition:

$\forall z \in \C: \exists -z \in \C: z + \paren {-z} = 0 = \paren {-z} + z$

## Proof

From Complex Addition Identity is Zero, the identity element for $\struct {\C, +}$ is $0 + 0 i$.

Then:

 $\displaystyle$  $\displaystyle \paren {x + i y} + \paren {-x - i y}$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \paren {x - x} + i \paren {y - y}$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle 0 + 0 i$ $\quad$ $\quad$

Similarly for $\paren {-x - i y} + \paren {x + i y}$.

$\blacksquare$