Inverse for Complex Addition

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

$\blacksquare$


Sources