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:

$\forall z \in \C: \exists -z \in \C: z + \left({-z}\right) = 0 = \left({-z}\right) + z$


Proof

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

Then:

\(\displaystyle \) \(\) \(\displaystyle \left({x + i y}\right) + \left({-x - i y}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({x - x}\right) + i \left({y - y}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 0 + 0 i\) $\quad$ $\quad$

Similarly for $\left({-x - i y}\right) + \left({x + i y}\right)$.

$\blacksquare$


Sources