Complex Addition Identity is Zero

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Theorem

Let $\C$ be the set of complex numbers.

The identity element of $\left({\C, +}\right)$ is the complex number $0 + 0 i$.


Proof

We have:

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

$\blacksquare$


Sources