Complex Addition Identity is Zero

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

Similarly for $\left({0 + 0 i}\right) + \left({x + i y}\right)$.