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