Complex Addition Identity is Zero

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Theorem

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

The identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$.


Proof

We have:

$\paren {x + i y} + \paren {0 + 0 i} = \paren {x + 0} + i \paren {y + 0} = x + i y$
$\paren {0 + 0 i} + \paren {x + i y} = \paren {0 + x} + i \paren {0 + y} = x + i y$

$\blacksquare$


Sources