Addition of Real and Imaginary Parts

Theorem
Let $z_0, z_1 \in \C$ be two complex numbers.

Then:


 * $\operatorname{Re} \left({ z_0 + z_1 }\right) = \operatorname{Re} \left({ z_0 }\right) + \operatorname{Re} \left({ z_1 }\right)$

and:


 * $\operatorname{Im} \left({ z_0 + z_1 }\right) = \operatorname{Im} \left({ z_0 }\right) + \operatorname{Im} \left({ z_1 }\right)$

Here, $\operatorname{Re} \left({ z_0 }\right) $ denotes the real part of $z_0$, and $\operatorname{Im} \left({ z_0 }\right) $ denotes the imaginary part of $z_0$.

Proof
We have:

The equation shows that $\operatorname{Re} \left({ z_0 + z_1 }\right) = \operatorname{Re} \left({ z_0 }\right) + \operatorname{Re} \left({ z_1 }\right)$, and $\operatorname{Im} \left({ z_0 + z_1 }\right) = \operatorname{Im} \left({ z_0 }\right) + \operatorname{Im} \left({ z_1 }\right)$.