# 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:

 $\displaystyle z_0 + z_1$ $=$ $\displaystyle \left({\operatorname{Re} \left({z_0}\right) + i \operatorname{Im} \left({z_0}\right)}\right) + \left({\operatorname{Re} \left({z_1}\right) + i \operatorname{Im} \left({z_1}\right)}\right)$ Definition of Complex Number $\displaystyle$ $=$ $\displaystyle \left({\operatorname{Re} \left({z_0}\right) + \operatorname{Re} \left({z_1}\right) }\right) + i \left({\operatorname{Im} \left({z_0}\right) + \operatorname{Im} \left({z_1}\right)}\right)$ Definition of Complex Addition $\displaystyle \implies \ \$ $\displaystyle \operatorname{Re} \left({z_0 + z_1}\right)$ $=$ $\displaystyle \operatorname{Re} \left({z_0}\right) + \operatorname{Re} \left({z_1}\right)$ Definition of Real Part $\displaystyle \operatorname{Im} \left({z_0 + z_1}\right)$ $=$ $\displaystyle \operatorname{Im} \left({z_0}\right) + \operatorname{Im} \left({z_1}\right)$ Definition of Imaginary Part

$\blacksquare$