Addition of Real and Imaginary Parts

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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

$\blacksquare$