Addition of Real and Imaginary Parts

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Theorem

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


Then:

$\map \Re {z_0 + z_1} = \map \Re {z_0} + \map \Re {z_1}$

and:

$\map \Im {z_0 + z_1} = \map \Im {z_0} + \map \Im {z_1}$


Here, $\map \Re {z_0}$ denotes the real part of $z_0$, and $\map \Im {z_0}$ denotes the imaginary part of $z_0$.


Proof

We have:

\(\displaystyle z_0 + z_1\) \(=\) \(\displaystyle \paren {\map \Re {z_0} + i \, \map \Im {z_0} } + \paren {\map \Re {z_1} + i \, \map \Im {z_1} }\) Definition of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \paren {\map \Re {z_0} + \map \Re {z_1} } + i \paren {\map \Im {z_0} + \map \Im {z_1} }\) Definition of Complex Addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \Re {z_0 + z_1}\) \(=\) \(\displaystyle \map \Re {z_0} + \map \Re {z_1}\) Definition of Real Part
\(\displaystyle \map \Im {z_0 + z_1}\) \(=\) \(\displaystyle \map \Im {z_0} + \map \Im {z_1}\) Definition of Imaginary Part

$\blacksquare$