# 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 |

$\blacksquare$