Addition of Real and Imaginary Parts

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:

\(\ds z_0 + z_1\)

\(=\)

\(\ds \paren {\map \Re {z_0} + i \, \map \Im {z_0} } + \paren {\map \Re {z_1} + i \, \map \Im {z_1} }\)

Definition of Complex Number

\(\ds \)

\(=\)

\(\ds \paren {\map \Re {z_0} + \map \Re {z_1} } + i \paren {\map \Im {z_0} + \map \Im {z_1} }\)

Definition of Complex Addition

\(\ds \leadsto \ \ \)

\(\ds \map \Re {z_0 + z_1}\)

\(=\)

\(\ds \map \Re {z_0} + \map \Re {z_1}\)

Definition of Real Part

\(\ds \map \Im {z_0 + z_1}\)

\(=\)

\(\ds \map \Im {z_0} + \map \Im {z_1}\)

Definition of Imaginary Part

$\blacksquare$