# Multiplication of Real and Imaginary Parts

## Theorem

Let $w, z \in \C$ be complex numbers.

$(1)$ If $w$ is wholly real, then:

$\map \Re {w z} = w \, \map \Re z$

and:

$\map \Im {w z} = w \, \map \Im z$

$(2)$ If $w$ is wholly imaginary, then:

$\map \Re {w z} = -\map \Im w \, \map \Im z$

and:

$\map \Im {w z} = \map \Im w \, \map \Re z$

Here, $\map \Re z$ denotes the real part of $z$, and $\map \Im z$ denotes the imaginary part of $z$.

## Proof

Assume that $w$ is wholly real.

Then:

 $\displaystyle w z$ $=$ $\displaystyle \map \Re w \, \map \Re z - \map \Im w \, \map \Im z + i \paren {\map \Re w \, \map \Im z + \map \Im w \, \map \Re z}$ Definition of Complex Multiplication $\displaystyle$ $=$ $\displaystyle w \, \map \Re z + i w \, \map \Im z$ as $\map \Re w = w$ and $\map \Im w = 0$

This equation shows that $\map \Re {w z} = w \, \map \Re z$, and $\map \Im {w z} = w \, \map \Im z$.

This proves $(1)$.

Now, assume that $w$ is wholly imaginary.

Then:

 $\displaystyle w z$ $=$ $\displaystyle \map \Re w \, \map \Re z - \map \Im w \, \map \Im z + i \paren {\map \Re w \, \map \Im z + \map \Im w \, \map \Re z}$ $\displaystyle$ $=$ $\displaystyle -\map \Im w \, \map \Im z) + i \, \map \Im w \, \map \Re z$ as $\map \Re w = 0$

This equation shows that $\map \Re {w z} = -\map \Im w \, \map \Im z$, and $\map \Im {w c} = \map \Im w \, \map \Re z$.

This proves $(2)$.

$\blacksquare$