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:

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:

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 $(2)$.