Multiplication of Real and Imaginary Parts

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

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


 * $\operatorname{Re} \left({ wz }\right) = w \operatorname{Re} \left({ z }\right) $

and:


 * $\operatorname{Im} \left({ wz }\right) = w \operatorname{Im} \left({ z }\right) $

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


 * $\operatorname{Re} \left({ wz }\right) = - \operatorname{Im} \left({ w }\right) \operatorname{Im} \left({ z }\right) $

and:


 * $\operatorname{Im} \left({ wz }\right) = \operatorname{Im} \left({ w }\right) \operatorname{Re} \left({ z }\right) $

Here, $\operatorname{Re} \left({ z }\right) $ denotes the real part of $z$, and $\operatorname{Im} \left({ z }\right) $ denotes the imaginary part of $z$.

Proof
Assume that $w$ is wholly real.

Then:

This equation shows that $\operatorname{Re} \left({ wz }\right) = w \operatorname{Re} \left({ z }\right)$, and $\operatorname{Im} \left({ wc }\right) = w \operatorname{Im} \left({ z }\right)$.

This proves $(1)$.

Now, assume that $w$ is wholly imaginary.

Then:

This equation shows that $\operatorname{Re} \left({ wz }\right) = - \operatorname{Im} \left({ w }\right) \operatorname{Im} \left({ z }\right)$, and $\operatorname{Im} \left({ wc }\right) = \operatorname{Im} \left({ w }\right) \operatorname{Re} \left({ z }\right)$.

This proves $(2)$.