Multiplication of Real and Imaginary Parts

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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:

\(\displaystyle wz\) \(=\) \(\displaystyle \operatorname{Re} \left({ w }\right) \operatorname{Re} \left({ z }\right) - \operatorname{Im} \left({ w }\right) \operatorname{Im} \left({ z }\right) + i \left({ \operatorname{Re} \left({ w }\right) \operatorname{Im} \left({ z }\right) + \operatorname{Im} \left({ w }\right) \operatorname{Re} \left({ z }\right) }\right)\) Definition of Complex Multiplication
\(\displaystyle \) \(=\) \(\displaystyle w \operatorname{Re} \left({ z }\right) + i w \operatorname{Im} \left({ z }\right)\) as $\operatorname{Re} \left({ w }\right) = w$, and $\operatorname{Im} \left({ w }\right) = 0 $

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:

\(\displaystyle w z\) \(=\) \(\displaystyle \operatorname{Re} \left({ w }\right) \operatorname{Re} \left({ z }\right) - \operatorname{Im} \left({ w }\right) \operatorname{Im} \left({ z }\right) + i \left({ \operatorname{Re} \left({ w }\right) \operatorname{Im} \left({ z }\right) + \operatorname{Im} \left({ w }\right) \operatorname{Re} \left({ z }\right) }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle - \operatorname{Im} \left({ w }\right) \operatorname{Im} \left({ z }\right) + i \operatorname{Im} \left({ w }\right) \operatorname{Re} \left({ z }\right)\) as $\operatorname{Re} \left({ w }\right) = 0$

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 $(2)$.

$\blacksquare$