Multiplication of Real and Imaginary Parts
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Two results -- or is it 4??
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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 $(2)$.
$\blacksquare$
