Addition and Multiplication of Complex Numbers Examples 1
Recall from the Addition and Multiplication of Complex Numbers page that if $z = a + bi, w = c + di \in \mathbb{C}$ then addition between $z$ and $w$ is defined by:
(1)Also, multiplication between $z$ and $w$ is defined by:
(2)We will now look at some example problems regarding these operations on $\mathbb{C}$.
Example 1
Prove that if $z, w \in \mathbb{C}$ then $\mathrm{Re} (z + w) = \mathrm{Re} (z) + \mathrm{Re} (w)$.
Let $z = a + bi$ and $w = c + di$. Then $\mathrm{Re} (z) = a$ and $\mathrm{Re} (w) = c$. So:
(3)Example 2
Prove that if $z, w \in \mathbb{C}$ then $\mathrm{Re} (z \cdot w) \neq \mathrm{Re} (z) \cdot \mathrm{Re} (w)$ in general by counter example.
Let $z = w = 1 + i$. Then $\mathrm{Re} (z) = 1$ and $\mathrm{Re} (w) = 1$. Note that:
(4)Therefore we have that:
(5)Example 3
Prove that if $z \in \mathbb{C}$ then $\mathrm{Re} (iz) = - \mathrm{Im} (z)$ and $\mathrm{Im} (iz) = \mathrm{Re} (z)$.
Let $z = a + bi \in \mathbb{C}$. Then $\mathrm{Re} (z) = a$ and $\mathrm{Im} (z) = b$. Also:
(6)Thus $\mathrm{Re} (iz) = -b = -\mathrm{Im} (z)$, and $\mathrm{Im}(z) = \mathrm{Re}(z)$.
Example 4
Prove that if $\lambda \in \mathbb{R}$ and $z \in \mathbb{C}$ then $\mathrm{Re} (\lambda z) = \lambda \mathrm{Re} (z)$ and that $\mathrm{Im} (\lambda z) = \lambda \mathrm{Im} (z)$
Let $z = a + bi \in \mathbb{C}$. Then $\mathrm{Re} (z) = a$, $\mathrm{Im} (z) = b$ and:
(7)So $\mathrm{Re} (\lambda z) = \lambda a = \lambda \mathrm{Re} (z)$, and $\mathrm{Im} (\lambda z) = \lambda b = \lambda \mathrm{Im}(z)$.