Adding and Multiplying Complex Numbers in Rectangular Form

Consider the complex numbers:

$$\begin{align*} z &= a + bi \\ w &= c + di \end{align*}$$

where $a, b, c, d \in \mathbb{R}$.

To add (or subtract) them, we simply add (or subtract) the real and imaginary parts separately. That is:

$$\begin{align*} z+w &= a + bi + c + di \\ &= (a + c) + (b + d)i. \end{align*}$$

Similarly,

\begin{align*} ...