4.4. Multiplication and division#

We previously saw how to multiply two complex numbers in rectangular form, but let’s see what happens when we use polar form instead.

Say we’re given two numbers \(z = r\cdot e^{\mathrm{j}\theta}\) and \(w = s\cdot e^{\mathrm{j}\phi}\). We can use the rules of exponents to simplify this:

\[\begin{align*} z\cdot w &= r\cdot e^{\mathrm{j}\theta} \cdot s\cdot e^{\mathrm{j}\phi}\\ &= (r \cdot s) \cdot e^{\mathrm{j}\theta} \cdot e^{\mathrm{j}\phi} & \text{Collect exponentials together}\\ &= (r \cdot s) \cdot e^{\mathrm{j}\theta + \mathrm{j}\phi} & \text{Use rule: } e^x \cdot e^y = e^{x+y}\\ &= (r \cdot s) \cdot e^{\mathrm{j}(\theta + \phi)} & \text{Pull out common factor of }\mathrm{j}. \end{align*}\]

The result in a new complex number in polar form, with magnitude \(r\cdot s\) and angle \(\theta + \phi\). There’s a helpful mnemonic device for remembering this:

Tip

Magnitudes multiply and angles add.

In the special case where \(s=1\), the magnitude is preserved, and all that happens is that the angle moves from \(\theta \rightarrow \theta + \phi\). But adding angles is nothing more than rotation!