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!

Fig. 4.8 Complex multiplication can also be understood as adding the angles of the two numbers and multiplying their magnitudes.#

4.4.1. Inversion (division)#

We skipped inversion and division when working in rectangular coordinates. It’s certainly not impossible to do these operations with rectangular coordinates, but it’s much simpler in polar form if we use the rule

\[ \frac{1}{e^x} = e^{-x}. \]

A complex number \(z\) can therefore be inverted to get \(z^{-1}\):

\[ \frac{1}{z} = \frac{1}{r \cdot e^{\mathrm{j}\theta}} = \frac{1}{r} \cdot e^{-\mathrm{j}\theta}. \]

That is, the radius is inverted, and the angle is negated.

Fig. 4.9 shows how a complex number and its inverse relate to each other. Some things to pay attention to here:

  • \(z\) and \(z^{-1}\) cross the unit circle at the same time, since \(r=1\) is its own inverse.

  • When \(z\) is outside the unit circle, the inverse is inside, and vice versa.

  • \(z\) and \(z^{-1}\) cross the both horizontal and vertical axes at the same time, but in general, both their horizontal and vertical positions can be different.

  • There are two points at which \(z\) and \(z^{-1}\) coincide exactly: \(z = 1\) and \(z=-1\).

Fig. 4.9 Inverting a complex number \(z \rightarrow z^{-1}\) can be understood as inverting its magnitude and negating its angle.#

If you want to divide two complex numbers \(z = r\cdot e^{\mathrm{j}\theta}\) and \(w = s\cdot e^{\mathrm{j}\phi}\) as \(z/w\), it’s equivalent to multiply by the inverse \(z\cdot w^{-1}\). This results in

\[ \frac{z}{w} = \frac{r \cdot e^{\mathrm{j}\theta}}{s\cdot e^{\mathrm{j}\phi}} = \frac{r}{s} \cdot e^{\mathrm{j}(\theta - \phi)} \]