Calculators and complex numbers (Part 16)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number $z = a+bi$ is

$z = r(\cos \theta + i \sin \theta) = r e^{i \theta}$

where $r = |z| = \sqrt{a^2 + b^2}$ and $\tan \theta = b/a$ , with $\theta$ in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

Real mathematicians use the notation $e^{i \theta}$ to represent $\cos \theta + i \sin \theta$ . I say this because I’ve seen textbooks that basically invented the non-standard notation $\hbox{cis} \, \theta$ (pronounced siss), where presumably the c represents $\cos$ and the s represents $\sin$ . I express my contempt for this non-standard notation by saying that this is a sissy way of writing it.

With this shorthand notation of $r e^{i \theta}$ , several of the theorems that we’ve discussed earlier in this series of posts become a lot more memorable.

First, the formula

$\left[ r_1 ( \cos \theta_1 + i \sin \theta_1) \right] \cdot \left[ r_2 ( \cos \theta_2 + i \sin \theta_2 ) \right] = r_1 r_2 \left[ \cos( \theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2) \right]$

can be rewritten as something that resembles the familiar Law of Exponents:

$r_1 e^{i \theta_1} r_2 e^{i \theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)}$

Similarly, the formula

$\displaystyle \frac{ r_1 ( \cos \theta_1 + i \sin \theta_1)}{ r_2 ( \cos \theta_2 + i \sin \theta_2 ) } = \displaystyle \frac{r_1}{ r_2} \left[ \cos( \theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2) \right]$

can be rewritten as

$\displaystyle \frac{r_1 e^{i \theta_1}}{ r_2 e^{i \theta_2}} = \displaystyle \frac{r_1 }{r_2} e^{i(\theta_1 - \theta_2)}$

Finally, DeMoivre’s Theorem, or

$\left[ r (\cos \theta + i \sin \theta) \right]^n = r^n (\cos n \theta + i \sin n \theta)$

can be rewritten more comfortably as

$\left( r e^{i \theta} \right)^n = r^n e^{i n \theta}$

When showing these to students, I stress that these are not the formal proofs of these statements… the formal proofs required trig identites and mathematical induction, as shown in previous posts. That said, now that the proofs have been completed, the $e^{i \theta}$ notation provides a way of remembering these formulas that wasn’t immediately obvious when we began this unit on the trigonometric form of complex numbers.

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For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.

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Calculators and complex numbers (Part 16)

Published by John Quintanilla

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