Calculators and complex numbers (Part 10)

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)

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.

There’s a shorthand notation for the right-hand side (r e^{i \theta}) that I’ll justify later in this series.

Today, I want to share some pedagogical thoughts about this series of posts. I’ll continue with the mathematical development of these ideas tomorrow.

My experience is that most math majors have never seen this particular application of trigonometry to find the nth roots of complex numbers… or even are familiar with the idea of expressing a complex number into trigonometric form at all. This personally surprises me, as this was just one of the topics that I had to learn when I took Precalculus (which was called Trig/Analysis when I took it). I really don’t know if I was fortunate to be exposed to these ideas in my secondary curriculum of the 1980’s or if this was simply a standard topic back then. However, at least in Texas, the trigonometric form of complex numbers does not appear to be a standard topic these days.

This certainly isn’t the most important topic in the mathematics secondary curriculum. That said, I really wish that this was included in a standard Pre-AP course in Precalculus to better serve the high school students who are most likely to take more advanced courses in mathematics and science in college. These ideas are simply assumed in, say, Differential Equations, when students are asked to solve

$y^{5} – 32 y = 0$.

The characteristic equation of this differential equation is r^5 - 32 = 0, and it’s really hard to find all five complex roots unless De Moivre’s Theorem is employed.

To give another example: In physics, even a cursory look at my old electricity and magnetism text reveals that familiarity with the trigonometric form of complex numbers can only facilitate student understanding of these physical concepts. Ditto for many concepts in electrical engineering. Stated another way, students who aren’t used to thinking of complex numbers in this way may struggle through physics and engineering in ways that could have been avoided with prior mathematical training.

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



2 thoughts on “Calculators and complex numbers (Part 10)

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