Calculators and complex numbers (Part 15)

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, at long last, I will explain in today’s post.

Definition. If z is a complex number, then we define

e^z = \displaystyle \sum_{n=0}^{\infty} \frac{z^n}{n!}

This of course matches the Taylor expansion of e^x for real numbers x.

Theorem. If \theta is a real number, then e^{i \theta} = \cos \theta + i \sin \theta.

e^{i \theta} = \displaystyle \sum_{n=0}^{\infty} \frac{(i \theta)^n}{n!}

= \displaystyle 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \frac{(i\theta)^4}{4!} + \frac{(i\theta)^5}{5!} + \frac{(i\theta)^6}{6!} + \frac{(i\theta)^7}{7!} + \dots

= \displaystyle \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} \dots \right) + i \left( \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} \right)

= \cos \theta + i \sin \theta,

using the Taylor expansions for cosine and sine.

This theorem explains one of the calculator’s results:

e^{i \pi} = \cos \pi + i \sin \pi = -1 + 0i = -1.

That said, you can imagine that finding something like e^{4-2i} would be next to impossible by directly plugging into the series and trying to simply the answer. The good news is that there’s an easy way to compute e^z for complex numbers z, which we develop in the next few posts.

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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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