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 is

where and , with 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 () that, at long last, I will explain in today’s post.

Definition. If is a complex number, then we define

This of course matches the Taylor expansion of for real numbers .

Theorem. If is a real number, then .

,

using the Taylor expansions for cosine and sine.

This theorem explains one of the calculator’s results:

.

That said, you can imagine that finding something like 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 for complex numbers , which we develop in the next few posts.

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