# Calculators and complex numbers (Part 13)

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’m about to justify.

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 $x$.

For example,

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

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

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

$= \cos 1 + i \sin 1$,

using the Taylor expansions for cosine and sine (and remembering that this is 1 radian, not 1 degree).

This was a lot of work, and raising $i$ to successive powers is easy! 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. Eventually, this will lead to the calculation of $e^{\pi i}$ which is demonstrated in the video below.

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

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