Calculators and complex numbers (Part 18)

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.

Definition. If is a complex number, then we define

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

In the last few posts, we proved the following theorem.

Theorem. If and are complex numbers, then $e^z e^w = e^{z+w}$.

This theorem allows us to compute without directly plugging into the above infinite series.

Theorem. If , where and are real numbers, then

Proof. With the machinery that’s been developed over the past few posts, this one is actually a one-liner:

.

For example,

Notice that, with complex numbers, it’s perfectly possible to take to a power and get a negative number. Obviously, this is impossible when using only real numbers.

Another example:

In this answer, we have to remember that the angle is 3 radians and not 3 degrees.

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