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.
Theorem. If , where
and
are real numbers, then
Definition. Let be a complex number so that
. Then we define
.
At long last, we are now in position to explain the last surprising results from the calculator video below.
Definition. Suppose that and
are complex numbers so that
. Then we define
Naturally, this definition makes sense if and
are real numbers.
For example, let’s consider the computation of . For the base of
, we note that
.
Therefore,
,
which is (surprisingly) a real number.
As a second example, let’s compute . To begin,
.
Therefore,
In other words, a problem like this is a Precalculus teacher’s dream come true, as it contains , and
in a single problem.
For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.
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