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
Definition. Let and be complex numbers so that . Then we define
Technical point: for the latter two definitions, these are the principal values of the functions. In complex analysis, these are usually considered multiply-defined functions. But I’m not going to worry about this technicality here and will only consider the principal values.
This is the last post in this series, where I state some generalizations of the Laws of Exponents for complex numbers.
In yesterday’s post, we saw that as long as . This prevents something like , since is undefined.
Theorem. Let , , and . Then .
As we saw in a previous post, the conclusion could be incorrect outside of the above hypothesis, as .
Theorem. Let and . Then .
Theorem. Let be real numbers and . Then .
Again, the conclusion of the above theorem could be incorrect outside of these hypothesis, as .
For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.