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.
In the remaining posts in this series, I want to explore which properties of exponential functions remain true when complex numbers are used.
To begin, if is a real rational number, then there is an alternative definition of that matches De Moivre’s Theorem. Happily, the two definitions agree. Suppose that with . Then
Next, one of the Laws of Exponents remains true even for complex numbers:
However, in previous posts, we’ve seen that the rules and may not be true if nonpositive bases, let alone complex bases, are used.
We can also derive the usual rules and . First,
Next, we think like an MIT freshman and use the above Law of Exponents to observe that
Dividing, we see that .
For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.