Calculators and complex numbers (Part 20)

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) = r e^{i \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.

Over the past few posts, we developed the following theorem for computing e^z in the case that z is a complex number.

Theorem. If z = x + i y, where x and y are real numbers, then

e^z = e^x (\cos y + i \sin y)

As a consequence, there are infinitely many complex solutions of the equation

e^z = -2 - 2i,

namely, z = \ln 2\sqrt{2} - \displaystyle \frac{3\pi}{4} + 2 \pi n i.

Choosing the solution that has an imaginary part in the interval (-\pi,\pi] leads to the definition of the complex logarithm.

Definition. Let z = r e^{i \theta} be a complex number so that -\pi < \theta \le \theta. Then we define

\log z = \ln r + i \theta.

Of course, this looks like what the definition ought to be if one formally applies the Laws of Logarithms to r e^{i \theta}. So, for example,

\log (-2-2i) = \ln 2\sqrt{2} - \displaystyle \frac{3\pi}{4}

A technicality: this is the principal value of the complex logarithm. In complex analysis, this is technically thought of as a multiply-defined function.

The complex version of the natural logarithm function matches the ordinary definition when applied to real numbers. For example,

\log 6 = \log \left( 6 e^{0i} \right) = \ln 6 + 0 i = \ln 6.

A couple of observations. In high school, the symbol \log is usually dedicated to base 10. However, in higher-level mathematics courses, \log always means natural logarithm. That’s because, for the purposes of abstract mathematics, base-10 logarithms are practically useless. They are helpful for us people since our number system uses base 10; it’s easy for me to estimate \log_{10} 9000, but \ln 9000 requires a little more thought. But nearly all major theorems that involve logarithms specifically employ natural logarithms. Indeed, when I first become a professor, I had to remind myself that my students used \ln for natural logarithms and not \log. Still, I write \log_{10} for base-10 logarithms and not \log as a silent acknowledgment of the use of the symbol in higher-level courses.

This use of the logarithm explains the final results of the calculator in the video below. When \ln(-5) is entered, it assumes that a real answer is expected, and so the calculatore returns an error message. On the other hand, when \ln(-5+0i) is entered, it assumes that the user wants the principal complex logarithm. Since -5+0i = 5 e^{i \pi}, the calculator correctly returns \ln 5 + \pi i as the answer. (Of course, the calculator still uses \ln and not \log to mean natural logarithm.)

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



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