Calculators and complex numbers (Part 23)

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.

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

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

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.

Definition. Let z and w be complex numbers so that z \ne 0. Then we define

z^w = e^{w \log z}

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 w is a real rational number, then there is an alternative definition of z^w that matches De Moivre’s Theorem. Happily, the two definitions agree. Suppose that z = r e^{i \theta} with -\pi < \theta \le \pi. Then

z^w = e^{w \log z}

= e^{w [\ln r + i \theta]}

= e^{w \ln r + i w \theta}

= e^{w \ln r} e^{i w \theta}

= r^w (\cos w\theta + i \sin \theta)

Next, one of the Laws of Exponents remains true even for complex numbers:

z^{w_1} z^{w_2} = e^{w_1 \log z} e^{w_2 \log z}

= e^{w_1 \log z + w_2 \log z}

= e^{(w_1 + w_2) \log z}

= z^{w_1 + w_2}.

However, in previous posts, we’ve seen that the rules (x^y)^z = x^(yz) and x^z y^z = (xy)^z may not be true if nonpositive bases, let alone complex bases, are used.

We can also derive the usual rules z^0 = 1 and z^{-w} = \displaystyle \frac{1}{z^w}. First,

z^0 = e^{0 \log z} = e^0 = 1.

Next, we think like an MIT freshman and use the above Law of Exponents to observe that

z^w z^{-w} = z^{w-w} = z^0 = 1.

Dividing, we see that z^{-w} = \displaystyle \frac{1}{z^w}.

green line

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



2 thoughts on “Calculators and complex numbers (Part 23)

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