Calculators and complex numbers (Part 18)

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.

Definition. If z is a complex number, then we define

e^z = \displaystyle \sum_{n=0}^{\infty} \frac{z^n}{n!}

This of course matches the Taylor expansion of e^x for real numbers x.

In the last few posts, we proved the following theorem.

Theorem. If z and w are complex numbers, then $e^z e^w = e^{z+w}$.

This theorem allows us to compute e^z without directly plugging into the above infinite series.

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

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

Proof. With the machinery that’s been developed over the past few posts, this one is actually a one-liner:

e^z = e^{x+iy} = e^x e^{iy} = e^x (\cos y + i \sin y).

For example,

e^{4+\pi i} = e^4 (\cos \pi + i \sin \pi) = -e^4

Notice that, with complex numbers, it’s perfectly possible to take e to a power and get a negative number. Obviously, this is impossible when using only real numbers.

Another example:

e^{-2+3i} = e^{-2} (\cos 3 +i \sin 3)

In this answer, we have to remember that the angle is 3 radians and not 3 degrees.

green line

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



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