Complex Multiplication

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Textbook Reference: Section 8.5 Polar Form of Complex Numbers

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Only one day studying complex numbers, and we've already seen so many connections to other things we've learned this year.

• The powers of \(i\) start at \(i^0=1\) and rotate counterclockwise ... just like coterminal angles on the unit circle.
• Addition and subtraction of complex numbers are just like what we do with vectors — add the \(x\) and \(y\) components separately.
• The absolute value of a complex number — its distance from zero — is calculated using the Pythagorean theorem, again just like the magnitude (length) of vectors.
• In fact, since every complex number has a magnitude \(r\) and a direction \(\theta\), we can write them in polar form as \(r\,\text{cis}\,\theta\), which is short for \(r(\cos\theta+i\sin\theta)\).
• Electrical engineers write complex numbers in an even shorter form \(r\,\angle\,\theta\) — which is almost exactly what we've been using for polar coordinates.
• Speaking of electricity, complex numbers are often used to describe alternating current electricity, in which the current and voltage fluctuate in a sinusoidal wave.

...whoa. That's a LOT of connections.

But we're just getting started.

The main thing we left out of our connections was complex multiplication (and its cousin, division).

And in this final(!) Preview Activity, you'll start to see how something as basic as multiplication can explain the entirety of why complex numbers work the way they do.

Preview Activity 30

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

1. Let \(z=2+2i\) and \(w=-3\).

   1. Calculate the product \(zw\).
   2. Rewrite each of \(z\), \(w\), and \(zw\) in polar form \(r\,\text{cis}\,\theta\).
      Keep your angles in degrees, and make sure each angle is between \(0^\circ\) and \(360^\circ\).
   3. What seems to be the relationship between the magnitudes of \(z\), \(w\), and \(zw\)?
   4. What seems to be the relationship between the angles of \(z\), \(w\), and \(zw\)?

2. Repeat #1 with \(z=-3+4i\) and \(w=-12-5i\).
   Do your same observations from parts c and d in #1 still seem to hold true?

   (The magnitudes will be "nice", but the angles won't be. Remember that the inverse tangent function only gives angles on the right half of the unit circle, so you'll need to adjust your angle accordingly if you use it.)

3. Let \(z=-3+3i\) and \(w=1+i\). This time you're going to investigate division.

   1. Rewrite each of \(z\) and \(w\) in polar form \(r\,\text{cis}\,\theta\).
   2. Thinking about what you saw in the previous two problems, what do you guess will be the magitude and direction of \(\dfrac zw\)?
   3. Calculate \(\dfrac zw\), and put it in polar form. Were you correct in part b?
      (Hint: Multiply the numerator and denominator by \(1-i\).)
4. Answer AT LEAST one of the following questions:
   1. What was something you found interesting about this activity?
   2. What was an "a-ha" moment you had while doing this activity?
   3. What was the muddiest point of this activity for you?
   4. What question(s) do you have about anything you've done?

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Posted by Bill Shillito