Imaginary Numbers are Real [Part 6: The Complex Plane]

Imagine you had an art class in which they taught you how to paint a fence, but never showed you the great masters. Of course, you would say; ‘I hate art.’ You were bad at painting the fence but you wouldn’t know what else there is to art. Unfortunately, that is exactly what happens with mathematics. What we study at school is a tiny little part of mathematics. I want people to discover the magic world of mathematics, almost like a parallel universe, that most of us aren’t aware even exists.
— Edward Frenkel

Let's Do Some Real Math

This week I'm leaving you with a challenge. The quote above is from Edward Frenkel, a Berkeley mathematics professor, author of Love and Math, and generally cool guy. Frenkel draws a really interesting analogy about how we teach math. People who do math for a living exist in an uncertain world of creativity and discovery, while math classes are typically quite the opposite. Math is, by nature, a highly technical subject - meaning that just wrapping your head around things can take quite a bit of time. This leaves many students too tired for "creative discovery". The unfortunate side effect here is that, just as Mr. Frenkel says, many students do end up feeling like fence painters, and not explorers. 

With this in mind, between this episode and the next, I'm leaving you with some actual math. A real problem. No fence painting.

Forgive me in advance for the vague nature of the problem statement - but when you come across a real problem in STEM, this is what it feels like. This is the nature of mathematical discovery - we don't know what we're searching for. The upside is that when we do find something, this makes it all the more exciting. And after all, it wouldn't really be searching or discovery if we knew what we are going to find beforehand. When you hop on a plane to fly to Albuquerque, you aren't "discovering it".

Your Assignment

So this is your job. I would like you to discover for yourself what it means to multiply complex numbers on the complex plane. 

There's a very specific, and very useful interpretation of complex multiplication using the complex plane, and I want you to find it.

Figure 1. Tools you need.

Figure 2. Examples to try.

The tools you need our colorfully summarized in figure 1. Our approach next time will make use of four examples, these are (less colorfully) shown in figure 2. I recommend for each example plotting the two numbers we're multiplying and the result on the complex plane. From there, look for patterns, make theories, test your assumptions, and do try to have some fun. Good luck!