Saturday, January 16, 2016

One Good Thing: Complex Numbers

Snazzy Image, eh?I had twins this summer, so I'm only teaching part time this year (which turned out to be an excellent idea, babies-wise). So what better way to transition myself back into the world of teaching than the reflection of the MathTwitterBlogosphere Blogging initiative?!

This week: One Good Thing.

This week, we introduced Complex Numbers in my Math 3B (Algebra 2 level) class. I've been loving the opportunity to dive into what numbers are all about with this. First off, I love the "get ready to have your mind blown" aspect of complex numbers. I sometimes record pieces of class from my Smartboard using ScreenFlow, either for students who are absent or for topics that students might want to review later. This Intro to Complex Numbers lesson starts with reviewing the types of numbers they have experienced thus far, with hand-wavey tie-ins to the history of human civilization & their individual histories of learning math. And then, dah, dah dum... Imaginary Numbers! Go to 12:06 to watch my fun smartboard trick about it at 13:15.

But I also love getting into why the rules of complex addition/subtraction/everything else work, and going back and forth between algebraic and graphical representations, and between complex and real number worlds. For example, the rule of multiplication is totally bizarre (and will remain unproven in our class) - it involves addition, for pete's sake.
Multiply the radii, add the angles, in case you're rusty.
Which raises the question of whether this is totally divorced from regular multiplication. So we look at it. Spoiler alert: NO. IT TOTALLY WORKS.
Do you see that pretty connection between polar representation from the graph on the complex plane?

And then there is the idea of One, which we need if we are going to talk about reciprocals of complex numbers, which we need in order to divide. What is "One" in complex numbers?" What does "One" even do? And the best thing is that they don't need to believe me that "one" should be 1+0i as opposed to 1+1i, or anything else. 
So if we are talking about "one" in terms of what one does, then we can talk about reciprocals in terms of what a reciprocal does (result in One when multiplied by the original number), rather than their middle school definition of 1/original number. And then we can divide! 

My students ask me about why we are learning complex numbers, and rightly so. And I can talk about how complex numbers are useful in higher math, and with computers (so they don't just stop when they get to the even root of a negative), and how they make many things possible (like laser eye surgery machines) because the coding requires computers to not freak out over the square root of a negative number. But why do we, right now, care about them? Because this is cool. Because math can be intrinsically interesting all by itself. We are not going to get to "real world" applications, and sure, this lays the groundwork if you want to go there later on in your life. But the purpose of this, right now, is just talking about numbers and how they work, and how they would work if we just changed the rules and had to think differently. Like, what do you mean by "one"? 


And of course, after doing these lessons this term, I realized that I didn't discuss Zero as being the Additive Identity, which would have really grounded our discussion about subtracting complex numbers as adding the opposite. Next time!