In the video below I try to give some of the "patter" I use when I introduce the idea of "i" for the first time. I think that there is a great benefit in doing it this way because it helps strengthen their understanding of what real numbers do as well. It also emphasizes that numbers often get paired with operations. I actually could make that clearer. Anyway, feel free to steal this introduction to use in your own classes.
In my continued quest to spread an understanding of complex numbers I put together this little dance sequence with my Summer School Algebra 2 students. I used the function f(x)=(x-1)^2+1 determine the dance sequence. This function has two complex roots (1+i) and (1-i). I had students stand at 1+i, i, -1+i, 1, -1, 1-i, -i, and -1-i, then we went through the three steps of the function. These were "minus 1", "squared", and "plus 1". The most important visual here is to get a sense of what squaring does to the complex plane. This includes some expansion of the numbers with distance greater than 1 and a wrapping of plane on top of itself. Watch the video and let me know what you think?
I was talking about function notation with my students, and trying hard to differentiate between f, f(x), and f(x)=x+3. The metaphor that I tried was recipe. Does anyone else have a good metaphor that helps to distinguish these concepts?
Teaching summer school today. I had a student not understand what f(f^(-1)(x))=x was trying to say. I was searching for a process that would help her understand doing and undoing. And then it hit me. Sylvester McMonkey McBean.
I have been reading the book Conics by Kevin Kendig. This movie was made with Grapher on my Mac, and displays the viewpoint of taking conics and projecting them down on the sphere.
This post is continuation of this rant.
We do such a bad job with complex numbers that students often assume that the word Imaginary in front of Number is an adjective that i is not really a number. It is some sort of imaginary thing.
One important reason that students find this easy to believe is the fact that it is not possible to order the complex numbers. Ordering is a basic and fundamental concept for numbers. Ordering is first thing that students are to learn about numbers. We stress the idea of ordering of the numbers, we use the ordering of numbers to help explain addition and subtraction. So if we can neither tell if i is neither bigger or smaller than 0, then it might not really exist.
So makes imaginary and complex numbers real. What makes them real is that can represent things that happen in the real world. Real numbers can represent length and direction (direction in one dimension, left or right, up or down). Complex numbers can represent length and direction in two dimensions. So 3+4i can represent 3 steps to the right and 4 steps up. You can represent movement in two dimensions with complex numbers. But that is not different than vectors. What makes complex numbers so different is the existence of a coherent multiplication rule, one that even allows a coherent definition of division.
So it is multiplication that makes complex numbers so special. What does multiplication in complex numbers do. It does two things, it is capable of doing what multiplication of real numbers does, which is dilation. Multiplication by a positive real number performs a dilation on the points of the complex plane. Multiplication by a negative real number performs a dilation and a 180° rotation. In particular, multiplication by -1 is just a 180° rotation. Now how about multiplication by i. If you take any complex number and multiply it by i the result is a 90° rotation about 0. And from all that we know about i, this makes tremendous sense. We know that i is the square root of negative one, so that multiplying by i does half of what multiplying by -1 does. We know that i^4=1 so that multiplying by i four times does nothing to a number, just as rotating by 90° four times returns things to where they started. It turns out then that multiplying by complex numbers models rotation! This is what is important. Turns are everywhere. Everywhere in life there are rotations of all sorts and these numbers allow you to do computations with coordinates that affect the desired rotations. Want to turn 90°, multiply by i. Want to turn 45°, multiply by sqrt(i). This is why polar form for a complex number exists, and it is the importance of DeMoivre's Theorem.
As a final note tonight. Let's bring it back to algebra. Let's say we have to solve x^6=1. Then we are looking for 6 solutions. Well then these are complex numbers that multiplication is the same as rotating 0°, 60°, 120°, 180°, 240°, and 300°. In other words 1, 1/2 +sqrt(3)/2 i, -1/2 +sqrt(3)/2 i, -1, -1/2 -sqrt(3)/2 i, 1/2 -sqrt(3)/2 i. But what is better is seeing it on the complex plane.
That is right the solutions to this equation are vertices of regular polygon, and more this generalizes. So that the regular polygons can be associated to the polynomials x^n-1.
Are you a math teacher? If you are then you probably have built up a tolerance to the funny looks; a tolerance to the subtle comments that are meant to infer that you are VERY different from everyone else. You must have this tolerance or how else could you continue in your profession. I understand. I have it too.
But here is the thing. It dulls us to the messages around us. Something is wrong in in the state of math education. Most of us know this, though some still fight it. Most of us know that it is the charge of math teachers to make it better. We must raise an entire generation of children that meet math teachers and say, "Yeah, math was always too easy for me so I decided to do something difficult like teach grammar instead." That is our charge, and to make it happen we need to assiduously evaluate everything that we do. We must find the mistakes and fix them. I don't know if what I am about to propose a solution to is the most pressing problem, but it definitely is one. We may be able to learn something from the problem, and I hope something from my proposed solution. Here goes...
Complex Numbers
Let's take what a child is supposed to know about complex numbers after Algebra 2. Algebra 2 is or was a terminating course in high school math. So this may be all that any one every learns about these crazy things. Here are the objectives from the Algebra 2 book I am teaching out of during summer school:
Our latest post on determinants made the carnival of mathematics at catsynth.com. Which reminds me of one of the things that makes me happy in the world... Cats falling compilation videos on youtube. One of my AP Calc BC kids showed me this one. Those kids are so smart.
I really love the music. I always wondered what the purpose "Man in motion" was.
