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.
Wednesday, June 18, 2008
This post is continuation of this rant.
Posted by Matthew Bardoe at 9:21 PM
Tuesday, June 17, 2008
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...
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:
- Simplify radicals containing negative radicands
- multiply pure imaginary numbers
- solve quadratic equations that have pure imaginary solutions
- add, subtract, and multiply complex numbers
- simplify rational expressions containing complex numbers in the denominator
This is really the first time that students have been exposed to these kinds of numbers. Some have heard about them and wondered what they meant. And what do get to find out about them. That the purpose of complex numbers is to be added, multiplied, subtracted and simplified. Imagine that you were trying to sell and innumerate person on the integers.
Stone Age Math Teacher: Hey, have you heard about this great new number -1?
Stone Age English Major: Cool what can you do with it?
Stone Age Math Teacher: You can add, subtract, multiply and divide with it. Pretty cool huh? And you can solve any subtraction problem if you allow a whole class of new numbers called the integers!
Stone Age English Major: Wow. Major. Let's go invent beer, so I can invent poetry.
A purpose for complex numbers must be the ground work to any introduction to these numbers. Many of you may feel that these numbers have no purpose. Or that maybe the purpose is to allow us to say that yes, every quadratic does have two solutions (if you count multiplicities). But no, the purpose is real. It is part of our everyday existence. Just as the reason that 1, 2, 3 came about was to help sheep herders keep track of the flock. We are talking real.
What is it. Yawn.... I will write that tomorrow.
Sunday, June 15, 2008
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.
Friday, June 13, 2008
So far we have found that we can use the determinant to find area and volume, and that they can also help us reform the concept of proportion. Determinants can provide a way to calculate this quantity in some cases where other methods could be much more difficult. We have seen that the determinant is a generalization of proportion. What more?
One of the most important properties of determinants is that if two rows (or columns) are the same then the determinant is zero. This can be easily seen from the area interpretation of the determinant, because if two columns are the same then the area defined is degenerate for that dimension. The box has been flattened.
This is very similar to another way in which we discriminate things in math. In algebra we often see the quantity (x-a). This comes up in factoring or even in creating polynomials that have specific quantities, such as Lagrange Interpolation.
Using determinants to discriminate
Just like the expression (x-a) has the ability to test if the value of the variable x is equal to a, Discriminant will allow us to test if many variables have desired quantities simultaneously.
Example: Finding the equation of a line
In this example uses the multi-linearity of the determinant to find the equation of a line through two points. For example let’s try to find an equation of a line through (3, 5) and (-2, 3). To do this with determinants we set up the following 3x3 determinant equation.
Clearly, the points (3, 5) and (-2, 3) satisfy this equation, so that solution set does go through those two points. Is it a line? Well from any method that you choose to evaluate the determinant you can see that the coefficient of x and y will be numbers. Therefore it is a line.
Why are the in the third column? Well we need to make sure that we have a square matrix to take the determinant of, and we need to make sure that the three rows have the same number. So that we can get the determinant to evaluate to zero when we insert the points we know.
Example: Equations for other curves
One problem that we often ask students in Algebra 2 is to find the equation through three points. Let’s do one now, for instance find an equation through the points (3, 5), (-2, 3), and (6, 9). Then we set up the following determinant equation.
The numbers in the first column are the squares of the numbers in the second column. In fact, we can find an equation for a circle through those three points by the following equation.
Determinants may have more use and meaning that we have given them credit for in the standard high school curriculum. Can we bring them to the students in a meaningful way? Can we show them their usefulness and their beauty? I encourage you to explore this with students, friends