We take any Euclidean line, mark point 0 on it, choose a unit length and mark point 1, and then the non-negative "numbers" are associated (identified) with all the points on the ray from 0 through 1in the usual way. The negative numbers are then identified with the remaining points on the opposite ray in the usual way so that we have now a Cartesian coordinate number line L. Now choose a second line M through 0 and mark a 1 on it at the same unit distance as we used on L, and complete this so M is also a Cartesian coordinate line. Now for any number b (point) on L, and any nonzero number c on M, the product point b*c can be found: let J be the line from 1 on L to c on M, and K the line parallel to J through b on L, then b*c is the intersection of K and M.
Here is a picture that shows L as x-axis, M as y-axis, and we are looking for 4*3.
The only problem with this explanation is that the students need to know about similar triangles! Regardless, this was way cool. If I had thought about this on time, I would have given it to my students on their final that had them think about similar triangles. Next time!
9 comments:
Could this be what you were looking for?
Nice! I wasn't necessarily looking for anything, but I liked that post. I would like to find this lesson plan on ILoveMath. I'll work on that. The only problem I see with lesson: has no student asked "What is a rotation by 37 degrees?" (or any other number).
The complex number cos(37pi/180)+i*sin(37pi/180)! Or, the number such that, if you raise it to the power of 180/37, you get -1. Or did I misunderstand your question?
I think so. I'll quote from that post: "multiplying by what number gives you a 90 degree rotation?" So, I was asking the same question, for 37 degree rotation. What I'm saying, I suppose, is that from these two examples that were given, a student may attempt to extrapolate a general rule that says: multiplication by x is rotation by x', which isn't there. They need to know more. This is one of the issues, that often to describe some thing really cool, you need to know more math. Unless they know little linear algebra, vectors, matrices and transformations. I may not be making sense, in which case I apologize.
A rotation by 90 degrees corresponds to a multiplication by i, not by 90, and a rotation through 37 degrees corresponds to a multiplication by .8+.6i. So while not every number corresponds to a rotation, every rotation corresponds to multiplication by a number. Multiplying a positive real number by another positive real number x involves stretching the number by a factor x in the postive direction, multiplying it by -x involves rotating it through 180 degrees and stretching it by a factor of x, and multiplying with a complex number of absolute value x and argument a involves rotating it through an angle a and stretching it by a factor x. I guess it all hinges on the notion of a number as a place on the number line, so that multiplication must be conceived as some kind of motion from one place to another. But, yeah, my students did not find these ideas thrilling or clarifying when I introduced them, but then that was at the very end of last semester and we did not spend nearly the necessary kind of time on it. Maybe next year... It is possible that it's too hard, as you're suggesting, but it's so awfully neat I want to try it out with better planning and scheduling at least once more.
A rotation by 90 degrees corresponds to a multiplication by i, not by 90,... I think I said multiplication by x is rotation by x', where x in x' was indicating that it depends on x, and ' in x' that it's not x (i should have said a(x) or something like that) :)
I guess the answer to the question above depends on what they know. If you're only introducing complex numbers then I can see how it could be confusing. I was by no means arguing that you shouldn't attempt teaching this! In fact I think it's a great idea, I just wonder how one might take them to the multiplication by a complex number without just telling them. Also, do they even learn the representation of a complex number with the absolute value and an angle in high school?
You're absolutely right; sorry about my misreading:)
The students are supposed to learn to plot complex numbers and find their absolute value in Algebra II, which would be in 10th grade according to the optimistic CA standards (Algebra I in 8th grade, Geometry in 9th). The neatest way to teach the absolute value formula for complex numbers is to remind the students that the absolute value of a number is its distance from the origin - and then using the familiar distance formula gives the rest. As for the angle part - it's in the Math Analysis standards, so college-bound students are expected to see it junior or senior year.
So you're a math teacher teacher with a Ph.D. in math, and here I was telling you about them complex numbers. I feel silly, but since you were so nice about it I guess I have permission to laugh and move on. I'm glad to know about your blog, it's been a good read.
You should definitely not feel silly. I enjoyed the conversation, and I am always happy to talk about math. Besides, I always learn something, too.
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