Thursday, July 23, 2009

Visualizing Complex Numbers

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?


Kate Nowak said...

I see the value in helping understand function transformations, but I'm not sure the effect on the roots is coming through clearly.

Do you think it helped the students understand?

Matt said...

This didn't work as well as other similar things I had tried, and I didn't get the chance to show students the video. I think this view of what is going on is somewhat helpful.

Here is what is very helpful though. Physically demonstrating what +1, -1, *2, /2, *i, +i, -i all mean. I would love students to leave algebra 2 with a better understanding of those, and I think that similar activities to this activity do a good job of that.

What isn't demonstrated enough here is that squaring has this effect of wrapping the complex numbers over itself. The two girls on the left are the only two that ended up in the same spot, and that was due to poor planning, and divergent goals. Those two girls both end up back at zero. What I should try to do next time is use a tripod. Assign students based on height better. Include more visible labels. Steady the camera so I could overlay a grid. Etc.

The main point continues to be that complex numbers have real physical meanings that generally are forcibly concealed in the way we currently teach the subject. That the major thrust of complex numbers needs to be based on geometry and not on the fundamental theorem of algebra.

In fact, I think that this lesson is an effort on my part to bring the fundamental theorem of algebra back in through the geometry.

Thanks for your comment Kate. I love your blog, and I am very excited to have your comment on my page.

Kate Nowak said...

I am with you that the geometric interpretation of complex numbers is important and frequently neglected. This year I plan to dwell longer, at least, on the transformations on the complex plane accomplished by multiplying by -1 and i. I'm also toying with the idea of introducing i as the geometric mean of 1 and -1.

If you revise this activity and re-do it, I hope you will post your reflection of the results.

Thanks for the kind words about my humble efforts! :)

Matt said...

I am interested in how you will demonstrate this idea that i is the geometric mean between 1 and -1. I see it algebraically/formulaicly but how does that help. What insight can be gained from thinking of i in this way?



Kate Nowak said...

Think of (1,0) (-1,0) and (0,i) as vertices of a right triangle, with an altitude drawn from (0,i) to (0,0).

Again I'm not sure it's the best way to introduce it. I'm just thinking about whether it makes sense to incorporate it.

Anonymous said...

It was certainly interesting for me to read that article. Thanks for it. I like such themes and everything that is connected to them. I definitely want to read more soon.