Thursday, February 12, 2015

Could This Introduce Kids To Complex Numbers?


Does this strike you as a problem that leads to complex numbers? If not, then that's precisely why we should start teaching complex numbers in this way.

Complex numbers were invented for algebraic reasons (solving cubics!) and then collectively disparaged by mathematicians for a few hundred years. What did it take for complex numbers to become widely accepted? Geometry.
It is not an unreasonable demand that operations used in geometry be taken in a wider meaning than that given to them in arithmetic.
There are many situations in which it is helpful to have precise algebraic ways to describe geometric situations. This is a problem that complex multiplication were born to help with. Beginning with a "Follow That Point!" activity drives at this intersection of rotations and algebra.

That's my case for introducing students to complex numbers with a rotation activity instead of with quadratic equations.

I wrote a lesson (as part of a unit) headed towards this understanding of complex numbers. I got help from Max, Malke and Bridget. I would looove feedback and critical questions on all this. Be in touch here or (even better) on twitter.

Lesson Materials

6 comments:

  1. I remain much more algebraic in my initial thoughts that you, Michael.

    Here's what an old post of yours inspired last year:

    https://mikesmathpage.wordpress.com/2014/04/19/imaginary-numbers/

    I followed that up a few weeks later with a similar talk with the boys inspired by Ed Frenkel:

    https://mikesmathpage.wordpress.com/2014/05/17/ed-frenkel-the-square-root-of-2-and-i/

    Don't get me wrong, though, I think the geometric point of view with complex numbers is incredibly important and I'm excited to see where you take this.

    My gut feeling is that the algebraic ideas will aide with the geometric ideas and that fuzzy feeling is probably motivating my approach.

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  2. I rushed off to look at this. gepmetrical, yes, but the algebra (of transformations) rather took over. Here are my thoughts:
    https://howardat58.files.wordpress.com/2015/02/complex-numbers-by-rotations.doc

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  3. "Be in touch here or (even better) on twitter."

    Pardon?

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    Replies
    1. 1. As far as I can tell, men are more comfortable leaving comments on my blog than women are. Men seem comfortable on twitter too, so it seems like the more balanced medium.

      2. Comments have certain apparent advantages over tweets. For one, they can be as long as we want. But lately it seems like the norms of twitter have made it OK to just bust out a long series of tweets to get your point across. Comments are less important than they once were.

      3. It seems to me that I get sharper pushback through comments, but more feedback on twitter. I think that makes sense, given the very-public nature of twitter. I value that too.

      4. I treasure pushback (and praise, when it comes) in any format. It doesn't really make a great difference to me. But pushbackers (and praisers) have their favored mediums.

      5. Now that I think about it, maybe that "even better" was sort of dumb.

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  4. I got this from a link on a fellow blogger's blog. Wham! It gets right to the heart of this matter:
    http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

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  5. Realizing that I am late to the party, here's what I have.

    https://drive.google.com/a/gscs.org/file/d/0BzsW7IB5qQOZVi1USHpncTRjUnc/view?usp=sharing

    I'm not sure this would be the best way to introduce complex numbers to high school students but if definitely serves as an opportunity to show why complex numbers are relevant.

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