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Thursday, September 30, 2010

A Whole New Kind of Number

I think I've finally got my introduction to imaginary numbers and the complex plane to a point where, let's say, the students can make room in their brain for the idea. They still don't like it, but they leave with some sense of what the heck we mean when we say i.

I start by asking them to place some real numbers on a number line.

Then I ask them to think about the lengths of sides of different squares. We try several fractions and terminating decimals to try to find one that we can square and get a 2, and we are unable to find one.

So that's why people needed to invent irrational numbers: to solve this problem. We just define radical 2 to be the number that gets you a 2 when you multiply it by itself.

Then we read through this story. I have them read the slides popcorn style (reader of this slide chooses reader of the next slide.)

They enjoy the story, except sometimes they make comments about how John and Betty are freakishly precocious, and sometimes they wonder what is up with Betty's hair. We don't read the whole thing. Just up to where they have to invent i.

The story gets them up to: i is the number people had to invent because there aren't any real numbers we can square and get -1. And if i2 = -1, it stands to reason that we can define i as the square root of -1.

This is the best, most grabby part of the lesson: I put the number line back up, and say

So if i is a number...where do we put it?

Stop and wait and let the room be silent for a little while. They're considering things, and deciding against them. They sometimes suggest putting it at both 1 and -1, but of course they don't really know. So I say:

i isn't on the line. But it is on the board.

Then I carefully measure with the thumb and finger of one hand the distance between 0 and 1, turn my hand, and put i the same distance above 0. Then they can tell me where 2i and -i are located, and they can pretty much figure out where we should put complex numbers like 3+2i and -1 - 3i.

This lesson goes on to consider what we might mean by things like 5i + 6i,  2(4i), and 25i/5. Having the graph to refer to really helps. It sets us up nicely for powers of i tomorrow, too.

With all three groups today, there was a moment of "ick." "I don't like this." "This is weird." I tried hard to acknowledge and legitimize that feeling. I told them that feeling of discomfort is normal when you're making room in your mind for a brand new idea. I likened it to that saying "Pain is weakness leaving the body."

Except I said that weird feeling is ignorance leaving the brain. They seemed to like that.


  1. Thanks for showing this. I did a little history of numbers to introduce i which was helpful. The kids were able to see that I just didn't pull i out of thin air. Next time, I'll also do the number line.

  2. That's kinda cool to have a visceral reaction to learning. It may not be the best feeling, but your head knows you are getting it, while your body is somehow confused by it.

    Welcome to the SECOND DIMENSION, kiddies! If you have them read Flatland, this is probably the feeling that A. Square feels when he leaves flatland for spaceland.

    Then we sing "A Whole New World" and everyone flies home on their magic carpet.

  3. I love your quote at the end, but I might change it for myself to "Confusion is ignorance leaving the brain."

  4. Looks great. I especially like the number line part, and where are we gonna put this i thing.

    I think the ick comes from why put i above the number line?

    In my post here I attempt to give some reason for this.

    The added benefit of this, is that it not only explains why we would put i there, it also helps explain why would use at i at all. We can use it to model rotations. And lots of things turn. And that makes more useful/relevant.

    Which is the opposite of what you might think would happen if you used "imaginary numbers".

  5. I think that quote may have to go on a poster in my room. Thanks!

  6. There's another way of seeing where to place the imaginary numbers, based on looking at multiplication on the number line geometrically.

    What happens when we multiply by -1? We turn the entire number line end over end.

    So what happens when we multiply by i? We turn e number line halfway around. That is, it's at a right angle to the real number line!

  7. @Matt and John -

    You'd approve of tomorrow's lesson, because it's all about how multiplying by -1 and i are transformations on the vectors. I'd love to have done it today, too, but you can only squeeze so much into 40 minutes.

  8. Kate,
    I assume you've seen this post?

    A visual-intuitive-guide-to-imaginary-numbers.

    Anything we can do to help kids see vectors more visually (before jumping into components) is awesome in my book.

  9. Hey Kate! Maybe the kiddies would enjoy those quotes from Ars Magna in which Cardano deals with imaginaries for the first time in print, and is like "putting aside the mental tortures involved..." I quoted them in my history of negatives post. (They're about halfway down the post. Search for "mental tortures.")

    That's just for fun, so they know that they can relate in their sense of confusion to the people who actually did the development.

    But perhaps, for the purpose of seeing the point, it would also be cool to show them that imaginary numbers can be used to find a real-number answer to a real-number question. A historically important exmaple is from Bombelli (1572) who showed that if you apply Cardano's cubic formula to solve

    x^3 = 15x + 4

    you get

    x = cube root(2+sqrt(-121)) + cube root(2-sqrt(-121))

    Now this "doesn't make any sense" because you're taking square roots of -121, but if you pretend that this makes sense, i.e. will imaginary numbers into being, then

    cube root(2+sqrt(-121)) = cube root(2+11i) = 2+i

    and likewise

    cube root(2-sqrt(-121)) = 2-i

    so that Cardano's formula gives you

    x = 2+i + 2-i = 4

    as the solution to the original equation, and sure enough,

    4^3 = 15*4 + 4

    so that x=4 really is a solution. How rad is that.

  10. Kate,

    I am teaching 6th and 7th graders algebra. They are seeing most of this stuff for the first time and it is REALLY strange to them. So, I LOVE this...

    "I told them that feeling of discomfort is normal when you're making room in your mind for a brand new idea. I likened it to that saying "Pain is weakness leaving the body."
    Except I said that weird feeling is ignorance leaving the brain. "

    That is exactly what I have been trying to say to them, however poorly. Thank you so much for the right words. I am sharing this with my dear sweet students who are currently awed, amazed (and sometimes confused) by algebra.


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