In the video below I try to give some of the "patter" I use when I introduce the idea of "i" for the first time. I think that there is a great benefit in doing it this way because it helps strengthen their understanding of what real numbers do as well. It also emphasizes that numbers often get paired with operations. I actually could make that clearer. Anyway, feel free to steal this introduction to use in your own classes.
Friday, July 24, 2009
Visualizing Numbers Real and Imaginary
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14 comments:
I like your video and attempt to make an abstract subject a little more concrete. I often used my hands in similar ways although I have not done it with i numbers. Did you find it helped them understand the concept of imaginery numbers?
www.realmathinaminute.com
I'm so impressed with my nephew! I didn't even know there were i numbers.
Sandra - I do think it helped them understand it, and in the end the most important message here is that these numbers have a purpose. Too often we present mathematics as a meaningless mental game. Without purpose we reinforce every negative stereotype that exists about mathematics and mathematics education.
Nancy Hill- So glad you stopped by.
I am really impressed. I wish you were my teacher 20 yrs ago :-)
I don't know math. I don't like math. All I want to know is how (a) 2 x 0 equal 0, but (b) 0 x 2 equal 0 and not 2? That is, (a) if I begin with nothing, and multiply it twice (or ten times, etc.), my answer must remain zero, because it doesn't matter how many times nothing is multiplied--I'm left with nothing. But (b) if I begin with, say, two, and multiply it zero times, it seems I'm left with my original two, not with nothing. Go figure.
What a fascinating way of thinking about i! I had never heard this explanation before... Do solutions involving i take us into a fourth dimension other than time (I am thinking of antenna impedance calculations, etc.)?
I am still pondering the connection between this view of i and vectors/rotation matrices. I suppose there should be a one to one correspondence between the two, in that I could represent a quadratic in one variable using a two dimensional (x-dimension, i-dimension) vector for the variable where the imaginary components are always zero initially. If I did this, then the answers to a problem such as
X^2 = -4
should be the same when graphed using either your rotational approach or vector multiplication.
Hmmmm - I'll play with this a bit after refreshing my memory about vector multiplication (it has been a long time).
Whit--
yes there is a strong connection between rotational matrices and complex numbers. Complex numbers are somewhat more compact way to write those operators. For instance, as you asked about X^2=-4. To think about these in terms of 2x2 matrices the equation would more accurately be written X^2=-4I where I is the 2x2 identity matrix. In this case, [[0,2],[-2,0]] is a solution to this equaiton. There is a correspondence between complex numbers and 2x2 matrices that respects addition and multiplication by sending a+bi --> [[a, b],[-b,a]]. As an interesting aside, checkout the connection between the norm of complex number and the determinant of the matrix.
This is so great, Matt.
Okay that I post this video (with attribution and praise, of course) to the ol' blog?
I can also just do the link, if you prefer.
Joshua,
I am so glad that you like it. And ever since I found out that this is the case I have been really interested in spreading the word. So please place it on your blog with attribution. Thanks so much.
Matt
I used this worksheet with my Alg2 kids this year and it *really* helped to give some meaning to complex operations. And they really had fun with it!
http://untilnextstop.blogspot.com/2010/01/encouraging-things.html
When I tought i to kids, I had a discussion with them about whether they think we should study things that we can't see and touch. Their responses were fascinating. (My honors class was vehemently split down the middle.)
This presentation also makes the conceptual difference between addition and multiplication almost laughably clear.
So, how might one visualize the impact of exponentiation on the number line?
I realize that you're aiming this explanation at high school students and hence it has to be simplified and gloss over theoretical details, but nonetheless I have a problem with the explanation. Treating multiplication by a negative number as an operation involving a rotation presupposes that numbers can be modelled via a plane and not just via a number line. Aren't there any students that ever question this assumption? I'm sure most wouldn't because your explanation is smooth and the average high school mathematics student doesn't have the reflective ability or interest to notice this explanatory gap, but I think that the brighter ones would want to know why numbers should be represented two-dimensionally as opposed to points on a line, which is how we are accustomed to thinking about them. When I studied complex numbers, the concept of numbers as points in a plane was to me just as intuitively troublesome as the idea that -1 could have a square root.
MFC,
You are absolutely correct. Most of my students don't challenge this, and I would say that you are right that in many ways this explanation is for the less mathematically confident/interested. I feel it is very important to make complex numbers have meaning for these students and therefore I give this explanation. More mathematically savvy students are generally more willing to accept added abstraction.
But it is very interesting to hear that 2-dimensionality of the number was a "stumbling block". This gets to the meaning of "number". Which is again a deep and abstract topic.
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