"I don't get why I am teaching imaginary/complex roots and calculations to second year HS students. I am struggling to answer the ever present comment from my students about 'Why do I need to know this?'" --@misterpatterson
This is the single biggest problem with the most common approaches to teaching complex numbers.
"Complex numbers are solutions to previously unsolvable equations." Well, who needed to solve them?
"Complex numbers are the square roots of negative numbers."
Well, who needs that?
"Complex numbers are an arbitrary extension of the number line that turns out -- shockingly! -- to be consistent."
What's the need to extend? Why would we want to do this?
"Complex numbers were invented to aid in the finding of real solutions to real problems."
Well, what are those real problems? Cubic equations?! We don't need to solve cubic equations in math class!
I propose that the solution to this problem of need is in the geometric interpretation of complex numbers. Complex numbers are a precise algebraic language for geometric transformations. As such, they can give us precise answers to questions that, previously, we could only solve imprecisely.
What questions can complex numbers help with?
"Rotate the (1,0) vector by 90 degrees. Where does the vector end up?"
"What if you rotate it by 45 degrees. What point is that vector going to end up closest to?"
"OK, now how about 30 degrees?"
Those are all questions that can be solved imprecisely using sloppy, inefficient techniques. But they can also be solved much more efficiently using complex multiplication.
I just recently learned that the Italian 16th-century mathematicians who first did arithmetic with square roots of negative numbers were trying to solve problems like this:
Find the intersection between the cubic y = x^3 and a line of the form y = 3px + 2q (that form is, apparently, special because any cubic can be written in the form x^3 = 3px + 2q.)
One of the Italian mathematicians said you could always find the intersection of x^3 and 3px + 2q at x = (q + (q^2 - p^3)^(1/2))^(1/3) + (q - (q^2 + p^3)^(1/2))^(1/3)
So here's the thing about this formula. It didn't seem to always work... it seemed to fail in pretty obvious cases, and give no reasonable solution when there clearly was a solution.
For example, x^3 = 15x + 4. y = x^3 intersects y = 15x + 4 at x = 4. A good formula should spit out x = 4.
But what the long formula above spits out is
x = (2 + (2^2 - 5^3)^(1/2))^(1/3) + (2 - (2^2 - 5^3)^(1/2))^(1/3)
x = (2 + (-121)^(1/2))^(1/3) + (2 - (-121))^(1/2))^(1/3)
So he tried to figure out if it was possible for (2 + (-121)^(1/2))^(1/3) + (2 - (-121))^(1/2))^(1/3) to somehow equal 4 once all the addition was complete. To do that he had to figure out ways to work with (or around) (-121)^1/2 and sensibly add and multiply such square roots of negative numbers.
As to whether that would motivate Algebra II students, who had no great faith in the formula in the first place, I remain skeptical... But I thought it was cool and I liked knowing that the arithmetic rules we know and love came from a problem with a known answer and a weird set-up, and the process of trying to force the weird set-up to yield the known answer.
Reply
Chris Brownell
3/25/2014 12:34:51 pm
I think two books every teacher should read, well every Mathematics teacher anyway, are "An Imaginary Tale: the story of i" or something like that; and "The Story of e" both of these values have a rich human history that some students find fascinating (I certainly do). Brilliant thinkers thinking at the edge of established knowledge create/discover new things to describe what they see, and then we later get to discover the consistencies and inconsistencies with other known stuff.
That's why we teach math, it sure ain't so kids can find the roots of the cubic polynomial 5x^3 - 2x^2 - 4x + 1 blech...
Reply
Ms Maths
6/21/2014 12:18:02 am
Fractals are so beautiful. Show your students animations of the Mandelbrot Set on youtube. Show them the Mandelbox Trip as well. Applications are in analysing extreme weather and research on the brain. I always try to find real life applications to what I teach. Also, hell, why can't we just create Maths for fun? That is what Mandelbrot did and only now we are seeing applications of his work.