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August 10, 2007

A great example

What good are imaginary numbers to a 7th grader?

DRDR gives a great answer:

In answering this question in the simplest way, I don't think it's enough to say "complex numbers are important because they're sometimes solutions to equations with real coefficients." We need to find an example where the complex solutions yield real results.

The simplest college-level example is probably the solution to the second-order differential equation that governs spring oscillation, but 7th-graders don't know differential eqs. Difference equations, however, might be more understandable to a seventh-grader.

For instance, consider the recursive system a(t+2) = a(t+1) + a(t), with a(0)=a(1)=1. This is just the Fibonacci sequence, which I think most grade-school kids can understand or might have been exposed to (I first saw it in third grade at a public school).
Now supposed you want evaluate a(1000) without having to calculate a(999),...,a(1). You can derive a closed-form non-recursive solution to this system of the form a(t) = A*r1^t+B*r2^t where r1 and r2 are the irrational roots of the equation r^2=r+1, and A and B are determined by the initial conditions. What's beautiful about this example is that though the equation involves irrational numbers, the solution yields integer results for integer initial conditions.

From there, it's not hard to leap to the example of the difference equation a(t+2) = a(t+1)-a(t) which exhibits oscillitory behavior, and it's not too hard to think of a practical application for this equation. Here the solution is the same form as above, except the roots are now complex. Here even though the closed form solution involves all sorts of irrational and complex numbers, you still get integer results when you evaluate the equation given integer initial conditions. I think this is about the simplest application of complex roots leading to real results that you can get.

Posted by OneEyedMan at August 10, 2007 5:04 PM

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