Approaches to introducing complex numbers

1. “Some quadratic equations don’t have solutions. But imagine that they did. Then we would need complex numbers. Now, here are some complex numbers.” (See Discovering Advanced Algebra, from Key Press)

2. “We need something that can be the square root of a negative number. So we need to invent a new kind of number for that. Now, here are some complex numbers.” (See A Whole New Kind of Number, from Kate)

3. “Historically, people needed complex numbers to come up with something like the Quadratic Formula for Cubic equations. That’s why we first cared about complex numbers. Now, here are some complex numbers.” (Source?)

4. [This one is a bit complicated. Hold on tight.] “In a monic quadratic equation $x^2 + bx + c$ the sum of the roots is $b$ and the product is $c$. Convince yourself that $x^2 + y^2 = (x + y)^2 - 2xy$. Now, if I tell you that $a + b = 10$ and $ab=29$ then you can use the identity to find the value of $a^2 + b^2$, and you get 42. But you could also pass to a quadratic equation and find the roots $a, b$ directly, using the quadratic formula or something. But, wait, now we’ll end up with square roots of negative numbers appearing. Can we still get the answer we got from the first method, 42?” (CME Project)

2 thoughts on “Approaches to introducing complex numbers”

1. I first saw these approaches, I think, at http://betterexplained.com/

1. Compare Complex Numbers with negative numbers. Before the 1700s, or some ancient time, negative numbers didn’t exist. But they were invented to answer a question….

2. Look at the representation on the complex plane. Multiplying by i rotates a vector 90 degrees counterclockwise. Multiplying by i again gives you a total of 180 degrees which is the same as multiplying by -1. And adding two complex numbers gives you the same result as vector addition.

• Hey Damon,

I like both of these points. Your first point, about the comparison between negatives and complex numbers, fits well with approach 2 above. During my first year I had a discussion with my students about the historical resistance to treating the irrationals, negatives, and complex numbers as legitimate numbers. (The discussion flopped, for the record.) I also really like your second suggestion, and somewhere on this blog I have an activity posted that deals with complex rotations.

Thanks so much, Damon!