Discovering a Complex Relationship
A discussion of complex roots of a parabolas leads to a connection between the modulus of the solution and the vertex of the parabola.
Consider the following problem: Find all roots of the function y = (x-3)^2 + 4.
Now, most of us will know this as the equation of a parabola in the xy-plane; one whose vertex is at the point (3,4).
And most of us would be happy noting that this equation does not have any roots over the real numbers. For those of you who want to travel down the rabbit hole of complex numbers though, let’s take a walk.
Assume that we now have the equation y = (z-3)^2 +4, where z = a + bi is permitted to be a complex number. We can do a bit of algebra to find the two complex roots of this equation:
Now, one of my clever calculus students was trying to make the connection between distance and the modulus of a complex number. He was trying to connect the modulus of these particular solutions to the distance from the origin (0,0) to the vertex of the parabola (3,4). He wrote down |z| < 5? and |z| > 3?
Notice that he was thinking about the right triangle formed below, where the 5 is the hypotenuse and the 3 and 4 are the legs:
I believe he was on this track because I had mentioned that the modulus formula for
z = a + bi is given by |z| = sqrt(a^2 + b^2). So it makes sense that he was thinking about the distance to the origin here (just not on the correct plane). After a bit more discussion, he was still adamant about 3 < |z| < 5, which is certainly true for this particular example, since |z| = sqrt(3^2 + 2^2) = sqrt(13).
Then I stopped and thought about this. I found it weird that the modulus of the solution was greater than the length of the smaller leg, yet smaller than the length of the hypotenuse. I dove in to see if it would work in general.
Assume y = (x – a)^2 + b is a parabola in the xy-plane that lies above the x-axis with b > 1. Extend this parabola naturally over the complex numbers and find its roots:
Consider the modulus of these complex roots:
Now, since b > 1, we can see that
and this is fascinating to me because it tells me that the complex modulus of our roots will lie on some kind of ring in the complex plane! In fact, we know a < |z| < c, where a is the y-coordinate of the vertex of the parabola and c is the distance of the vertex to the origin (see the blue triangle and parabola given above). In the Re/Im plane we would get a region that looks like this:
All in all we didn’t get too far discussing the complex modulus, but it was definitely still a bull’s-eye in my books.