The other day I was investigating a problem involving complex numbers when I was hit by a flash of insight. Here is the problem as it was originally stated:

If is a complex number such that , compute:

Initially, the problem seemed very dry. My first solution involved multiplying out a bunch of terms, combining like terms, and arriving at an answer of 4. However, as I was drawing a diagram of the problem, something amazing happened.

I realized that the expression could be interpreted as the distance from the point to the number . But the constraint that , meant had to lie on a circle of radius 1 centered at the origin.

Further, the expression could be interpreted as the distance from the point to the number , which would be directly opposite the original .

From here, all I had to do was connect the dots and observe the solution. I had formed a right-triangle! If you want to play around with the diagram, feel free to click on the interactive link: https://ggbm.at/bJucvN5B

Hence, the expression: , could be rewritten as:

And using Pythagoras’s theorem, we get:

Now the hypotenuse is the diameter of the circle. Since the circle has radius 1, the diameter is 2, and:

The question ultimately boiled down to using Pythagoras’s theorem. I continue to be amazed at how many times Pythagoras’ theorem is used. A simple relationship, taught to students in grade 7 or 8, continues to be relevant when evaluating complex valued expressions. Remarkable.