# Complex Rotations

This is an activity where students repeatedly multiply a complex number by i and graph each product. There’s a version of it here, from the NRICH site:

The complex number a+ib is represented in the plane by the point with coordinates (a,b). This is called an Argand diagram. Make your own choice of some complex numbers, and mark them on a graph with lines joining the points to the origin. Now multiply your numbers by −1and join their images to the origin. Make and prove a conjecture about the geometric effect of multiplying complex numbers by −1.

Again make a choice of some complex numbers and multiply each one by i. Draw the complex numbers and their images on a graph and make and prove a conjecture about the effect of multiplying complex numbers by i.

What happens if you multiply a complex number by i twice, three times, four times, …, ntimes?

Here’s a version of this problem that I made in 2011:

A note about how I did this. On pages 3 and 4 I ask students to divide by a complex number, but at this stage they don’t really know how to divide by a complex number in a way that returns something in a + bi form. That was on purpose, because I wanted to use the need to graph a quotient to motivate dividing in a way that returns something in a + bi form.* But at this stage kids did know how to rationalize the denominator, so I dropped that hint for students during class.

*The other option, in my mind, is to try to get kids to understand why mathematicians care about closure under an operation. That could be cool, but it’s not part of my approach right now.