When students find the nth roots of a complex number, they use de Moivre’s Theorem and a fair bit of calculation and trigonometry. In this blog post, I’m going to approach the topic from a more visual perspective and make use of the following geometric way to think about complex number multiplication: To multiply two complex numbers, multiply their lengths and add their arguments (the angle they make with the positive real axis).

In the Web Sketchpad model below (and here), drag point *z* around the unit circle. As you do, notice that its location is given in both rectangular and polar form. Point *z*^{2 }has the same length as *z*, but its argument is twice that of *z*. Thus when the argument of *z* is 45°, *z*^{2} is at *i, *making *z* = cos(45°) + *i* sin(45°) one solution to *z*^{2} = *i*. To leave a record of this solution, press *Mark Location of z*.

If you continue dragging point z counterclockwise around the unit circle, *z*^{2 }moves twice as fast as *z* since its argument is twice as large. Thus when *z* travels 180° from 45° to 225°, *z*^{2 }has moved 360°, back to its same location at *i*. This gives us the second solution to *z*^{2} = *i. *Press* Mark Location of z *again to leave a record of this additional solution.

To solve *z*^{3}= *i*, *z*^{5}= *i*, *z*^{6}= *i*, *z*^{10}= *i*, and *z*^{15}= *i*, use the page navigation arrows in the bottom-right corner of the websketch to move from problem to problem. For *z*^{6}= *i, *notice* *that when* z *rotates 60*°, **z*^{6 }rotates* *360*°, *explaining why the solutions to *z*^{6}*= i *form the vertices of a regular hexagon*. *Below are the patterns formed for different powers of* z, *and for each, you can apply similar reasoning to make sense of the spacing of the points on the unit circle.