Complex Analysis, Video #23 (Complex Arithmetic, Part 23).
Main Topic: Finding the nth roots of a complex number and visualizing them as a regular polygon using Mathematica
Review that we found the 12th roots of unity in the last video and visualized them with ListPlot. Add lines between the vertices in this video to create a regular dodecagon (12-sided regular polygon). Discuss how to define the general nth roots of an arbitrary complex number, given in polar form (n is a positive integer). Write a proposed nth root in polar form and consider what the possibilities for the modulus and argument would be. Use the radical notation to indicate non-negative real nth roots of non-negative real numbers. The ambiguity in the argument of the original number gives n possibilities for the nth root...every nonzero complex number has exactly n nth roots. The nth root function is therefore multi-valued. Finish the video by making an animation using Manipulate and Locator to graph the nth roots of an arbitrary complex number (and Locator will allow us to move the starting point around)...initially accidentally raise the modulus to the 12th power, but then I catch the mistake and fix it (should raise it to the 1/12th power).