Complex nth roots - Department of Mathematics and Statistics 复杂的n次方的数学与统计学系.pdfVIP

Math 421 November 2009 Complex nth roots Copyright © 2004–2009 by Murray Eisenberg. All rights reserved. Prerequisites Mathematica Most of this notebook requires David Park Mathematica add-on application Presen- tations. If this is not already available to you, see Parks Mathematica page for how to obtain it. In order to reproduce all the graphics output from this notebook, you must have installed that application and then initialized it by evaluating, for example, the expres- sion: Needs[Presentations`Master`] That initialization is done below when it is first needed. Mathematics You should already know the basics of the algebra of complex numbers—how to add and multiply them—and the representation of complex numbers by points in the plane. 2 nthRoots.nb The nth roots of a complex number For a positive integer n=1, 2, 3, …, a complex number w 0 has n different com- plex roots z . That is, for a given w 0, the equation zn w has n different solutions z . This is the case, in particular, when w 1. In this case, the n different values of z are called the nth roots of unity. This notebook shows how to use Mathematica to calculate such roots as well as how to visualize them geometrically. It also includes material about expressing complex roots of unity in polar form. The cube roots of unity For an example, work with the cube roots of unity. By definition, a cube root of unity is a solution of the equation z3 1. Surely Mathematica can solve this equation directly. Try it: 3 In[1]:= Solvez 1, z Out[1]= z 1, z 113, z 123 Thats not useful! What are the solutions, really? Use ComplexExpand on the result of Solve: 3 In[2]:= cubeRootsUnity ComplexExpand z .Solvez 1, z 1