The Road to Reality: A Complete Guide to the Laws of the Universe

by Roger Penrose

When z = n is a positive integer, things are much simpler, and we get just n roots. A situation of particular interest occurs, in this case, when w = 1. Then, specifying some possible values of log 1 successively, namely 0, 2πi, 4πi, 6πi, . . . , we get 1 = e0, e2πi/n, e4πi/n, e6πi/n, . . . for the possible values of 11/n. We can write these as 1, є, є2, є3, . . . , where є = e2πi/n. In terms of the complex plane, we get n points equally spaced around the unit circle, called nth roots of unity. These points constitute the vertices of a regular n-gon (see Fig. 5.10). (Note that the choices, −2πi, − 4πi, − 6πi, etc., for log 1 would merely yield the same nth roots, in the reverse order.)