and so forth. The reasons were that (1) the absolute value |i| of i was one, so all its powers also have absolute value 1 and, therefore, lie on the unit circle, and (2) the argument arg(i) of i was 90°, so its nth power will have argument n90°, and those angles will repeat in a period of length 4 since 4·90° = 360°, a full circle.
More generally, you can find zn as the complex number (1) whose absolute value is |z|n, the nth power of the absolute value of z, and (2) whose argument is n times the argument of z.
In the figure you see a complex number z whose absolute value is about the sixth root of 1/2, that is, |z| = 0.89, and whose argument is 30°. Here, the unit circle is shaded black while outside the unit circle is gray, so z is in the black region. Since |z| is less than one, it's square is at 60° and closer to 0. Each higher power is 30° further along and even closer to 0. The first six powers are displayed, as you can see, as points on a spiral. This spiral is called a geometric or exponential sprial.
Roots. Note that in the last example, z6 is on the negative real axis at about -1/2. That means that z is just about equal to one of the sixth roots of -1/2. There are, in fact, six sixth roots of any complex number. Let z be a complex number, and w any of its sixth roots. Since z6 = w, it follows that (1) the absolute value of |z| is |w|6, and (2) arg(z) is 6 arg(w). Actually, the second statement isn't quite right since 6 arg(w) could be any multiple of 360° more than arg(z).
For example, take z to be -1/2, the green dot in the figure to the right. Then |z| is 1/2, and arg(z) is 180°. Let w be a sixth root of z. Then (1) |w| is |z|1/6 which is about 0.89. Also, (2) the argument of z is arg(z) = 180°. But the same angle could be named by any of
If we take 1/6 of each of these angles, then we'll have the possible arguments for w:
Since each of the angles for z differs by 360°, therefore each of the possible angles for w will differ by 60°. These six sixth roots of -1/2 are displayed in the figure as blue dots.
More roots of unity. Recall that an "nth root of unity" is just another name for an nth root of one. The fourth roots are ±1, ±i, as noted earlier in the section on absolute value. We also saw that the eight 8th roots of unity when we looked at multiplication were ±1, ±i, and ±√2/2 ± i√2/2.
Let's consider now the sixth roots of unity. They will be placed around the circle at 60° intervals. Two of them, of course, are ±1. Let w be the one with argument 60°. The triangle with vertices at 0, 1, and w is an equilateral triangle, so it is easy to determine the coordinates of w. The x-coordinate is 1/2, and the y-coordinate is √3/2. Therefore, w is (1 + i√3)/2. The remaining sixth roots are reflections of w in the real and imaginary axes. In summary, the six sixth roots of unity are ±1, and (±1 ± i√3)/2 (where + and can be taken in any order).
Now some of these sixth roots are lower roots of unity as well. The number 1 is a square root of unity, (1 ± i√3)/2 are cube roots of unity, and 1 itself counts as a cube root, a square root, and a "first" root (anything is a first root of itself). But the remaining two sixth roots, namely, (1 ± i√3)/2, are sixth roots, but not any lower roots of unity. Such roots are called primitive, so (1 ± i√3)/2 are the two primitive sixth roots of unity.
It's fun to find roots of unity, but we've found most of the easy ones already.
Previoius section: Reciprocals, conjugation, and division
Table of Contents
David E. Joyce
Department of Mathematics and Computer Science
Worcester, MA 01610
These pages are located at http://www.clarku.edu/~djoyce/complex/