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 *n*th power will have argument *n*90°, and those angles will repeat in a period of length 4 since 4*·*90° = 360°, a full circle.

More generally, you can find *z*^{n} as the complex number (1) whose absolute value is |*z*|^{n}, the *n*^{th} 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.

There are, in fact, six sixth roots of any complex number. Let *w* be a complex number, and *z* any of its sixth roots. Since *z*^{6} = *w,* it follows that

- the absolute value of
*w*, |*w*| is |*z*|^{6}, so |*z*| = |*w*|^{1/6}, and - 6 arg(
*z*) is arg(*w*), so arg(*z*)=arg(*w*)/6.

For example, take *w* to be -1/2, the green dot in the figure to the right. Then |*w*| is 1/2, and arg(*w*) is 180°. Let *z* be a sixth root of *w.* Then (1) |*z*| is |*w*|^{1/6} which is about 0.89. Also, (2) the argument of *w* is arg(*w*) = 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 *z*:

Since each of the angles for *z* differs by 360°, therefore each of the possible angles for *z* will differ by 60°. These six sixth roots of -1/2 are displayed in the figure as blue dots.

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.