Roots of Complex Numbers
Drag the locator, which represents the complex number . The gray dots represent the solutions of the equation . As you drag notice these roots are always the vertices of a regular polygon. You can explore the powers of for each of the choices of .
Snapshot 1: As long as , there will always be different complex numbers that satisfy the equation . All the roots have the same magnitude and lie on the circle of radius . Shown here are the five fifth roots of . Holding the Alt key down refined the movement of the locator and allowed us to place the locator at exactly .
Snapshot 2: One of the roots is always easily found. Its argument is the argument of divided by . So in this example we divide the argument of by 5.
Snapshot 3: The five roots form the vertices of a regular pentagon.
Snapshot 4: These observations are true for any . Notice the five roots still form a regular pentagon, and one of the root's argument is 1/5 the argument of .
Snapshot 5: But that's the easy part. It is not so obvious imagining how the powers of each root expand and rotate to coincide with , because they lie on a spiral that sometimes overlaps itself. Here we tie the five powers of together as an "orbit" which is displayed as a set of orange lines and a spiral.
Snapshot 6: Here the arguments are shown as overlapping sectors. Since has an argument of 300 degrees, the root with the smallest argument has an argument that is degrees. Because each vertex of a regular pentagon is separated by 72 degrees from its neighbor, the has an argument equal to degrees. This is shown as a yellow sector. The green, blue, purple, and red sectors are also each degrees and track the unfolding powers of .
Snapshot 7: When is a positive real number without any imaginary component, it will have one root with no imaginary component whose powers all lie on the real number line.
Snapshot 8: -1 can now be easily seen as a real number with an argument equal to 180 degrees. So its square root, , must have arguments equal to degrees and degrees.
Snapshot 9: The unit circle is drawn to emphasize that all roots outside the unit circle expand during their orbit to , while all roots inside the unit circle shrink during their orbit, and roots exactly on the unit circle just bounce around on the unit circle.
Snapshot 10: The roots on the unit circle that are solutions to have a special place in number theory.
Snapshot 11: We stopped at 17th roots in honor of Gauss, who in 1796 at the age of 19 proved that the 17-sided heptadecagon was constructible by ruler and compass alone.
See also Michael Schreiber's Demonstration, "Powers Of Complex Numbers."