Finding roots of complex numbers can be … complex.
This post describes a way to compute roots of any number–real or complex–via systems of equations without any conversions to polar form or use of DeMoivre’s Theorem. Following a “traditional approach,” one non-technology example is followed by a CAS simplification of the process.
Most sources describe the following procedure to compute the roots of complex numbers (obviously including the real number subset).
- Write the complex number whose root is sought in generic polar form. If necessary, convert from Cartesian form.
- Invoke DeMoivre’s Theorem to get the polar form of all of the roots.
- If necessary, convert the numbers from polar form back to Cartesian.
As a very quick example,
Compute all square roots of -16.
Rephrased, this asks for all complex numbers, z, that satisfy . The Fundamental Theorem of Algebra guarantees two solutions to this quadratic equation.
The complex Cartesian number, , converts to polar form, , where . Unlike Cartesian form, polar representations of numbers are not unique, so any full rotation from the initial representation would be coincident, and therefore equivalent if converted to Cartesian. For any integer n, this means
Invoking DeMoivre’s Theorem,
For , this gives polar solutions, and . Each can be converted back to Cartesian form, giving the two square roots of -16: and . Squaring either gives -16, confirming the result.
I’ve always found the rotational symmetry of the complex roots of any number beautiful, particularly for higher order roots. This symmetry is perfectly captured by DeMoivre’s Theorem, but there is arguably a simpler way to compute them.
NEW(?) NON-TECH APPROACH:
Because the solution to every complex number computation can be written in form, new possibilities open. The original example can be rephrased:
Determine the simultaneous real values of x and y for which .
Start by expanding and simplifying the right side back into form. (I wrote about a potentially easier approach to simplifying powers of i in my last post.)
Notice that the two ends of the previous line are two different expressions for the same complex number(s). Therefore, equating the real and imaginary coefficients gives a system of equations:
Solving the system gives the square roots of -16.
From the latter equation, either or . Substituting into the first equation gives , an impossible equation because x & y are both real numbers, as stated above.
Substituting into the first equation gives , leading to . So, and -OR- and are the only solutions– and –the same solutions found earlier, but this time without using polar form or DeMoivre! Notice, too, that the presence of TWO solutions emerged naturally.
Higher order roots could lead to much more complicated systems of equations, but a CAS can solve that problem.
Determine all fourth roots of .
That’s equivalent to finding all simultaneous x and y values that satisfy . Expanding the right side is quickly accomplished on a CAS. From my TI-Nspire CAS:
Notice that the output is simplified to form that, in the context of this particular example, gives the system of equations,
Using my CAS to solve the system,
First, note there are four solutions, as expected. Rewriting the approximated numerical output gives the four complex fourth roots of : , , , and . Each can be quickly confirmed on the CAS:
Given proper technology, finding the multiple roots of a complex number need not invoke polar representations or DeMoivre’s Theorem. It really is as “simple” as expanding where n is the given root, simplifying the expansion into form, and solving the resulting 2×2 system of equations.
At the point when such problems would be introduced to students, their algebraic awareness should be such that using a CAS to do all the algebraic heavy lifting is entirely appropriate.
As one final glimpse at the beauty of complex roots, I entered the two equations from the last system into Desmos to take advantage of its very good implicit graphing capabilities. You can see the four intersections corresponding to the four solutions of the system. Solutions to systems of implicit equations are notoriously difficult to compute, so I wasn’t surprised when Desmos didn’t compute the coordinates of the points of intersection, even though the graph was pretty and surprisingly quick to generate.