Monthly Archives: August 2016

Roots of Complex Numbers without DeMoivre

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.

TRADITIONAL APPROACH:

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  z^2=-16.  The Fundamental Theorem of Algebra guarantees two solutions to this quadratic equation.

The complex Cartesian number, -16+0i, converts to polar form, 16cis( \pi ), where cis(\theta ) = cos( \theta ) +i*sin( \theta ).  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

-16 = 16cis( \pi ) = 16 cis \left( \pi + 2 \pi n \right)

Invoking DeMoivre’s Theorem,

\sqrt{-16} = (-16)^{1/2} = \left( 16 cis \left( \pi + 2 \pi n \right) \right) ^{1/2}
= 16^{1/2} * cis \left( \frac{1}{2} \left( \pi + 2 \pi n \right) \right)
= 4 * cis \left( \frac{ \pi }{2} + \pi * n \right)

For n= \{ 0, 1 \} , this gives polar solutions, 4cis \left( \frac{ \pi }{2} \right) and 4cis \left( \frac{ 3 \pi }{2} \right) .  Each can be converted back to Cartesian form, giving the two square roots of -16:   4i and -4i .  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 a+bi form, new possibilities open.  The original example can be rephrased:

Determine the simultaneous real values of x and y for which -16=(x+yi)^2.

Start by expanding and simplifying the right side back into a+bi form.  (I wrote about a potentially easier approach to simplifying powers of i in my last post.)

-16+0i = \left( x+yi \right)^2 = x^2 +2xyi+y^2 i^2=(x^2-y^2)+(2xy)i

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:

demoivre5

Solving the system gives the square roots of -16.

From the latter equation, either x=0 or y=0.  Substituting y=0 into the first equation gives -16=x^2, an impossible equation because x & y are both real numbers, as stated above.

Substituting x=0 into the first equation gives -16=-y^2, leading to y= \pm 4.  So, x=0 and y=-4 -OR- x=0 and y=4 are the only solutions–x+yi=0-4i and x+yi=0+4i–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.

CAS APPROACH:

Determine all fourth roots of 1+2i.

That’s equivalent to finding all simultaneous x and y values that satisfy 1+2i=(x+yi)^4.  Expanding the right side is quickly accomplished on a CAS.  From my TI-Nspire CAS:

demoivre1

Notice that the output is simplified to a+bi form that, in the context of this particular example, gives the system of equations,

demoivre6

Using my CAS to solve the system,

demoivre2

First, note there are four solutions, as expected.  Rewriting the approximated numerical output gives the four complex fourth roots of 1+2i-1.176-0.334i-0.334+1.176i0.334-1.176i, and 1.176+0.334i.  Each can be quickly confirmed on the CAS:

demoivre3

CONCLUSION:

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 (x+yi)^n where n is the given root, simplifying the expansion into a+bi 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.

demoivre4

Powers of i

I was discussing integer powers of i in my summer Algebra 2 last month and started with the “standard” modulus-4 pattern I learned as a student and have always taught.  While not particularly insightful, my students and I considered another approach that might prove simpler for some.

TRADITIONAL APPROACH:

I began with the obvious i^0 and i^1 before invoking the definition of i to get i^2.  From these three you can see every time the power of i increases by 1, you multiply the result by i and simplify the result if possible using these first 3 terms.  The result of i^3 is simple,  taking the known results to

i_1

But i^4=-i^2=-(-1)=1, cycling back to the value initially found with i^0.  Continuing this procedure creates a modulus-4 pattern:

i_2

They noticed that to any multiple of 4 was 1, and other powers were i, -1, or –i, depending on how far removed they were from a multiple of 4.  For an algorithm to compute a simplified form of to an integer power, divide the power by 4, and raise i to the remainder (0, 1, 2, or 3) from that division.

They got the pattern and were ready to move on when one student who had glimpsed this in a math competition at some point noted he could “do it”, but it seemed to him that memorizing the list of 4 base powers was a necessary requirement to invoking the pattern.

Then recalled a comment I made on the first day of class.  I value memorizing as little mathematics as possible and using the mathematics we do know as widely as possible.  His challenge was clear:  Wasn’t asking students to use this 4-cycle approach just a memorization task in disguise?  If I believed in my non-memorization claim, shouldn’t there be another way to achieve our results using nothing more the definition of i?

A POTENTIAL IMPROVEMENT:

By definition, i = \sqrt{-1}, so it’s a very small logical stretch with inverse operations to claim i^2=-1.

Even Powers:  After trying some different examples, one student had an easy way to handle even powers.  For example, if n=148, she invoked an exponent rule “in reverse” to extract an i^2 term which she turned into a -1.  Because -1 to any integer power is either 1 or -1, she used the properties of negative numbers to odd and even powers to determine the sign of her answer.

i_3

Because any even power can always be written as the product of 2 and another number, this gave an easy way to handle half of all cases using nothing more than the definition of i and exponents of -1.

A third student pointed out another efficiency.  Because the final result depended only on whether the integer multiplied by 2 was even or odd, only the last two digits of n were even relevant.  That pattern also exists in the 4-cycle approach, but it felt more natural here.

Odd Powers:  Even powers were so simple, they were initially frustrated that odd powers didn’t seem to be, too.  Then the student who’d issued the memorization challenge said that any odd power of i was just the product of i and an even power of i.  Invoking the efficiency in the last paragraph for n=567, he found

i_4

CONCLUSION:

In the end, powers of i had become nothing more complicated than exponent properties and powers of -1.  The students seemed to have greater comfort with finding powers of complex numbers, but I have begun to question why algebra courses have placed so much emphasis on powers of i.

From one perspective, a surprising property of complex numbers for many students is that any operation on complex numbers creates another complex number.  While they are told that complex numbers are a closed set, to see complex numbers simplify so conveniently surprises many.

Another cool aspect of complex number operations is the stretch-and-rotate graphical property of complex number multiplication.   This is the basis of DeMoivre’s Theorem and explains why there are exactly 4 results when you repeatedly multiply any complex number by i–equivalent to stretching by a factor of 1 and rotating \frac{\pi}{2}.  Multiplying by 1 doesn’t change the magnitude of a number, and after 4 rotations of \frac{\pi}{2}, you are back at the original number.

So, depending on the future goals or needs of your students, there is certainly a reason to explore the 4-cycle nature of repeated multiplication by i.  If the point is just to compute a result, perhaps the 4-cycle approach is unnecessarily “complex”, and the odd/even powers of -1 is less computationally intense.  In the end, maybe it’s all about number sense.

My students discovered a more basic algorithm, but I’m more uncomfortable.  Just because we can ask our students a question doesn’t mean we should.  I can see connections from my longer studies, but do they see or care?  In this case, should they?