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*Dear Matt,*

The topic of complex numbers is included in my Algebra II course and the text book just defines *i* to be the square root of negative one, and then goes on to develop the various rules for computation with these new numbers. It seems to me that there should be some better way to motivate the existence of complex numbers so that the definition and the computation make more sense. Any ideas?

*Struggling with complexities*

*Response: *

*Dear Struggling*:

First of all, it is important to remember that the way your students learn is by **doing** and by **thinking**. So the challenge is to develop activities for them to do both. Yours is also an interesting question in terms of Minnesota standards. It points up the difference between the content included in typical textbooks, and the content specified in Minnesota standards. There is only one Standard and two Benchmarks that refer to complex numbers. The statement of these gives some clues to a possible approach.

MN STANDARD 9.2.4: Represent real-world and mathematical situations using equations and inequalities involving linear, quadratic, exponential and nth root functions. Solve equations and inequalities symbolically and graphically. Interpret solutions in the original context

Benchmark 9.2.4.1: Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.

Benchmark 9.2.4.3: Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers, and from real numbers to complex numbers. In particular, non-real complex numbers are needed to solve some quadratic equations with real coefficients.

In other words, the context suggested by the Minnesota standards is twofold: to focus on finding solutions to all quadratic equations, and extending the number system to include complex numbers. These are the only aspects of complex numbers that are required. They arise naturally when you teach the Quadratic Formula. Let’s back up, though, to see how this idea fits into the whole of the study of number.

One of the most fundamental properties of numbers is related to taking a set of numbers, and an operation, and checking to see if the result is part of the original set. If the answer is in the set, the set has **closure** for that operation. For example, the set of Whole Numbers has closure for addition and multiplication, since any sum or product of any two whole numbers is also a whole number. But to have closure for subtraction, it is necessary to expand the set to the Integers, since some differences are less than zero. That is, the answer when you subtract any two whole numbers is not always a whole number, for example 5 – 9 = -4. For division, we require the set of Rational Numbers, since many quotients are fractions. If the operation is taking roots of Whole Numbers, we need to expand our original set to the Real Numbers, since many roots are not rational numbers. If the operation is taking roots of Integers, we need to develop the set of Imaginary Numbers, which include even roots of negatives. Finally, when numbers include both real and imaginary parts, we have the set of Complex Numbers.

Another approach suggested in Benchmark 9.2.4.1 is to see the closure occurring in efforts to solve polynomials. All polynomials with real coefficients are solvable using the set of Complex Numbers. The place in the curriculum where this arises most naturally is related to the use of the Quadratic Formula, where it is evident that not all solutions produced by the formula represent real numbers. In addition, not all solutions may be applicable for a given problem context (see Standard above.) Here are some ideas for activities. Many lend themselves to individual student investigations and presentations.

- Students could explore the history of the development of Complex Numbers. Mathematicians to study might include Heron, Cardano, Tartaglia, Bombelli, DesCartes, Hamilton, Euler, Gauss, and/or DeMoivre.
- They might have a quick “spell-off” for finding powers of i, which contains some nice cyclic patterns. You could begin with i, i
^{2}, i^{3}, i^{4}, i^{5}, i^{6}, and then start skipping around with powers of i as students begin to see the pattern of repetition. - They could struggle in pairs or groups with how to represent a complex number graphically. Some might arrive at Argand’s method, some may find other interesting approaches.
- They could investigate uses for complex numbers, applications that would not have been possible without them. (For example, much of electricity!) Advanced or highly motivated students might be interested in exploring additional (optional) topics such as moduli and conjugates of complex numbers. Students might think about the naming for number sets: Natural, Whole, Integer, Rational, Irrational, Real, Imaginary, Complex. Are any of these numbers any more imaginary than others? (Did you ever meet a 2 walking down the street?)
- Another topic that will be of interest to some students, depending on their prior studies, would be the similarities, relationships, and isomorphisms related to complex numbers, vectors, and trigonometry. Others might explore the group properties, similarities and differences for fields of real and complex numbers.
- Euler is worth an entire exploration of his own, extending beyond complex numbers. He is responsible for much of the notation used daily in the mathematics classroom, including e, the base of the natural logarithm (e for Euler), π, omnipresent in trigonometry, geometry and measurement, and i to represent √-1. Furthermore, he is responsible for the celebrated equation

e^{πi}+ 1 = 0

which unites the five most fundamental constants in an astonishingly simple identity.

Though much of this may seem theoretical, you should be able to generate lessons that require students to do some exploration and thinking, making connections with history, other parts of mathematics, and applications of mathematics through these ideas.

NOTE: Currently, in early 2012, Minnesota has not adopted the Common Core State Standards for Mathematics, used by many other states. These standards call for more computation and operations with complex numbers. If at some point in the future, Minnesota decides to adopt those standards, additional content, perhaps similar to that in your textbook, will be needed. For now, you may deemphasize or omit parts of the unit on complex numbers from your textbook that are not related to Minnesota Standards.

Happy moving, talking and thinking!

Matt