Complex numbers don’t make any sense. How can such weird numbers have any real use? The term “imaginary part” suggests that complex numbers are fake, cooked up by a bunch of crackpot mathematicians. That’s what I thought when I was in high school. But since my high school days, I’ve learned to use complex numbers to solve AC circuit problems, to do 2D vector math, to really appreciate the fundamental theorem of algebra, and to explore the famous Mandelbrot set. Complex numbers are now an essential tool in almost every branch of mathematics, science, and engineering.
In previous posts, I discussed how teachers can help students better understand and use the quadratic formula. But in order to have a complete understanding of the quadratic formula, it’s necessary to have a basic understanding of complex numbers.
I begin my introduction to complex numbers by asking my students to imagine that they are 3rd grade students who know the basic whole number addition and multiplication facts. I then have them consider how they, a 3rd grade student, would answer the six questions below.
After some discussion, my students agree that a 3rd grader would correctly answer questions 1, 2, and 5, but would not be able to answer questions 3, 4, and 6, because they don’t know about negative numbers and fractions. When those 3rd graders grow older and learn about fractions and negative numbers, they will be able to answer questions 3 and 4 correctly.
My students, not the 3rd graders, can correctly answer questions 1 – 5 but can’t correctly answer question 6, because they don’t know about the strange complex number i where i = √(-1) and i2 = -1. I explain that 7i * 7i = 49i2 = 49(-1) = -49. I tell students that all numbers after the counting numbers (1, 2, 3, . . .) are inventions of the human intellect and were invented to solve specific types of equations. It has been said, “God gave man the counting numbers, and man invented all the other numbers.”
In the next part of the lesson, I develop a list of the powers of the complex number i. The list of powers and the graph below enable students to easily see the circular pattern in the powers of i. (Note: i3 = i2 *i = (-1)i = -i and i4 = i2 * i2 = (-1)(-1) = 1)
After they learn about the powers of the complex number i, I show students how to plot a complex number and how to graph a complex number as a vector because all complex numbers have a magnitude and direction. Initially, students find it strange that complex numbers don’t have a negative property like some real numbers. Example: If the complex number z = 6 – 12i, then –z = -6 + 12i. I tell students that they should say, “the opposite of z,” for the symbol –z. The graph below shows complex number -9 + 6i and its conjugate -9 – 6i graphed as vectors. The other complex numbers in the graph below are graphed as a single point. Of course, 0 + 8i = 8i.
If time allows after the main lesson, I show students some interesting geometric patterns generated by the powers and roots of complex numbers. Students will learn how these pattern come about when they study De Moivre’s Theorem in a later course. It’s fun to make conjectures about the patterns. The left graph shows z, z2, z3, . . . , z20 where z = 1.15Cos(350) + 1.15Sin(350)i. The right graph shows the 12 12th roots of -4,096.
You can download the student and teacher versions of the free handout Introduction to Complex Numbers from www.mathteachersresource.com/instructional-content.html. This handout has two pages of exercises and student activities that I use to introduce my students to complex numbers. We usually work about a third of the problems together and the remaining exercises are left as homework. To make your presentations more dynamic, project graphs on a screen and use simple mouse control clicks to plot points and draw vectors.
Teaching Points: (Of course, teachers can modify the lesson to meet the needs of their class.)
- Read and study the free handout Introduction to Complex Numbers. As the lesson progresses, students should be taking notes and writing on a teacher provided student version of the handout.
- Some of the exercises involve calculating the absolute value of a complex number. Remind students that the absolute value of any number equals the positive distance of the number from zero, and therefore the theorem of Pythagoras can be used to calculate the absolute of a complex number. The absolute value of any nonzero number is always a positive real number, and i is never used to describe the absolute value of a complex number.
- Point out the geometric relationship between a complex number and its conjugate. After doing the exercises in the handout, many students see a way to use the conjugate to calculate the absolute value of a complex number.
- The handout Introduction to Complex Numbers covers all of the basic types of complex number arithmetic problems that an advanced algebra, trig, or precalculus student would be expected to handle. When appropriate, the polar form of a complex number can be explained at a later time.
- A geometric understanding of complex numbers is very important. Graphing complex numbers makes complex numbers more real to students. On homework and tests, have students graph various complex number expressions. Example: Let z = -8 + 4i. Graph and label each of the following as a vector: z, -z, 1.5z, -0.5z, and the conjugate of z.
- If time allows, show students interesting geometric patterns generated by the powers and roots of a complex number. It is interesting to see what pattern observations that students come up with. Tell students that they will learn the details of how these patterns come about in a later course. In most elementary math courses, students are never exposed to the really cool and interesting aspects of mathematics.
- Some students will claim that they can use their graphing calculators to get the answer in a matter of seconds. They are right. Remind them that they will not be allowed to use their graphing calculator on a test or quiz until they have demonstrated that they can do basic complex number arithmetic.
The above graphics were created with the program, Basic Trig Functions, which is offered by Math Teacher’s Resource. In addition to graphing x-y variable relations and polar functions, users can graph the powers or roots of a complex number, and view a list of the powers or roots which appears to the right of the graphic output. Segments or vectors can be drawn by left-clicking and dragging the mouse. The Edit/ Edit Graphics menu provides options for setting segment color, pen width, and head/tail parameters.
The user interface for all program modules is simple and intuitive. When graphing equations, users can select a sample equation which is automatically pasted into the active equation edit box. When appropriate, the program provides comments and suggestions for setting screen parameters to achieve best results. After an equation is graphed, you can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. With simple mouse control clicks, you can find relative minimum points, relative maximum points, x-intercepts, and intersection points. A Help menu provides a quick summary of all magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. To view multiple screen shots of the program’s modules, go to www.mathteachersresource.com. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.
Surely ? * ? =-49 is just 7 x -7 or -7 x 7.
The symbol “?” is used to indicate that the number(s) used to satisfy the conditions must be equivalent. It might seem more clear to display the problem as n x n = -49. The solutions that you propose would fit the question n x m =-49 or ? x ! = -49. Hope that helps.