[WW HOME] [TEACHING] [MATH] [SEARCH] [FEEDBACK]

Complex Numbers

Rose Caldwell, Carolyn Foster, Grace Williams,

Barbara Ziegenhals, and Marcia Zrubek

Abstract: Chaos and dynamical systems are current topics in mathematics, and complex numbers are critical to their study. This unit develops the graphing of complex numbers, uses iterations, and culminates in a construction of the Mandelbrot Set. It is composed of three activities appropriate for an Algebra II or precalculus class and is intended to be used as a group activity to complement the study of complex numbers. It is recommended that the students have prior knowledge of operations with complex numbers and have experience iterating functions.The three activities are:
1. Graphing Products of Complex Numbers
2. Iterative Multiplication of Complex Numbers
3. Creating the Mandelbrot Set

Activity I: Graphing Products of Complex Numbers

Supplies: protractor, ruler, and cm grid paper

Any complex number of the form x+yi can be graphed as the point (x, y) on the coordinate plane where the normal x-axis is considered the real-axis and the normal y-axis is considered the imaginary-axis. To plot a complex number on this new version of the coordinate plane, locate the point whose coordinates are the real part and the coefficient of i in the imaginary part of the complex number. For example, the complex number 1.5+2i is plotted as the point (1.5,2). You are to investigate how multiplication by a complex number affects the location of a point.

1. On a piece of graph paper mark off four grids.

a. On the first grid graph the point 3+4i and draw a line segment connecting this point to the origin. (This number will be used as the multiplier in the following three problems.)

b. On the second grid graph the point 2+3i and draw a line segment connecting this point to the origin. Multiply 2+3i by 3+4i and graph your answer on the grid with 2+3i. Again draw a segment connecting this point with the origin.

c. On the third grid graph the point -1+2i and draw a line segment connecting this point to the origin. Multiply -1+2i by 3+4i and graph your answer on the grid with -1+2i. Again draw a segment connecting this point with the origin.

d. On the fourth grid graph the point 4i and draw a line segment connecting this point to the origin. Multiply 4i by 3+4i and graph your answer on the grid with 4i. Again draw a segment connecting this point with the origin.

e. Use your protractor to measure the angles formed by the two segments drawn on the last three grids and label angle measurements on the grids. Now look at the first grid and measure the angle formed between the segment and the x-axis. Compare these measurements.

2. Mark off four more grids on your graph paper and repeat the above steps, replacing 3+4i with .5+.5i each time.

3. Mark off four more grids and repeat the procedure using i in place of 3+4i.

What generalization can you make about the angles formed by products?

With your ruler, measure the lengths of all of the segments that you have drawn and complete the table below.

(a+bi)(c+di)

Length of the
a+bi segment

Length of the
c+di segment

Length of the product segment

(2+3i)(3+4i)



(-1+2i)(3+4i)



(4i)(3+4i)



(2+3i)(.5+.5i)



(-1+2i)(.5+.5i)



(4i)(.5+.5i)



(2+3i)(i)



(-1+2i)(i)



(4i)(i)



Compare the length of the original segment, the length of the multiplier segment and the length of the product segment. Generalize your results.

There is an analytical way of finding the length of each of these segments. Determine a method and explain why it works.

The absolute value of a complex number a+bi, written as |a +bi|, is defined to be the distance between the point and the origin; therefore,

|a +bi| = ___________________.

Activity II: Iterative Multiplication of Complex Numbers
(An Introduction to Orbits)

Supplies: TI-82, link cord, and at least one TI-82 with the program COMPLEX already entered. The program named COMPLEX is found on the last page of this unit.

This lesson investigates orbits of complex numbers created by iterating the function f(z)=kz in which k and z are complex numbers. An orbit of a point is a sequence of points {z0, z1, z2, ...,zn, ...} formed by iterating a function starting at a value z0, where zn=f(zn-1). The z0 is referred to as the seed. An orbit is converging if the sequence approaches a fixed point, an orbit is diverging if the sequence increases without bound, and it is periodic if it cycles.

Exercise 1:

Use the iteration function f(z)=kz with k=1+i and z0=2i to complete the following table.

z0 = ______2i________ z1 = (1+i)(2i)=-2+2i__ z2 = (1+i)(-2+2i)=-4i__

z3 = _______________ z4 = ______________ z5 = _______________

Plot these six points on the grid provided below.

Without further calculations, project the location of the next two points and locate these points on your graph. What you have calculated and graphed is called an orbit of the point 2i.

You will now use the TI-82 program COMPLEX to see the orbits of the points under the function f(z)=kz. For this activity you will be entering the values of k and z0 given below.

Program Operating Instructions:

1. Before you begin, set the window for the program using ZStandard and ZSquare .

2. After the program plots the points, if the points are difficult to see, adjust the window using Zoom In or Zoom Out.

3. The program allows you to choose the number of iterations, N. A choice of N=10 gives a nice sampling of points.

4. After the program sketches the graph, use the TRACE key to see the coordinates of the points in order.

Note: If you want a list of the coordinates, the program stores the values in L1 and L2.

Group Task:

Run the program COMPLEX entering the numbers given below for z0 and k . Make a sketch of the points from the calculator screen onto the grids provided below.

1. z0 = 1+1i and k = 1.5 + 2i 2. z0 = 1 - 1i and k = -1.2 + 1i

3. z0 = 2 + 2i and k = 0.5 + 0.5i 4. z0 = -2 - 2i and k = 0.6 - 0.6i

5. z0 = 3 + 4i and k = 0.6 + 0.8i 6. z0 = 1 + 0i and k =

Which of the above orbits are

diverging? _____________ converging?_____________ periodic?_____________

Do you see any differences among the periodic orbits? If so, describe your conjecture.

Find |k| in each of the previous six exercises and record the results below.

1. _____________ 2. _____________ 3. _____________

4. _____________ 5. _____________ 6. _____________

Generalize a rule for determining when the orbit of a point formed by the function f(z)=kz will be diverging, converging, or periodic. Explain your rule.

Activity III: Creating the Mandelbrot Set

Supplies: Overhead transparency of the grid for the Mandelbrot Set, overhead pens in four colors (red, blue, green, and black), TI-82 calculators, link cords, at least one TI-82 with the program ITERATE already entered, and a copy of one part of the point table for each group. The program ITERATE is found on the last page of this unit.

Note to the teacher: Divide the point table among the students so that the iteration of every point is done.

In the previous lesson you iterated the complex function f(z) = kz and discovered that the iteration produced orbits that were either converging, diverging, or periodic. One of the most interesting discoveries of the twentieth century is the Mandelbrot Set discovered by Benoit Mandelbrot in 1980. This set is created by iterating the function f(z) = z2+c where z0 = 0 + 0i and c is a complex point on the plane. You are now going to create your own picture of this set; however, iterating every point on the plane is a task no one can accomplish, so you will be working together to produce an approximation of the set.

A point c belongs to the Mandelbrot Set if its orbit with seed 0 + 0i does not diverge when iterated by the function f(z) = z2+c. Your task is to decide which points on the complex plane are in the Mandelbrot Set. To do this you are to use the program ITERATE on the TI-82 to determine the fate of the orbits for different values of c. In order to give more definition to the set, a color scheme will be used to illustrate how quickly a point escapes to infinity.

Program Instructions:

1. At the prompt asking for the value of the constant, enter the real and imaginary coefficients of your point c. The program will give you the next value in the orbit sequence and a count of the number of iterations.

2. Press ENTER to get the next iteration.

3. Continue the process until the calculator gives an overflow error message. Select the QUIT option and press ENTER, which will take you back to the home screen.

4. The value of N on the screen is actually the previous count so the overflow () occurs at the (N+1)st iteration.

5. Record the value of N+1 in the point table beside your number.

6. If an overflow message does not occur by the 35th iteration, record an M in the point table.

7. Press ENTER again and repeat the program for the remainder of your points.

Examples:

Try the point 1.2+1i, enter 1.2 for C and 1 for D. When the overflow message occurs, select Quit and press ENTER to return to your previous screen and you will see N=9. By the point 1.2+1i in the point table record the number 10. Press ENTER again to restart the program for your next point.

Now try the point 0.2-0.2i, enter 0.2 for C and -0.2 for D. This time you iterate for 35 times and still do not receive an overflow message. By the point 0.2-0.2i in the point table, record an M. Press ON and choose Quit then press ENTER twice to restart the program for your next point.

Instructions for coloring the overhead transparency for the Mandelbrot Set: Consider each point as the center of a square on the grid rather than as a point on the intersection of two lines, i.e., treat the points like pixels on the calculator screen. Color each pixel according to the chart provided below.

Number of Iterations

Pen color

1 - 11

Red

12 - 23

Green

24 - 35

Blue

M or 36 and above

Black

When all of the groups have colored their pixels, display the finished transparency. The set of all points colored black is the Mandelbrot Set.

Note: A finer division of points will give better definition for the set. For further Investigation ask your librarian or teacher for books and articles on chaos, fractals, and dynamical systems.

Point Table

c value

N

c value

N

c value

N

1.2 - 2.0i

1.2 - 1.8i

1.2 - 1.6i

1.0 - 2.0i

1.0 - 1.8i

1.0 - 1.6i

0.8 - 2.0i

0.8 - 1.8i

0.8 - 1.6i

0.6 - 2.0i

0.6 - 1.8i

0.6 - 1.6i

0.4 - 2.0i

0.4 - 1.8i

0.4 - 1.6i

0.2 - 2.0i

0.2 - 1.8i

0.2 - 1.6i

0 - 2.0i

0 - 1.8i

0 - 1.6i

-0.2 - 2.0i

-0.2 - 1.8i

-0.2 - 1.6i

-0.4 - 2.0i

-0.4 - 1.8i

-0.4 - 1.6i

-0.6 - 2.0i

-0.6 - 1.8i

-0.6 - 1.6i

-0.8 - 2.0i

-0.8 - 1.8i

-0.8 - 1.6i

-1.0 - 2.0i

-1.0 - 1.8i

-1.0 - 1.6i

-1.2 - 2.0i

-1.2 - 1.8i

-1.2 - 1.6i

c value

N

c value

N

c value

N

1.2 - 1.4i

1.2 - 1.2i

1.2 - 1.0i

1.0 - 1.4i

1.0 - 1.2i

1.0 - 1.0i

0.8 - 1.4i

0.8 - 1.2i

0.8 - 1.0i

0.6 - 1.4i

0.6 - 1.2i

0.6 - 1.0i

0.4 - 1.4i

0.4 - 1.2i

0.4 - 1.0i

0.2 - 1.4i

0.2 - 1.2i

0.2 - 1.0i

0 - 1.4i

0 - 1.2i

0 - 1.0i

-0.2 - 1.4i

-0.2 - 1.2i

-0.2 - 1.0i

-0.4 - 1.4i

-0.4 - 1.2i

-0.4 - 1.0i

-0.6 - 1.4i

-0.6 - 1.2i

-0.6 - 1.0i

-0.8 - 1.4i

-0.8 - 1.2i

-0.8 - 1.0i

-1.0 - 1.4i

-1.0 - 1.2i

-1.0 - 1.0i

-1.2 - 1.4i

-1.2 - 1.2i

-1.2 - 1.0i

c value

N

c value

N

c value

N

1.2 - 0.8i

1.2 - 0.6i

1.2 - 0.4i

1.0 - 0.8i

1.0 - 0.6i

1.0 - 0.4i

0.8 - 0.8i

0.8 - 0.6

0.8 - 0.4i

0.6 - 0.8i

0.6 - 0.6i

0.6 - 0.4i

0.4 - 0.8i

0.4 - 0.6i

0.4 - 0.4i

0.2 - 0.8i

0.2 - 0.6i

0.2 - 0.4i

0 - 0.8i

0 - 0.6i

0 - 0.4i

-0.2 - 0.8i

-0.2 - 0.6i

-0.2 - 0.4i

-0.4 - 0.8i

-0.4 - 0.6i

-0.4 - 0.4i

-0.6 - 0.8i

-0.6 - 0.6i

-0.6 - 0.4i

-0.8 - 0.8i

-0.8 - 0.6i

-0.8 - 0.4i

-1.0 - 0.8i

-1.0 - 0.6i

-1.0 - 0.4i

-1.2 - 0.8i

-1.2 - 0.6i

-1.2 - 0.4i

c value

N

c value

N

c value

N

c value

N

1.2 - 0.2i

1.2

1.2 + 0.2i

1.2 + 0.4i

1.0 - 0.2i

1.0

1.0 + 0.2i

1.0 + 0.4i

0.8 - 0.2i

0.8

0.8 + 0.2i

0.8 + 0.4i

0.6 - 0.2i

0.6

0.6 + 0.2i

0.6 + 0.4i

0.4 - 0.2i

0.4

0.4 + 0.2i

0.4 + 0.4i

0.2 - 0.2i

0.2

0.2 + 0.2i

0.2 + 0.4i

0 - 0.2i

0

0 + 0.2i

0.4i

-0.2 - 0.2i

-0.2

-0.2 + 0.2i

-0.2 + 0.4i

-0.4 - 0.2i

-0.4

-0.4 + 0.2i

-0.4 + 0.4i

-0.6 - 0.2i

-0.6

-0.6 + 0.2i

-0.6 + 0.4i

-0.8 - 0.2i

-0.8

-0.8 + 0.2i

-0.8 + 0.4i

-1.0 - 0.2i

-1.0

-1.0 + 0.2i

-1.0 + 0.4i

-1.2 - 0.2i

-1.2

-1.2 + 0.2i

-1.2 + 0.4i













c value

N

c value

N

c value

N

1.2 + 0.6i

1.2 + 0.8i

1.2 + 1.0i

1.0 + 0.6i

1.0 + 0.8i

1.0 + 1.0i

0.8 + 0.6i

0.8 + 0.8i

0.8 + 1.0i

0.6 + 0.6i

0.6 + 0.8i

0.6 + 1.0i

0.4 + 0.6i

0.4 + 0.8i

0.4 + 1.0i

0.2 + 0.6i

0.2 + 0.8i

0.2 + 1.0i

0.6i

0.8i

1.0i

-0.2 + 0.6i

-0.2 + 0.8i

-0.2 + 1.0i

-0.4 + 0.6i

-0.4 + 0.8i

-0.4 + 1.0i

-0.6 + 0.6i

-0.6 + 0.8i

-0.6 + 1.0i

-0.8 + 0.6i

-0.8 + 0.8i

-0.8 + 1.0i

-1.0 + 0.6i

-1.0 + 0.8i

-1.0 + 1.0i

-1.2 + 0.6i

-1.2 + 0.8i

-1.2 + 1.0i

1.2
















1.0
















0.8
















0.6
















0.4
















0.2
















0
















-0.2
















-0.4
















-0.6
















-0.8
















-1.0.
















-1.2

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1.0

Program Lists for the TI-82

Program: COMPLEX

ClrDraw: Func: GridOff: PlotsOff: FnOff:

ClrList L1,L2: AxesOn

Disp "ENTER STARTING"

Disp "POINT A+BI"

Prompt A,B

Disp "ENTER MULTIPLIER"

Disp "POINT C+DI"

Prompt C,D

Disp "HOW MANY"

Disp "ITERATIONS?

Prompt N

A-->L1(1):B-->L2(1)

For(I,2,N)

A*C - B*D-->L1(I)

B*C+A*D-->L2(I)

L1(I)-->A

L2(I)-->B

Plot1(Scatter,L1,L2,•)

End

DispGraph

(Using a square in the scatter plot is better than the •)

Program: ITERATE

ClrList L1,L2

Disp "INPUT CONSTANT"

Disp "C+DI"

Prompt C

Prompt D

0--> N

0--> A

0--> B

Lbl 1

A2 - B2 + C --> X

2*A*B+D --> Y

X-->A

Y-->B

X-->L1(N+1)

Y-->L2(N+1)

Disp "New A",A

Disp "New B",B

N+1--> N

Disp "COUNT IS", N

Pause

Goto 1

[WW HOME] [TEACHING] [MATH] [SEARCH] [FEEDBACK]

Woodrow Wilson Leadership Program in Mathematics * lpt@www.woodrow.org
The Woodrow Wilson National Fellowship Foundation * webmaster@woodrow.org
CN 5281, Princeton NJ 08543-5281 * Tel:(609)452-7007 * Fax:(609)452-0066