David N. Bannard

Abstract

This project uses a disguised form of Newton's Method to introduce students to the arithmetic of complex numbers. Students can use a spreadsheet or calculator to explore a powerful technique for finding both the real and complex cube roots of one. A True Basic program to draw the fractal picture of basins of attraction of the roots is included. The idea for this lesson was first shown to me by Rick Paris of Phillips Exeter Academy.

Introduction

The first question that might be posed is to ask students to design an algorithm to find the square root of a number, say 8. In particular, if you start with a guess, say 2, how could you determine if the number is in fact the square root of 8? A common response would be to square 2, determine that it was too low, and so try 3. Since 3 is too large, our next guess might be 2.5. Since this is too low, 2.5² < 8, we might try 2.75 as our next guess. We might recognize this as the bisection method. Students might also recognize that this method converges on the square root relatively slowly.

As an alternative to this method, we might suggest that they take their first guess, say 2, and divide this into 8. Since the result is 4, we can see that the actual square root must be between 2 and 4. In fact we might try the average, 3, as our next guess. Again, we divide 8 by 3 and average the quotient with 3 to obtain our next estimate. On a TI81 or TI82, this can be done by typing 2, enter, and then typing (Ans+8/Ans)/2. Keep hitting enter. Notice after the fourth iteration, we have achieved 9 decimal places of accuracy which is quite impressive.

How can we extend this process to find the cube root of, say, 7? Again, we start with a guess, say 2. However, this time 2³ is our first estimate for the cube root of 7. To find an appropriate quotient, we must first square our guess and divide this into 7. The result is 1.75, indicating that our guess was too high. Since we have said that 2*2*1.75 = 7, we should obtain our next guess by using the average of these three numbers as our next guess. On the calculator, the steps would be: 2 enter; (2*Ans+7/Ans²)/3. Again, when this is iterated, the answer is found to 9 decimal places in 4 iterations. It turns out that the real bonus we have obtained with this algorithm is that it still works if our starting values are complex numbers.

The Cube Roots of 1

Our task is to use our algorithm to find the three cube roots of 1, or the solutions to the equation x³ - 1 = 0. Algebraically, we can find the roots to be 1, - , - . We can see that if we start with a real number other than zero as our guess, we will obtain the real root 1. However, suppose we start with a complex number as our initial guess.

Let us express our algorithm recursively.

T₀ = a + bi ; T_n+1 =

Since T₀ can be a complex number, it motivates us to learn how to compute (a + bi)² (multiply) and . Each result must be expressed in the form x + yi, where x and y are real numbers.

(a + bi)² = a² + 2abi + b²i²

since i² = -1

(a + bi)² = a² - b² + 2abi

now =

To put this in the form x + yi, we must multiply numerator and denominator by the conjugate of the denominator (a² - b² ) - 2abi, giving

^. =

Simplifying we obtain =

This gives + i

where the real coefficient is x =

and the imaginary coefficient is y =

Exploring the Cube Roots of 1

We can ask the question: given a non-zero complex number a + bi, if we iterate this number using the recursive formula,

T_n+1 = ,

can we predict which of the three complex cube roots of 1 we will find? The spreadsheet below allows you to explore this question.

If we select the last row of this spreadsheet, and fill down, say, 30 rows, we should be able to iterate to a root for virtually any starting value in the complex plane.

A sample output for the initial value -2 + 1 i is shown below.

Note for this selection, after 7 iterations, we have approached the root - + .

Students can now take a graph of the complex plane from say -2 to 2 on the real axis and -2 to 2 on the imaginary axis and iterate various points. If a point iterates to the real root, they should color the point red. If the point iterates to the root in quadrant 2, they should color the point blue. Finally, if the point iterates to the root in quadrant 3, they should color the point green. The question is what will this picture look like as they take many points in the complex plane. Your experimentation might start with points close to the roots. However, as they explore more, they may be surprised by the fact that some points do not go where they would predict. They should be sure to try points in the second and third quadrants near the real axis. To see a complete picture, write a program like the true basic below. The increments that you choose for x and y should be quite small, (.01 for example). The maximum # of iterations can usually be a small as 25. Be warned, on a slow computer, this program will probably have to run overnight.

True Basic Program

!Pictures the cube roots of 1

Input prompt "left x":sx

input prompt "right x":ex

input prompt "bottom y":sy

input prompt "top y ":ey

input prompt "increment for x":dx

input prompt "increment for y":dy

input prompt "max number of iterations":n

call setup !Sets proper windows

for x=sx to ex step dx !Loops to choose points over the complex plane

for y=sy to ey step dy

let a=x

let b=y

for I=1 to n

!print a,b

if (a^2+b^2)=0 then exit for

let qa = (a^2-b^2)/(a^2+b^2)^2

let qb=-2*a*b/(a^2+b^2)^2

let a=(2*a+qa)/3

let b=(2*b+qb)/3

if abs(a-1)<.01 and abs(b)<.01 then !Real root

call red

exit for

end if

if abs(a+.5)<.01 and abs(b-.866)<.01 then !Second quadrant root

call blue

exit for

end if

if abs(a+.5)<.01 and abs(b+.866)<.01 then !Third quadrant root

call green

exit for

end if

next i

next y

next x

sub setup

open #1: screen 0, 1, 0, 1

set window sx, ex, sy, ey

set background color "white"

end sub

sub red

set color 0

plot area: x,y;x+dx,y;x+dx,y+dy;x,y+dy

end sub

sub blue

set color 1

plot area: x,y;x+dx,y;x+dx,y+dy;x,y+dy

end sub

sub green

set color 2

plot area: x,y;x+dx,y;x+dx,y+dy;x,y+dy

end sub

get key z

end

Connection to Newton's Method

The method that we have been using to find cube roots is often called the Mechanic's algorithm, and it is actually a special case of Newton's Method. According to Newton's Method, the roots of f(z) = z³ - 1 can be found by iterating N(z) = z - . This gives

N(z) = z - = = ,

which is precisely the form we have used throughout the process. For those who are interested, this method can be extended to fourth roots and beyond. It has provided an interesting application that introduces students to a powerful iteration algorithm, requires students to learn how to multiply and divide complex numbers, and to plot points in the complex plane. The fractal that is produced by the program is a spectacular conclusion to the exercise.

A Graphical View of the Convergence of Newton's Method

In calculus, Newton's Method provides a wonderful algorithm to study, not just because it is an extremely efficient way to find roots if one selects a good starting guess, but also because of its erratic behavior if one is unlucky enough to pick a poor starting guess. In order to explore this behavior, one can use a spreadsheet like the one below. The graph in this spreadsheet plots the number of the iterate on the x-axis and the orbit of the root, i.e. the path the iterates take towards a root. Figure 1 contains the relevant formulas while figures 2 and 3 provide interesting examples. If you wish to graph the function in addition to the iterates, you must enter the function under a particular domain in a separate section of the spreadsheet. Two examples are provided of interesting orbits.

Figure 1

Figure 2

In this example, there is only one real root, but the path to that root can be very bizarre, depending on your starting value. The problem here is caused by the relatively flat portion of the curve to the left of the real root.

Figure 3

Notice in this case, the polynomial has three roots and a starting value to the left of the middle root goes to the right hand root. It is interesting to try to predict which root you will find for a given starting value, and then to check your conjecture.

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