# Application of complex numbers:

by Stephanie Reynolds

Here is an activity that I did with my Pre-Calculus students.  My textbook has a brief overview of fractals, which are pictures created by plotting numbers on the complex plane (real axis & imaginary axis). This program creates a graph that looks like a fern leaf. I entered the program "Fractal Fern" into my calculator, and then linked the calculators. (See your calculators manual if you are unfamiliar with creating a program.  E-mail me if you are having trouble finding some of the commands.  A student of mine helped me to find items that I couldn't find)  *Note - your calculator must be in radian mode or the graph will not look like a fern leaf.

The program takes about 30 minutes to complete the graph, so we started the program at the beginning of class, and as the students were working on other problems, they kept checking the progress of their graph.  If you have students who are interested in entering the programming code into their calculator, it will take them about 20-30 minutes to type in all the commands.

Note:  The attached document has several other programs that may be of interest to you.

Let me know if you tried this and how it went.  My students were really amazed at what types of graphs a calculator can graph.

TI-82

TI-83

TI-83 Plus

Quadratic Formula Program

This program will display the solutions of a quadratic equation or the words “No Real Solution.” To use the program, write the quadratic equation in general form and enter the values of a, b, and c.

PROGRAM:QUADRAT

1. :Disp “AX2_BX_C_0”
2. :Prompt A
3. :Prompt B
4. :Prompt C
5. :B2_4ACD
6. :If D³0
7. :Then
8. :(-B_ (D)) (2A)M
9. :Disp M
10. :(-B_ (D)) (2A)N
11. :Disp N
12. :Else
13. :Disp “NO REAL SOLUTION”
14. :End

Graph Reflection Program

This program will graph a function f and its reflection in the line To use this program, enter the function in Y1 and set a viewing window.

PROGRAM:REFLECT

1. :63Xmin 95Ymin
2. :63Xmax 95Ymax
3. :XsclYscl
4. :“X”Y2
5. :DispGraph
6. :(Xmax_Xmin) 94I
7. :XminX
8. :While X²Xmax
9. :Pt-On(Y1,X)
10. :X_IX
11. :End

Systems of Linear Equations Program

This program will display the solution of a system of two linear equations in two variables of the form if a unique solution exists.

PROGRAM:SOLVE

1. :Disp “AX_BY_C”
2. :Prompt A
3. :Prompt B
4. :Prompt C
5. :Disp “DX_EY_F”
6. :Prompt D
7. :Prompt E
8. :Prompt F
9. :If AE_BD_0
10. :Then
11. :Disp “NO UNIQUE”
12. :Disp “SOLUTION”
13. :Else
14. :(CE_BF) (AE_BD)X
15. :(AF_CD) (AE_BD)Y
16. :Disp X
17. :Disp Y
18. :End

Evaluating an Algebraic Expression

Program

This program can be used to evaluate an algebraic expression in one variable at several values of the variable. To use this program, enter an expression in Y1.

PROGRAM:EVALUATE

1. :Lbl A
2. :Input “ENTER X”,X
3. :Disp Y1
4. :Goto A
5. _ax _ by _ c
6. dx _ ey _ f
7. y _ x.

Fractal Fern Program

This program draws a fractal that is in the shape of a fern leaf. To use this program, make sure your calculator is in radian mode and enter a starting point for the fractal. This program will take several minutes to execute. For the TI-82, press to quit the program. For the TI-83and TI-83 Plus, ON 2 , press ON 1 to quit the program.

_x, y_

PROGRAM:FERN

1. :0Xmin
2. :50Xmax
3. :10Xscl
4. :0Ymin
5. :50Ymax
6. :10Yscl
7. :ClrDraw
8. :25A
9. :25B
10. :0J
11. :Disp “START (X, Y)”
12. :Disp “ENTER X”
13. :Input C
14. :Disp “ENTER Y”
15. :Input D
16. :Lbl 1
17. :8(A+D) 1U
18. :16(B+C)+20V
19. :Pt-On(U, V)
20. :randZ
21. :0.85R
22. :0.85S
23. :-0.0436A
24. :-0.0436B
25. :1K
26. :If Z<0.005
27. :Goto 2
28. :If Z<0.1025
29. :Goto 3
30. :If Z<0.2
31. :Goto 4
32. :Lbl 5
33. :R*C*cos(A) S*D*sin(B)E
34. :R*C*sin(A)+S*D*cos(B)+KF
35. :EC
36. :FD
37. :J+1J
38. :If J<10000
39. :Goto 1
40. :End
41. :Lbl 2
42. :0R
43. :0.16S
44. :0A
45. :0B
46. :0K
47. :Goto 5
48. :Lbl 3
49. :0.3R
50. :0.34S
51. :0.8552A
52. :0.8552B
53. :1.6K
54. :Goto 5
55. :Lbl 4
56. :0.3R
57. :0.37S
58. :2.0944A
59. :-0.8552B
60. :0.44K
61. :Goto 5
62. :End

PROGRAM: ROWOPS

1. :Disp “ENTER A”
2. :Disp “2 BY 3 MATRIX:”
3. :Disp “A B C”
4. :Disp “D E F”
5. :Prompt A,B,C
6. :Prompt D,E,F
7. :A[A](1,1):B[A](1,2)
8. :C[A](1,3):D[A](2,1)
9. :E[A](2,2):F[A](2,3)
10. :ClrHome
11. :Disp “ORIGINAL MATRIX:”
12. :Pause [A]
13. :“B_1(C_AX)”Y2
14. :“E_1(F_DX)”Y1
15. :ZStandard:Pause:ClrHome
16. :Disp “OBTAIN LEADING”
17. :Disp “1 IN ROW 1”
18. :*row(A_1,[A],1)[A]
19. :Pause [A]:ClrDraw
20. :“(A B)(C A_X)”Y2
21. :DispGraph:Pause:ClrHome
22. :Disp “OBTAIN 0 BELOW”
23. :Disp “LEADING 1 IN”
24. :Disp “COLUMN 1”
25. :*row+(-D,[A],1,2)[A]
26. :Pause [A]:ClrDraw
27. :“(E_(BD A))_1(F_(DC A))”Y1
28. :DispGraph:Pause:ClrHome
29. :[A](2,2)G
30. :If G_0
31. :Goto 1
32. :*row(G_1,[A],2)[A]
33. :Disp “OBTAIN LEADING”
34. :Disp “1 IN ROW 2”
35. :Pause [A]:ClrDraw
36. :DispGraph:Pause:ClrHome
37. :Disp “OBTAIN 0 ABOVE”
38. :Disp “LEADING 1 IN”
39. :Disp “COLUMN 2”
40. :[A](1,2)H
41. :*row+(-H,[A],2,1)[A]
42. :Pause [A]:ClrDraw:FnOff 2
43. :Vertical -(B A)(E_(BD A))_1(F_DC A)_C A
44. :DispGraph:Pause:ClrHome
45. :Disp “THE POINT OF”
46. :Disp “INTERSECTION IS”
47. :Disp “X=”,[A](1,3),“Y=”,[A](2,3)
48. :Stop
49. :Lbl 1
50. :If [A](2,3)_0
51. :Then
52. :Disp “INFINITELY MANY”
53. :Disp “SOLUTIONS”
54. :Else
55. :Disp “INCONSISTENT”
56. :Disp “SYSTEM”
57. :End

Visualizing Row Operations Program

This program demonstrates how elementary matrix row operations used in Gauss-Jordan elimination may be interpreted graphically. It asks the user to enter a 2 _ 3 matrix that corresponds to a system of two linear equations. (The matrix entries should not be equivalent to either vertical or horizontal lines. This emonstration is also most effective if the y-intercepts of the lines are between _10 and 10.) While the demonstration is running, you should notice that each elementary row operation creates an equivalent system. This equivalence is reinforced graphically because, although the equations of the lines change with each elementary row operation, the point of intersection remains the same. You may want to run this program a second time to notice the relationship between the row operations and the graphs of the lines of the system. To use this program, dimension matrix [A] as a 2 _ 3 matrix. Press ENTER after each screen display to continue the program.

Graphing a Sine Function Program

This program will simultaneously draw a unit circle and the corresponding points on the sine curve. After the circle and sine curve are drawn, you can connect the points on the unit circle with their corresponding points on the sine curve by pressing .

PROGRAM:SINESHOW

1. :Radian
2. :ClrDraw:FnOff
3. :Param:Simul
4. :-2.25Xmin
5. :_ 2Xmax
6. :3Xscl
7. :-1.19Ymin
8. :1.19Ymax
9. :1Yscl
10. :0Tmin
11. :6.3Tmax
12. :.15Tstep
13. :“-1.25_cos (T)”X1T
14. :“sin (T)”Y1T
15. :“T 4”X2T
16. :“sin (T)”Y2T
17. :DispGraph
18. :For(N,1,12)
19. :N_ 6.5T
20. :-1.25_cos (T)A
21. :sin(T)B
22. :T 4C
23. :Line(A,B,C,B)
24. :Pause
25. :End
26. :Pause :Func
27. :Sequential:Disp

Finding the Angle Between Two Vectors Program

This program will graph two vectors and calculate the measure of the angle between the vectors. Be sure to set an appropriate viewing window. After the vectors are drawn, press to view the angle between the vectors.

PROGRAM:VECANGL

1. :ClrHome
2. :Degree
3. :Disp “ENTER (A,B)”
4. :Input “ENTER A”,A
5. :Input “ENTER B”,B
6. :ClrHome
7. :Disp “ENTER (C,D)”
8. :Input “ENTER C”,C
9. :Input “ENTER D”,D
10. :Line(0,0,A,B)
11. :Line(0,0,C,D)
12. :Pause
13. :AC_BDE
14. : (A2 _B2)U
15. : (C2 _D2)V
16. :cos-1(E (UV))_
17. :ClrDraw:ClrHome
18. :Disp “_=”,_
19. :Stop

Adding Vectors Graphically Program

This program will graph two vectors in standard position. Using the parallelogram law for vector addition, the program also graphs the vector sum. Be sure to set an appropriate viewing window.

PROGRAM:ADDVECT

1. :ClrDraw
2. :Input “ENTER A”,A
3. :Input “ENTER B”,B
4. :Input “ENTER C”,C
5. :Input “ENTER D”,D
6. :Line(0,0,A,B)
7. :Line(0,0,C,D)
8. :A_CE
9. :B_DF
10. :Line(0,0,E,F)
11. :Line(A,B,E,F)
12. :Line(C,D,E,F)
13. :Pause
14. :Stop
15. ENTER
16. ENTER