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 ÒNo Real Solution.Ó To use the program, write the quadratic equation in general form and enter the values of a, b, and c.

PROGRAM:QUADRAT

        1. :Disp ÒAX2_BX_C_0Ó
        2. :Prompt A
        3. :Prompt B
        4. :Prompt C
        5. :B2_4AC→D
        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 ÒNO REAL SOLUTIONÓ
        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 95→Ymin
        2. :63Xmax 95→Ymax
        3. :Xscl→Yscl
        4. :ÒXÓ→Y2
        5. :DispGraph
        6. :(Xmax_Xmin) 94→I
        7. :Xmin→X
        8. :While X²Xmax
        9. :Pt-On(Y1,X)
        10. :X_I→X
        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 ÒAX_BY_CÓ
        2. :Prompt A
        3. :Prompt B
        4. :Prompt C
        5. :Disp ÒDX_EY_FÓ
        6. :Prompt D
        7. :Prompt E
        8. :Prompt F
        9. :If AE_BD_0
        10. :Then
        11. :Disp ÒNO UNIQUEÓ
        12. :Disp ÒSOLUTIONÓ
        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 ÒENTER XÓ,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. :0→Xmin
        2. :50→Xmax
        3. :10→Xscl
        4. :0→Ymin
        5. :50→Ymax
        6. :10→Yscl
        7. :ClrDraw
        8. :25→A
        9. :25→B
        10. :0→J
        11. :Disp ÒSTART (X, Y)Ó
        12. :Disp ÒENTER XÓ
        13. :Input C
        14. :Disp ÒENTER YÓ
        15. :Input D
        16. :Lbl 1
        17. :8(A+D) 1→U
        18. :16(B+C)+20→V
        19. :Pt-On(U, V)
        20. :rand→Z
        21. :0.85→R
        22. :0.85→S
        23. :-0.0436→A
        24. :-0.0436→B
        25. :1→K
        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)+K→F
        35. :E→C
        36. :F→D
        37. :J+1→J
        38. :If J<10000
        39. :Goto 1
        40. :End
        41. :Lbl 2
        42. :0→R
        43. :0.16→S
        44. :0→A
        45. :0→B
        46. :0→K
        47. :Goto 5
        48. :Lbl 3
        49. :0.3→R
        50. :0.34→S
        51. :0.8552→A
        52. :0.8552→B
        53. :1.6→K
        54. :Goto 5
        55. :Lbl 4
        56. :0.3→R
        57. :0.37→S
        58. :2.0944→A
        59. :-0.8552→B
        60. :0.44→K
        61. :Goto 5
        62. :End

PROGRAM: ROWOPS

        1. :Disp ÒENTER AÓ
        2. :Disp Ò2 BY 3 MATRIX:Ó
        3. :Disp ÒA B CÓ
        4. :Disp ÒD E FÓ
        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 ÒORIGINAL MATRIX:Ó
        12. :Pause [A]
        13. :ÒB_1(C_AX)Ó→Y2
        14. :ÒE_1(F_DX)Ó→Y1
        15. :ZStandard:Pause:ClrHome
        16. :Disp ÒOBTAIN LEADINGÓ
        17. :Disp Ò1 IN ROW 1Ó
        18. :*row(A_1,[A],1)→[A]
        19. :Pause [A]:ClrDraw
        20. :Ò(A B)(C A_X)Ó→Y2
        21. :DispGraph:Pause:ClrHome
        22. :Disp ÒOBTAIN 0 BELOWÓ
        23. :Disp ÒLEADING 1 INÓ
        24. :Disp ÒCOLUMN 1Ó
        25. :*row+(-D,[A],1,2)→[A]
        26. :Pause [A]:ClrDraw
        27. :Ò(E_(BD A))_1(F_(DC A))Ó→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 ÒOBTAIN LEADINGÓ
        34. :Disp Ò1 IN ROW 2Ó
        35. :Pause [A]:ClrDraw
        36. :DispGraph:Pause:ClrHome
        37. :Disp ÒOBTAIN 0 ABOVEÓ
        38. :Disp ÒLEADING 1 INÓ
        39. :Disp ÒCOLUMN 2Ó
        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 ÒTHE POINT OFÓ
        46. :Disp ÒINTERSECTION ISÓ
        47. :Disp ÒX=Ó,[A](1,3),ÒY=Ó,[A](2,3)
        48. :Stop
        49. :Lbl 1
        50. :If [A](2,3)_0
        51. :Then
        52. :Disp ÒINFINITELY MANYÓ
        53. :Disp ÒSOLUTIONSÓ
        54. :Else
        55. :Disp ÒINCONSISTENTÓ
        56. :Disp ÒSYSTEMÓ
        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. :-2.25→Xmin
        5. :_ 2→Xmax
        6. :3→Xscl
        7. :-1.19→Ymin
        8. :1.19→Ymax
        9. :1→Yscl
        10. :0→Tmin
        11. :6.3→Tmax
        12. :.15→Tstep
        13. :Ò-1.25_cos (T)Ó→X1T
        14. :Òsin (T)Ó→Y1T
        15. :ÒT 4Ó→X2T
        16. :Òsin (T)Ó→Y2T
        17. :DispGraph
        18. :For(N,1,12)
        19. :N_ 6.5→T
        20. :-1.25_cos (T)→A
        21. :sin(T)→B
        22. :T 4→C
        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 ÒENTER (A,B)Ó
        4. :Input ÒENTER AÓ,A
        5. :Input ÒENTER BÓ,B
        6. :ClrHome
        7. :Disp ÒENTER (C,D)Ó
        8. :Input ÒENTER CÓ,C
        9. :Input ÒENTER DÓ,D
        10. :Line(0,0,A,B)
        11. :Line(0,0,C,D)
        12. :Pause
        13. :AC_BD→E
        14. : (A2 _B2)→U
        15. : (C2 _D2)→V
        16. :cos-1(E (UV))→_
        17. :ClrDraw:ClrHome
        18. :Disp Ò_=Ó,_
        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 ÒENTER AÓ,A
      3. :Input ÒENTER BÓ,B
      4. :Input ÒENTER CÓ,C
      5. :Input ÒENTER DÓ,D
      6. :Line(0,0,A,B)
      7. :Line(0,0,C,D)
      8. :A_C→E
      9. :B_D→F
      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