## 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 ProgramThis 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, andc.PROGRAM:QUADRAT

- :Disp ÒAX2_BX_C_0Ó
- :Prompt A
- :Prompt B
- :Prompt C
- :B2_4AC→D
- :If D³0
- :Then
- :(-B_ (D)) (2A)→M
- :Disp M
- :(-B_ (D)) (2A)→N
- :Disp N
- :Else
- :Disp ÒNO REAL SOLUTIONÓ
- :End

Graph Reflection ProgramThis program will graph a function

fand its reflection in the line To use this program, enter the function in Y1 and set a viewing window.PROGRAM:REFLECT

- :63Xmin 95→Ymin
- :63Xmax 95→Ymax
- :Xscl→Yscl
- :ÒXÓ→Y2
- :DispGraph
- :(Xmax_Xmin) 94→I
- :Xmin→X
- :While X²Xmax
- :Pt-On(Y1,X)
- :X_I→X
- :End

Systems of Linear Equations ProgramThis 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

- :Disp ÒAX_BY_CÓ
- :Prompt A
- :Prompt B
- :Prompt C
- :Disp ÒDX_EY_FÓ
- :Prompt D
- :Prompt E
- :Prompt F
- :If AE_BD_0
- :Then
- :Disp ÒNO UNIQUEÓ
- :Disp ÒSOLUTIONÓ
- :Else
- :(CE_BF) (AE_BD)→X
- :(AF_CD) (AE_BD)→Y
- :Disp X
- :Disp Y
- :End

Evaluating an Algebraic Expression

ProgramThis 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

- :Lbl A
- :Input ÒENTER XÓ,X
- :Disp Y1
- :Goto A
- _
*ax*_*by*_*c* *dx*_*ey*_*f**y*_*x*.

Fractal Fern ProgramThis program draws a fractal that is in the shape of a fern leaf. To use this program, make sure your calculator is in

radianmode and enter a starting point for the fractal. This program will take several minutes to execute. For theTI-82, press to quit the program. For theTI-83andTI-83 Plus, ON 2 , press ON 1 to quit the program._

x,y_PROGRAM:FERN

- :0→Xmin
- :50→Xmax
- :10→Xscl
- :0→Ymin
- :50→Ymax
- :10→Yscl
- :ClrDraw
- :25→A
- :25→B
- :0→J
- :Disp ÒSTART (X, Y)Ó
- :Disp ÒENTER XÓ
- :Input C
- :Disp ÒENTER YÓ
- :Input D
- :Lbl 1
- :8(A+D) 1→U
- :16(B+C)+20→V
- :Pt-On(U, V)
- :rand→Z
- :0.85→R
- :0.85→S
- :-0.0436→A
- :-0.0436→B
- :1→K
- :If Z<0.005
- :Goto 2
- :If Z<0.1025
- :Goto 3
- :If Z<0.2
- :Goto 4
- :Lbl 5
- :R*C*cos(A) S*D*sin(B)→E
- :R*C*sin(A)+S*D*cos(B)+K→F
- :E→C
- :F→D
- :J+1→J
- :If J<10000
- :Goto 1
- :End
- :Lbl 2
- :0→R
- :0.16→S
- :0→A
- :0→B
- :0→K
- :Goto 5
- :Lbl 3
- :0.3→R
- :0.34→S
- :0.8552→A
- :0.8552→B
- :1.6→K
- :Goto 5
- :Lbl 4
- :0.3→R
- :0.37→S
- :2.0944→A
- :-0.8552→B
- :0.44→K
- :Goto 5
- :End

PROGRAM: ROWOPS

- :Disp ÒENTER AÓ
- :Disp Ò2 BY 3 MATRIX:Ó
- :Disp ÒA B CÓ
- :Disp ÒD E FÓ
- :Prompt A,B,C
- :Prompt D,E,F
- :A→[A](1,1):B→[A](1,2)
- :C→[A](1,3):D→[A](2,1)
- :E→[A](2,2):F→[A](2,3)
- :ClrHome
- :Disp ÒORIGINAL MATRIX:Ó
- :Pause [A]
- :ÒB_1(C_AX)Ó→Y2
- :ÒE_1(F_DX)Ó→Y1
- :ZStandard:Pause:ClrHome
- :Disp ÒOBTAIN LEADINGÓ
- :Disp Ò1 IN ROW 1Ó
- :*row(A_1,[A],1)→[A]
- :Pause [A]:ClrDraw
- :Ò(A B)(C A_X)Ó→Y2
- :DispGraph:Pause:ClrHome
- :Disp ÒOBTAIN 0 BELOWÓ
- :Disp ÒLEADING 1 INÓ
- :Disp ÒCOLUMN 1Ó
- :*row+(-D,[A],1,2)→[A]
- :Pause [A]:ClrDraw
- :Ò(E_(BD A))_1(F_(DC A))Ó→Y1
- :DispGraph:Pause:ClrHome
- :[A](2,2)→G
- :If G_0
- :Goto 1
- :*row(G_1,[A],2)→[A]
- :Disp ÒOBTAIN LEADINGÓ
- :Disp Ò1 IN ROW 2Ó
- :Pause [A]:ClrDraw
- :DispGraph:Pause:ClrHome
- :Disp ÒOBTAIN 0 ABOVEÓ
- :Disp ÒLEADING 1 INÓ
- :Disp ÒCOLUMN 2Ó
- :[A](1,2)→H
- :*row+(-H,[A],2,1)→[A]
- :Pause [A]:ClrDraw:FnOff 2
- :Vertical -(B A)(E_(BD A))_1(F_DC A)_C A
- :DispGraph:Pause:ClrHome
- :Disp ÒTHE POINT OFÓ
- :Disp ÒINTERSECTION ISÓ
- :Disp ÒX=Ó,[A](1,3),ÒY=Ó,[A](2,3)
- :Stop
- :Lbl 1
- :If [A](2,3)_0
- :Then
- :Disp ÒINFINITELY MANYÓ
- :Disp ÒSOLUTIONSÓ
- :Else
- :Disp ÒINCONSISTENTÓ
- :Disp ÒSYSTEMÓ
- :End

Visualizing Row Operations ProgramThis 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 ProgramThis 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

- :Radian
- :ClrDraw:FnOff
- :Param:Simul
- :-2.25→Xmin
- :_ 2→Xmax
- :3→Xscl
- :-1.19→Ymin
- :1.19→Ymax
- :1→Yscl
- :0→Tmin
- :6.3→Tmax
- :.15→Tstep
- :Ò-1.25_cos (T)Ó→X1T
- :Òsin (T)Ó→Y1T
- :ÒT 4Ó→X2T
- :Òsin (T)Ó→Y2T
- :DispGraph
- :For(N,1,12)
- :N_ 6.5→T
- :-1.25_cos (T)→A
- :sin(T)→B
- :T 4→C
- :Line(A,B,C,B)
- :Pause
- :End
- :Pause :Func
- :Sequential:Disp

Finding the Angle Between Two VectorsProgramThis 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

- :ClrHome
- :Degree
- :Disp ÒENTER (A,B)Ó
- :Input ÒENTER AÓ,A
- :Input ÒENTER BÓ,B
- :ClrHome
- :Disp ÒENTER (C,D)Ó
- :Input ÒENTER CÓ,C
- :Input ÒENTER DÓ,D
- :Line(0,0,A,B)
- :Line(0,0,C,D)
- :Pause
- :AC_BD→E
- : (A2 _B2)→U
- : (C2 _D2)→V
- :cos-1(E (UV))→_
- :ClrDraw:ClrHome
- :Disp Ò_=Ó,_
- :Stop

Adding Vectors Graphically ProgramThis 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

- :ClrDraw
- :Input ÒENTER AÓ,A
- :Input ÒENTER BÓ,B
- :Input ÒENTER CÓ,C
- :Input ÒENTER DÓ,D
- :Line(0,0,A,B)
- :Line(0,0,C,D)
- :A_C→E
- :B_D→F
- :Line(0,0,E,F)
- :Line(A,B,E,F)
- :Line(C,D,E,F)
- :Pause
- :Stop
- ENTER
- ENTER