Texas Instruments TI-89 user guide: Base

Base

Lets you perform calculations by entering numbers in decimal, binary, or hexadecimal form.

Unit System

Lets you enter a unit for values in an expression, such as 6_m * 4_m or 23_m/_s * 10_s, convert values from one unit to another within the same category, and create your own user-defined units.

Custom Units

Lets you select custom defaults. This mode is dimmed until you select Unit System, 3:CUSTOM.

Language

Lets you localize the TI-89 / TI-92 Plus into one of several languages, depending on which language Flash applications are installed.

554 Appendix B: Reference Information

TI-89 / TI-92 Plus Character Codes

The char() function lets you refer to any character by its numeric character code. For example, to display ◆ on the Program I/O screen, use Disp char(127). You can use ord() to find the numeric code of a character. For example, ord("A") returns 65.

Appendix B: Reference Information 555

TI-89 Key Codes

The getKey() function returns a value that corresponds to the last key pressed, according to the tables shown in this section. For example, if your program contains a getKey() function, pressing [2nd] [F6] will return a value of 273.

Table 1: Key Codes for Primary Keys

556 Appendix B: Reference Information

Table 1: Key Codes for Primary Keys (Continued)

Table 2: Arrow Keys (including diagonal movement)

Appendix B: Reference Information 557

Table 3: Greek Letters (prefixed by ◆ [α])

558 Appendix B: Reference Information

TI-92 Plus Key Codes

The getKey() function returns a value that corresponds to the last key pressed, according to the tables shown in this section. For example, if your program contains a getKey() function, pressing [2nd] [F1] will return a value of 268.

Table 1: Key Codes for Primary Keys

Appendix B: Reference Information 559

Table 1: Key Codes for Primary Keys (Continued)

560 Appendix B: Reference Information

Table 2: Arrow Keys

Note: The Grab (☐) modifier only affects the arrow keys.

Table 3: Grave Letters (prefixed by 2nd A)

Table 4: Cedilla Letters (prefixed by 2nd C)

Table 5: Acute Accent Letters (prefixed by 2nd E)

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Table 6: Greek Letters (prefixed by 2nd G)

Table 7: Tilde Letters (prefixed by 2nd N)

Table 8: Caret Letters (prefixed by 2nd O)

Table 9: Umlaut Letters (prefixed by 2nd U)

562 Appendix B: Reference Information

Entering Complex Numbers

You can enter complex numbers in the polar form (r∠θ), where r is the magnitude and θ is the angle, or polar form re^iθ. You can also enter complex numbers in rectangular form a+bi

Overview of Complex Numbers

A complex number has real and imaginary components that identify a point in the complex plane. These components are measured along the real and imaginary axes, which are similar to the x and y axes in the real plane.

The point can be expressed in rectangular form or in either of two polar forms.

The i symbol represents the imaginary number √-1.

As shown below, the form that you can enter depends on the current Angle mode.

Use the following methods to enter a complex number.

Note: To get the i symbol, press [2nd] [i], do not simply type an alphabetic i.

Appendix B: Reference Information 563

Important: Do not use the re^i\theta polar form in Degree angle mode. It will cause a Domain error.

Note: To get the $e$ symbol, press: TI-89: [♦] [eˣ]. TI-92 Plus: [2nd] [eˣ] Do not simply type an alphabetic e.

Tip: To get the \angle symbol, press [2nd] [\angle].

Tip: To enter \theta in degrees for (r\angle\theta), you can type a ° symbol (such as 45°). To get the ° symbol, press [2nd] [°]. You should not use degrees for re^i\theta.

Parentheses are required for the (r\angle\theta) form.

For example:

Results are shown in rectangular form, but you can select polar form.

Complex Format Mode for Displaying Results

Use [MODE] to set the Complex Format mode to one of three settings.

You can enter a complex number at any time, regardless of the Complex Format mode setting. However, the mode setting determines how results are displayed.

Note: You can enter complex numbers in any form (or a mixture of all forms) depending on the Angle mode.

564 Appendix B: Reference Information

Using Complex Variables in Symbolic Calculations

Regardless of the Complex Format mode setting, undefined variables are treated as real numbers. To perform complex symbolic analysis, you can use either of the following methods to set up a complex variable.

Method 1: Use an underscore _ (TI-89: [●] [] TI-92 Plus: [2nd] []) as the last character in the variable name to designate a complex variable. For example:

Note: For best results in calculations such as cSolve() and cZeros(), use Method 1.

z_ is treated as a complex variable (unless z already exists, in which case it retains its existing data type).

Method 2: Define a complex variable. For example:

x+yi \Rightarrow z

Then z is treated as a complex variable.

Complex Numbers and Degree Mode

Radian angle mode is recommended for complex number calculations. Internally, the TI-89 / TI-92 Plus converts all entered trig values to radians, but it does not convert values for exponential, logarithmic, or hyperbolic functions.

Note: If you use Degree angle mode, you must make polar entries in the form (r∠θ). In Degree angle mode, an re^i\theta entry causes an error.

In Degree angle mode, complex identities such as e^(i\theta) = \cos(\theta) + i \sin(\theta) are not generally true because the values for cos and sin are converted to radians, while those for $e^{()}$ are not. For example, e^(i45) = \cos(45) + i \sin(45) is treated internally as e^(i45) = \cos(\pi/4) + i \sin(\pi/4). Complex identities are always true in Radian angle mode.

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Accuracy Information

To maximize accuracy, the TI-89 / TI-92 Plus carries more digits internally than it displays.

Computational Accuracy

Floating-point (decimal) values in memory are stored using up to 14 digits with a 3-digit exponent.

• For min and max Window variables (xmin, xmax, ymin, ymax, etc.), you can store values using up to 12 digits. Other Window variables use 14 digits.
• When a floating-point value is displayed, the displayed value is rounded as specified by the applicable mode settings (Display Digits, Exponential Format, etc.), with a maximum of 12 digits and a 3-digit exponent.
• RegEQ displays up to 14-digit coefficients.

Integer values in memory are stored using up to 614 digits.

Graphing Accuracy

The Window variable xmin is the center of the leftmost pixel used, and xmax is the center of the rightmost pixel used. \Deltax is the distance between the centers of two horizontally adjacent pixels.

Note: For a table that lists the number of pixels in a full screen or split screen, refer to "Setting and Exiting the Split Screen Mode" in Chapter 14.

• \Deltax is calculated as (xmax – xmin) / (# of x pixels – 1).
• If \Deltax is entered from the Home screen or a program, xmax is calculated as xmin + \Deltax * (# of x pixels – 1).

The Window variable ymin is the center of the bottom pixel used, and ymax is the center of the top pixel used. \Deltay is the distance between the centers of two vertically adjacent pixels.

• \Deltay is calculated as (ymax – ymin) / (# of y pixels – 1).
• If \Deltay is entered from the Home screen or a program, ymax is calculated as ymin + \Deltay * (# of y pixels – 1).

Cursor coordinates are displayed as eight characters (which may include a negative sign, decimal point, and exponent). The coordinate values (xc, yc, zc, etc.) are updated with a maximum of 12-digit accuracy.

566 Appendix B: Reference Information

System Variables and Reserved Names

This section lists the names of system variables and reserved function names that are used by the TI-89 / TI-92 Plus. Only those system variables and reserved function names that are identified by an asterisk (*) can be deleted by using DelVar var on the entry line.

Graph

y1(x)–y99(x)*
xt1(t)–xt99(t)*
ui1–ui99*
tc
xfact
xmax
ymax
Δx
zscl
ncontour
tmin
tplot
Estep
nmax

y1'(t)–y99'(t)*
yt1(t)– yt99(t)*
xc
rc
yfact
xscl
yscl
Δy
eyeθ
θmin
tmax
ncurves
fldpic
plotStrt

yi1–yi99*
z1(x,y)–z99(x,y)*
yc
θc
zfact
xgrid
ygrid
zmin
eyeφ
θmax
tstep
dif tol
fldres
plotStep

r1(θ)–r99(θ)*
u1(n)–u99(n)*
zc
nc
xmin
ymin
xres
zmax
eyeψ
θstep
t0
dtime
nmin
sysMath

Graph Zoom

zxmin
zymin
zxres
ztmin
ztmaxde
zzmax
zeyeψ
zpltstep

zxmax
zymax
zθmin
ztmax
ztstepde
zzscl
znmin

zxscl
zyscl
zθmax
ztstep
ztplotde
zeyeθ
znmax

zxgrid
zygrid
zθstep
zt0de
zzmin
zeyeφ
zpltstrt

Statistics

x̄
Σx²
Σy²
medStat
medy1
minY
regCoef*
Sx

ȳ
Σxy
corr
medx1
medy2
nStat
regEq(x)*
Sy

Σx
Σy
maxX
medx2
medy3
q1
seed1
R²

σx
σy
maxY
medx3
minX
q3
seed2

Table

tblStart
Δtbl
tblInput

Data/Matrix

c1–c99
sysData*

Miscellaneous

main
ok
errornum

Solver

eqn*
exp*

Appendix B: Reference Information 567

EOS (Equation Operating System) Hierarchy

This section describes the Equation Operating System (EOS™) that is used by the TI-89 / TI-92 Plus. Numbers, variables, and functions are entered in a simple, straightforward sequence. EOS evaluates expressions and equations using parenthetical grouping and according to the priorities described below.

Order of Evaluation

\begin{tabular}{ll} Level & Operator \ \hline 1 & Parentheses ( ), brackets [ ], braces { } \ 2 & Indirection (#) \ 3 & Function calls \ 4 & Post operators: degrees-minutes-seconds (°,''), factorial (!), percentage (%), radian (^\Gamma), subscript ([ ]), transpose (^\intercal) \ 5 & Exponentiation, power operator (^) \ 6 & Negation (-) \ 7 & String concatenation (&) \ 8 & Multiplication (*), division (/) \ 9 & Addition (+), subtraction (-) \ 10 & Equality relations: equal (=), not equal (\neq or /=), less than (<), less than or equal (\leq or <=), greater than (>), greater than or equal (\geq or >=) \ 11 & Logical not \ 12 & Logical and \ 13 & Logical or, exclusive logical xor \ 14 & Constraint "with" operator (|) \ 15 & Store (\rightarrow) \ \end{tabular}

Parentheses, Brackets, and Braces All calculations inside a pair of parentheses, brackets, or braces are evaluated first. For example, in the expression 4(1+2), EOS first evaluates the portion of the expression inside the parentheses, 1+2, and then multiplies the result, 3, by 4.

The number of opening and closing parentheses, brackets, and braces must be the same within an expression or equation. If not, an error message is displayed that indicates the missing element. For example, (1+2)/(3+4 will display the error message "Missing )."

Note: Because the TI-89 / TI-92 Plus allows you to define your own functions, a variable name followed by an expression in parentheses is considered a "function call" instead of implied multiplication. For example a(b+c) is the function a evaluated by b+c. To multiply the expression b+c by the variable a, use explicit multiplication: a*(b+c).

568 Appendix B: Reference Information

Indirection

The indirection operator (#) converts a string to a variable or function name. For example, #("x"&"y"&"z") creates the variable name xyz. Indirection also allows the creation and modification of variables from inside a program. For example, if 10>r and "r">s1, then #s1=10.

Post Operators

Post operators are operators that come directly after an argument, such as 5!, 25%, or 60° 15' 45". Arguments followed by a post operator are evaluated at the fourth priority level. For example, in the expression 4^3!, 3! is evaluated first. The result, 6, then becomes the exponent of 4 to yield 4096.

Exponentiation

Exponentiation (^) and element-by-element exponentiation (.^) are evaluated from right to left. For example, the expression 2^3^2 is evaluated the same as 2^(3^2) to produce 512. This is different from (2^3)^2, which is 64.

Negation

To enter a negative number, press [(-)] followed by the number. Post operations and exponentiation are performed before negation. For example, the result of ¬x² is a negative number, and ¬9² = -81. Use parentheses to square a negative number such as (-9)² to produce 81. Note also that negative 5 (-5) is different from minus 5 (-5), and ¬3! evaluates as -(3!).

Constraint (l)

The argument following the "with" (l) operator provides a set of constraints that affect the evaluation of the argument preceding the "with" operator.

Appendix B: Reference Information 569

Regression Formulas

This section describes how the statistical regressions are calculated.

Least-Squares Algorithm

Most of the regressions use non-linear recursive least-squares techniques to optimize the following cost function, which is the sum of the squares of the residual errors:

$J = \sum_i=1^N [residualExpression]^2$

where: residualExpression is in terms of $x_i$ and $y_i$ $x_i$ is the independent variable list $y_i$ is the dependent variable list $N$ is the dimension of the lists

This technique attempts to recursively estimate the constants in the model expression to make $J$ as small as possible.

For example, y=a \sin(bx+c)+d is the model equation for SinReg. So its residual expression is:

a \sin(bx_i+c)+d - y_i

For SinReg, therefore, the least-squares algorithm finds the constants $a$, $b$, $c$, and $d$ that minimize the function:

$J = \sum_i=1^N [a \sin(bx_i + c) + d - y_i]^2$

Regressions

570 Appendix B: Reference Information

Appendix B: Reference Information 571

Contour Levels and Implicit Plot Algorithm

Contours are calculated and plotted by the following method. An implicit plot is the same as a contour, except that an implicit plot is for the z=0 contour only.

Algorithm

Based on your x and y Window variables, the distance between xmin and xmax and between ymin and ymax is divided into a number of grid lines specified by xgrid and ygrid. These grid lines intersect to form a series of rectangles.

For each rectangle, the equation is evaluated at each of the four corners (also called vertices or grid points) and an average value (E) is calculated:

$E = z_1 + z_2 + z_3 + z_4/4$

The E value is treated as the value of the equation at the center of the rectangle.

For each specified contour value ($C_i$):

• At each of the five points shown to the right, the difference between the point's z value and the contour value is calculated.
• A sign change between any two adjacent points implies that a contour crosses the line that joins those two points. Linear interpolation is used to approximate where the zero crosses the line.
• Within the rectangle, any zero crossings are connected with straight lines.
• This process is repeated for each contour value.

Each rectangle in the grid is treated similarly.

572 Appendix B: Reference Information

Runge-Kutta Method

For Runge-Kutta integrations of ordinary differential equations, the TI-89 / TI-92 Plus uses the Bogacki-Shampine 3(2) formula as found in the journal Applied Math Letters, 2 (1989), pp. 1–9.

Bogacki-Shampine 3(2) Formula

The Bogacki-Shampine 3(2) formula provides a result of 3rd-order accuracy and an error estimate based on an embedded 2nd-order formula. For a problem of the form:

$y' = f(x, y)$

and a given step size $h$, the Bogacki-Shampine formula can be written:

$F_1 = f(x_n, y_n)$

F_2 = f\left(x_n + h 1/2, y_n + h 1/2 F_1\right)

F_3 = f\left(x_n + h 3/4, y_n + h 3/4 F_2\right)

y_n+1 = y_n + h \left(2/9 F_1 + 1/3 F_2 + 4/9 F_3\right)

$x_{n+1} = x_n + h$

$F_4 = f(x_{n+1}, y_{n+1})$

errest = h \left(5/72 F_1 - 1/12 F_2 - 1/9 F_3 + 1/8 F_4\right)

The error estimate errest is used to control the step size automatically. For a thorough discussion of how this can be done, refer to Numerical Solution of Ordinary Differential Equations by L. F. Shampine (New York: Chapman & Hall, 1994).

The TI-89 / TI-92 Plus software does not adjust the step size to land on particular output points. Rather, it takes the biggest steps that it can (based on the error tolerance difftol) and obtains results for x_n ≤ x ≤ x_n+1 using the cubic interpolating polynomial passing through the point $(x_n, y_n)$ with slope $F_1$ and through $(x_{n+1}, y_{n+1})$ with slope $F_4$. The interpolant is efficient and provides results throughout the step that are just as accurate as the results at the ends of the step.

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574 Appendix B: Reference Information

Service and Warranty Information

C

• Battery Information ........ 576
• In Case of Difficulty ........ 579
• Support and Service Information ........ 580
• Warranty Information ........ 581

This appendix provides supplemental information that may be helpful as you use the TI-89 / TI-92 Plus. It includes procedures that may help you correct problems with the TI-89 / TI-92 Plus, and it describes the service and warranty provided by Texas Instruments.

When the BATT indicator appears in the status line, it is time to change the batteries.

Appendix C: Service and Warranty Information 575

Battery Information

The TI-89 / TI-92 Plus uses two types of batteries: four alkaline batteries, and a lithium battery as a backup for retaining memory while you change the alkaline batteries.

When to Replace the Batteries

Note: The TI-89 uses four AAA size alkaline batteries. The TI-92 Plus uses four AA size alkaline batteries.

As the alkaline batteries run down, the display will begin to dim (especially during calculations). To compensate for this, you will need to adjust the contrast to a higher setting. If you find it necessary to increase the contrast setting frequently, you will need to replace the alkaline batteries. To assist you, a BATT indicator (BATT) will display in the status line area when the batteries have drained down to the point when you should replace them soon. When the BATT indicator is displayed in reverse text (▓BATT▓), you must replace the alkaline batteries immediately.

Note: To avoid loss of information stored in memory, the TI-89 / TI-92 Plus must be off. Do not remove the alkaline batteries and the lithium battery at the same time.

BATT indicator

To avoid loss of data, do not remove the lithium battery unless four fresh alkaline batteries are installed. Replace the lithium backup battery about every three or four years.

Effects of Replacing the Batteries

If you do not remove both types of batteries at the same time or allow them to run down completely, you can change either type of battery without losing anything in memory.

Battery Precautions

Take these precautions when replacing batteries:

• Do not leave batteries within the reach of children.
• Do not mix new and used batteries. Do not mix brands (or types within brands) of batteries.
• Do not mix rechargeable and non-rechargeable batteries.
• Install batteries according to polarity (+ and -) diagrams.
• Do not place non-rechargeable batteries in a battery recharger.
• Properly dispose of used batteries immediately.
• Do not incinerate or dismantle batteries.

576 Appendix C: Service and Warranty Information

Replacing the Alkaline Batteries in the TI-89

1. If the TI-89 is on, turn it off (press [2nd] [OFF]) to avoid loss of information stored in memory.
2. Slide the protective cover over the keyboard.
3. Holding the calculator upright, push down on the battery cover latch, and then remove the cover.
4. Remove all four discharged AAA batteries.
5. Install four new AAA alkaline batteries, arranged according to the polarity (+ and -) diagram inside the battery compartment.

6. Replace the battery cover by inserting the two prongs into the two slots at the bottom of the battery compartment, and then push the cover until the latch snaps closed.

Replacing the Lithium Battery in the TI-89

To replace the lithium backup battery, remove the battery cover and unscrew the tiny screw holding the BACK UP BATTERY cover in place.

Remove the old battery and install a new CR1616 or CR1620 battery, positive (+) side up. Replace the cover and the screw.

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Replacing the Alkaline Batteries in the TI-92 Plus

1. If the TI-92 Plus is on, turn it off (press 2nd [OFF]) to avoid loss of information stored in memory.

2. Holding the TI-92 Plus unit upright, slide the latch on the top of the unit to the left unlocked position; slide the rear cover down about one-eighth inch and remove it from the main unit.

3. Remove all four discharged AA batteries.

4. Install four new AA batteries as shown on the polarity diagram located in the battery compartment.

5. Replace the rear cover, and slide the latch on the top of the TI-92 Plus to the locked position to lock the cover back in place.

6. Turn the TI-92 Plus on, and adjust the display contrast, if necessary.

Replacing the Lithium Battery in the TI-92 Plus

To replace the lithium backup battery, remove the back cover from the unit and unscrew the tiny screw holding the lithium battery cover in place.

Remove the old battery and install a new CR2032, positive (+) side up. Replace the cover and the screw.

578 Appendix C: Service and Warranty Information