HP 32SII
RPN Scientific Calculator
Owner’s Manual
HP Part No. 00032–90068
Printed in Singapore
Edition 5
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Contents
Part 1. Basic Operation
1. Getting Started
Important Preliminaries ................................................... 1–1
Turning the Calculator On and Off.............................. 1–1
Adjusting Display Contrast......................................... 1–1
Highlights of the Keyboard an Display.............................. 1–1
Shifted Keys............................................................. 1–1
Alpha Keys.............................................................. 1–2
Backspacing and Clearing......................................... 1–2
Using Menus ........................................................... 1–4
Exiting Menus .......................................................... 1–7
Annunciator............................................................. 1–7
Keying in Numbers ........................................................ 1–9
Making Numbers Negative...................................... 1–10
Exponent of Ten...................................................... 1–10
Understanding Digit Entry........................................ 1–11
Range Number and OVERFLOW ............................ 1–12
Doing Arithmetic.......................................................... 1–12
One–Number Functions........................................... 1–12
Two–Number Functions............................................ 1–13
Controlling the Display Format....................................... 1–14
Periods and Commas in Numbers ............................. 1–14
Contents
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Number of Decimal Places....................................... 1–15
SHOWing Full 12–Digit Precision ......................... 1–16
Fractions..................................................................... 1–17
Entering Fractions ................................................... 1–17
Displaying Fractions................................................ 1–19
Messages................................................................... 1–19
Calculator Memory ...................................................... 1–20
Checking Available Memory.................................... 1–20
Clearing All of Memory........................................... 1–20
2. The Automatic Memory Stack
What the Stack Is .......................................................... 2–1
The X–Register Is in the Display................................... 2–2
Clearing the X–Register ............................................. 2–2
Reviewing the stack................................................... 2–3
Exchanging the X– and Y–Registers in the Stack............ 2–4
Arithmetic–How the Stack Does It ..................................... 2–4
How ENTER Works................................................... 2–5
How CLEAR x Works................................................. 2–7
The LAST X Register ........................................................ 2–8
Correcting Mistakes with LAST X ................................. 2–9
Reusing Numbers with LAST X .................................. 2–10
Chain Calculations....................................................... 2–12
Work from the Parentheses Out................................. 2–12
Exercises ............................................................... 2–14
Order of Calculation............................................... 2–15
More Exercises....................................................... 2–16
2
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3. Storing Data into Variables
Storing and Recalling Numbers........................................ 3–1
Viewing a Variable without Recalling It ............................. 3–2
Reviewing Variables in the VAR Catalog............................ 3–3
Clearing Variables......................................................... 3–3
Arithmetic with Stored Variables....................................... 3–4
Storage Arithmetic .................................................... 3–4
Recall Arithmetic....................................................... 3–5
Exchanging x with Any Variable ...................................... 3–6
The Variable "i"............................................................. 3–7
4. Real–Number Functions
Exponential and Logarithmic Functions.............................. 4–1
Power Functions............................................................. 4–2
Trigonometry................................................................. 4–3
Entering π................................................................ 4–3
Setting the Angular Mode.......................................... 4–3
Trigonometric Functions.............................................. 4–4
Hyperbolic Functions ...................................................... 4–5
Percentage Functions ...................................................... 4–5
Conversion Functions...................................................... 4–7
Coordinate Conversions ............................................ 4–7
Time Conversions...................................................... 4–9
Angle Conversions.................................................. 4–10
Unit conversions ..................................................... 4–11
Probability Functions .................................................... 4–11
Contents
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Factorial................................................................ 4–11
Gamma ................................................................ 4–11
Probability Menu.................................................... 4–12
Parts of Numbers ......................................................... 4–14
Names of Function....................................................... 4–14
5. Fractions
Entering Fractions........................................................... 5–1
Fractions in the Display................................................... 5–2
Display Rules ........................................................... 5–2
Accuracy Indicators .................................................. 5–3
Longer Fractions ....................................................... 5–4
Changing the Fraction Display......................................... 5–5
Setting the Maximum Denominator.............................. 5–5
Choosing Fraction Format .......................................... 5–6
Examples of Fraction Displays..................................... 5–7
Rounding Fractions......................................................... 5–8
Fractions in Equations..................................................... 5–9
Fractions in Programs ................................................... 5–10
6. Entering and Evaluating Equations
How You Can Use Equations ........................................... 6–1
Summary of Equation Operations..................................... 6–3
Entering Equations into the Equation List ............................ 6–4
Variables in Equations............................................... 6–5
Number in Equations ................................................ 6–5
Functions in Equations ............................................... 6–6
4
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Parentheses in Equations............................................ 6–7
Displaying and Selecting Equations.................................. 6–7
Editing and Clearing Equations........................................ 6–9
Types of Equations ....................................................... 6–10
Evaluating Equations .................................................... 6–11
Using ENTER for Evaluation ..................................... 6–12
Using XEQ for Evaluation......................................... 6–14
Responding to Equation Prompts ............................... 6–14
The Syntax of Equations................................................ 6–15
Operator Precedence.............................................. 6–15
Equation Function ................................................... 6–17
Syntax Errors.......................................................... 6–20
Verifying Equations ...................................................... 6–20
7. Solving Equations
Solving an Equation ....................................................... 7–1
Understanding and Controlling SOLVE.............................. 7–5
Verifying the Result.................................................... 7–6
Interrupting a SOLVE Calculation ................................ 7–7
Choosing Initial Guesses for SOLVE............................. 7–7
For More Information.................................................... 7–11
8. Integrating Equations
Integrating Equations ( ∫ FN)............................................ 8–2
Accuracy of Integration................................................... 8–6
Specifying Accuracy ................................................. 8–6
Interpreting Accuracy ................................................ 8–7
Contents
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For More Information...................................................... 8–9
9. Operations with Comb Numbers
The Complex Stack ........................................................ 9–1
Complex Operations...................................................... 9–3
Using Complex Number in Polar Notation......................... 9–6
10. Base Conversions and Arithmetic
Arithmetic in Bases 2, 8, and 16.................................... 10–2
The Representation of Numbers...................................... 10–4
Negative Numbers ................................................. 10–4
Range of Numbers ................................................. 10–5
Windows for Long Binary Numbers........................... 10–6
SHOWing Partially Hidden Numbers ........................ 10–6
11. Statistical Operations
Entering Statistical Data................................................ 11–1
Entering One–Variable Data .................................... 11–2
Entering Two–Variable Data ..................................... 11–2
Correcting Errors in Data Entry ................................. 11–3
Statistical Calculations.................................................. 11–4
Mean ................................................................... 11–4
Sample Standard Deviation...................................... 11–6
Population Standard Deviation.................................. 11–7
Linear regression .................................................... 11–7
Limitations on Precision of Data.................................... 11–10
Summation Values and the Statistics Registers ................ 11–11
6
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Summation Statistics.............................................. 11–11
The Statistics Registers in Calculator Memory............ 11–12
Access to the Statistics Registers.............................. 11–13
Part 2. Programming
12. Simple Programming
Designing a Program ................................................... 12–2
Program Boundaries (LBL and RTN) ........................... 12–3
Using RPN and Equations in Programs....................... 12–4
Data Input and Output ............................................ 12–4
Entering a Program ...................................................... 12–5
Keys That Clear...................................................... 12–6
Function Names in Programs.................................... 12–7
Running a Program ...................................................... 12–8
Executing a Program (XEQ)...................................... 12–9
Testing a Program................................................... 12–9
Entering and Displaying Data...................................... 12–11
Using INPUT for Entering Data ............................... 12–11
Using VIEW for Displaying Data............................. 12–14
Using Equations to Display Messages...................... 12–14
Displaying Information without Stopping .................. 12–17
Stopping or Interrupting a Program............................... 12–18
Programming a Stop or Pause (STOP, PSE)................ 12–18
Interrupting a Running Program .............................. 12–18
Error Stops........................................................... 12–18
Editing Program......................................................... 12–19
Contents
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Program Memory....................................................... 12–20
Viewing Program Memory ..................................... 12–20
Memory Usage .................................................... 12–20
The Catalog of Programs (MEM)............................. 12–21
Clearing One or More Programs ............................ 12–22
The Checksum...................................................... 12–22
Nonprogrammable Functions....................................... 12–23
Programming with BASE ............................................. 12–23
Selecting a Base Mode in a Program ...................... 12–24
Numbers Entered in Program Lines.......................... 12–24
Polynomial Expressions and Horner's Method ................ 12–25
13. Programming Techniques
Routines in Programs .................................................... 13–1
Calling Subroutines (XEQ, RTN)................................ 13–2
Nested Subroutines................................................. 13–3
Branching (GTO).......................................................... 13–5
A Programmed GTO Instruction................................. 13–5
Using GTO from the Keyboard.................................. 13–6
Conditional Instructions................................................. 13–7
Tests of Comparison (x?y, x?0)................................. 13–8
Flags .................................................................... 13–9
Loops....................................................................... 13–16
Conditional Loops (GTO) ....................................... 13–16
Loops With Counters (DSE, ISG).............................. 13–17
Indirectly Addressing Variables and Labels .................... 13–20
The Variable "i" ................................................... 13–20
8
Contents
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The Indirect Address, (i)......................................... 13–21
Program Control with (i)......................................... 13–22
Equations with (i).................................................. 13–24
14. Solving and Integrating Programs
Solving a Program ....................................................... 14–1
Using SOLVE in Program............................................... 14–5
Integrating a Program................................................... 14–7
Using Integration in a Program ...................................... 14–9
Restrictions o Solving and Integrating............................ 14–10
15. Mathematics Programs
Vector Operations........................................................ 15–1
Solutions of Simultaneous Equations.............................. 15–12
Polynomial Root Finder................................................ 15–20
Coordinate Transformations......................................... 15–31
16. Statistics Programs
Curve Fitting ............................................................... 16–1
Normal and Inverse–Normal Distributions...................... 16–11
Grouped Standard Deviation....................................... 16–18
17. Miscellaneous Programs and Equations
Time Value of Money.................................................... 17–1
Prime Number Generator.............................................. 17–6
Contents
9
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Part 3. Appendixes and Regerence
A. Support, Batteries, and Service
Calculator Support.........................................................A–1
Answers to Common Questions ..................................A–1
Environmental Limits .......................................................A–2
Changing the Batteries ...................................................A–3
Testing Calculator Operation ...........................................A–4
The Self–Test .................................................................A–5
Limited One–Year Warranty ...........................................A–6
What Is Covered ......................................................A–6
What Is Not Covered................................................A–6
Consumer Transaction in the United Kingdom ...............A–7
If the Calculator Requires Service .....................................A–7
Service Charge ........................................................A–8
Shipping Instructions.................................................A–8
Warranty on Service.................................................A–8
Service Agreements..................................................A–9
Regulatory Information....................................................A–9
B. User Memory and the Stack
Managing Calculator Memory......................................... B–1
Resetting the Calculator .................................................. B–3
Clearing Memory .......................................................... B–3
The Status of Stack Lift .................................................... B–4
Disabling Operations................................................ B–5
10
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Neutral Operations................................................... B–5
The Status of the LAST X Register...................................... B–6
C. More about Solving
How SOLVE Finds a Root ................................................C–1
Interpreting Results .........................................................C–3
When SOLVE Cannot Find Root .......................................C–8
Round–Off Error ..........................................................C–14
Underflow...................................................................C–15
D. More about Integration
How the Integral Is Evaluated ..........................................D–1
Conditions That Could Cause Incorrect Results....................D–2
Conditions That Prolong Calculation Time..........................D–8
E. Messages
F. Operation Index
Index
Contents
11
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Part 1
Basic Operation
File name 32sii-Manual-E-0424
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Size : 17.7 x 25.2 cm
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1
Getting Started
Important Preliminaries
Turning the Calculator On and Off
To turn the calculator on, press
. ON is printed below the key.
†
To turn the calculator off, press
. That is, press and release the
{ ꢀ
{
shift key, then press
(which has OFF printed in blue above it). Since the
†
calculator has Continuous Memory, turning it off does not affect any
information you've stored, (You can also press z ꢀ to turn the
calculator off.)
To conserve energy, the calculator turns itself off after 10 minutes of no use. If
you see the low–power indicator (
) in the display, replace the batteries
¤
as soon as possible. See appendix A for instructions.
Adjusting Display Contrast
Display contrast depends on lighting, viewing angle, and the contrast setting.
To increase or decrease the contrast, hold down the
key and press
or
†
™
„.
Highlights of the Keyboard an Display
Shifted Keys
Each key has three functions: one printed on its face, a left–shifted
function (orange), and a right–shifted function (blue). The shifted function
Getting Started
1–1
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names are printed in orange and blue above each key. Press the appropriate
shift key ( or ) before pressing the key for the desired function. For
z
{
example, to turn the calculator off, press and release the
shift key, then
{
press
.
†
Pressing z or { turns on the corresponding
or ¡ annunciator
symbol at the top of the display. The annunciator remains on until you press
the next key. To cancel a shift key (and turn off its annunciator), press the
same shift key again.
Alpha Keys
Menu name
Letter for
Shifted
function
alphabetic key
Most keys have a letter written next to them, as shown above. Whenever
you need to type a letter (for example, a variable or a program label), the
A..Z annunciator appears in the display, indicating that the alpha keys
are "active".
Variables are covered in chapter 3; labels are covered in chapter 6.
Backspacing and Clearing
One of the first things you need to know is how to clear; how to correct
numbers, clear the display, or start over.
1–2
Getting Started
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Keys for Clearing
Description
Key
Backspace.
a
ꢀ
Keyboard–entry mode:
Erases the character immediately to the left of "_"
(the digit–entry cursor) or backs out of the current
menu. (Menus are described in "Using Menus" on
page 1–4.) If the number is completed (no cursor),
clears the entire number.
a
ꢀ
Equation–entry mode:
Erases the character immediately to the left of " "
¾
(the equation–entry cursor). If a number entry in
your equation is complete,
erases the entire
a
number. If the number is not complete,
erases
a
the character immediately to the left of "_" (the
number–entry cursor. "_" changes back to " "
¾
when number entry is complete.
also clears error messages, and deletes the
a
current program line during program entry.
Clear or Cancel.
†
Clears the displayed number to zero or cancels the
current situation (such as a menu, a message, a
prompt, a catalog, or Equation–entry or
Program–entry mode).
Getting Started
1–3
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Keys for Clearing (continued)
Description
Key
The CLEAR menu ({ } { } {Σ}
} {
º #ꢀꢁ ꢀꢂꢂ
zꢁb
Contains options for clearing x (the number in
the X–register), all Data, all variables, all of
memory, or all statistical data.
If you select {
ꢀꢂꢂ
& ꢄ
}, a new menu (
ꢃꢂꢁ ꢀꢂꢂ@
{ } { }) is displayed so you can verify your
decision before erasing everything in memory.
During program entry, {
} is replaced by
}, a new menu (
ꢀꢂꢂ
{
ꢅꢆꢇ
ꢅꢆꢇ @ & ꢄ
}. If you select {
ꢅꢆꢇ
ꢃꢂ
{ } { } ) is displayed, so you can verify
your decision before erasing all your programs.
During equation entry (either keyboard
equations or equations in program lines), the
{ } { } menu is displayed, so you
ꢃꢂꢁ ꢈꢉꢄ@ & ꢄ
can verify your decision before erasing the
equation.
If you are viewing a completed equation, the
equation is deleted with no verification.
Using Menus
There is a lot more power to the HP 32SII than what you see on the
keyboard. This is because 12 of the keys (with a shifted function name printed
on a dark–colored background above them) are menu keys. There are 14
menus in all, which provide many more functions, or more options for more
functions. Pressing a menu key (shifted) produces a menu in the
display–a series of choices.
1–5 PICTURE
1–4
Getting Started
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1. Menu choices.
2. Keys matched to menu choices.
3. Menu keys.
HP 32II Menus
Menu
Name
Menu
Description
Chapter
Numeric Functions
PARTS
4
ꢊꢅ ꢋꢅ ꢀꢌ
Number–altering functions: integer part,
fractional part, and absolute value.
PROB
L.R.
,
4
ꢃQ8T ꢅQ T ꢍ ꢁ
Probability functions: combinations,
permutations, seed, and random number.
ˆ ˆ
T P E
º ¸
Linear regression: curve fitting and linear
estimation.
º ¸ º·
11
11
11
y
x,
Arithmetic mean of statistical x– and y–values;
weighted mean of statistical x–values.
s,σ
σ
σ
Uº U¸ º ¸ꢎ
Sample standard deviation, population
standard deviation.
Q º ¸ ºꢏ ¸ꢏ º¸
SUMS
BASE
11
11
Statistical data summations.
ꢍꢈꢃ ꢐ% ꢑꢃ ꢌꢄ
Base conversions (decimal, hexadecimal,
octal, and binary).
Programming Instructions
FLAGS
13
13
13
ꢋ ꢃꢋ ꢋ @
Functions to set, clear, and test flags.
≠ ≤ > < ≥ =
Comparison tests of the X–and Y–registers.
≠ ≤ > < ≥ =
Comparison tests of the X–register and zero.
?
x y
?
x 0
Getting Started
1–5
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HP 32II Menus (continued)
Menu
Name
Menu
Description
Chapter
Other functions
MEM
1, 3, 12
QQQ)Q #ꢀꢁ ꢅꢆꢇ
Memory status (bytes of memory
available); catalog of variables; catalog
of programs (program labels).
MODES
DISP
4, 1
ꢍꢆ ꢁꢍ ꢆꢁ * 8
Angular modes and " ' or " " radix
(decimal point) convention.
)
8
I
ꢋ% ꢃ ꢈꢄ ꢀꢂꢂ
Fix, scientific, engineering, and ALL
display formats.
CLEAR
Functions to clear different portions of
memory—refer to z b in the
table on page 1–4.
1, 3,
6, 12
The following example shows you how to use a menu function:
Example:
How many permutations (n different arrangements) are possible
?
from 28 items taken four (r) at a time
Keys:
Display:
Description:
.
Displays r
28
4
š
ꢒ_
[PROB]
Displays the probability
menu.
{
FQ8T ꢅQ8T ꢍ ꢁꢎ
{
} (
)
-
Displays the result.
ꢅQ8T
ꢒꢓꢔ8ꢒꢕꢕ)ꢕꢕꢕꢕꢎ
Repeat the example for 28 items taken 2 at a time. (Result=756.)
Menus help you execute dozens of functions by guiding you to
them with menu choices. You don't have to remember the names of
1–6
Getting Started
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the functions built into the calculator nor search through the names printed on
its keyboard.
Exiting Menus
Whenever you execute a menu function, the menu automatically
disappears, as in the above example. If you want to leave a menu without
executing a function, you have three options:
ꢀ
Pressing
backs out of the 2–level CLEAR or MEM menu, one level
a
at a time. Refer to
in the table on page 1–4.
z b
ꢀ
Pressing
or
cancels any other menu.
a
†
Keys:
Display:
123
ꢔꢏꢖ_ꢎ
[PROB]
{
ꢃQ8T ꢅQ8T ꢍ ꢁꢎ
a or †
ꢔꢏꢖ)ꢕꢕꢕꢕꢎ
ꢀ
Pressing another menu key replaces the old menu with the new one.
Keys:
Display:
123
ꢔꢏꢖ_ꢎ
[PROB]
{
ꢃQ8T ꢅQ8T ꢍ ꢁꢎ
% #ꢀꢁ ꢀꢂꢂ ´ꢎ
ꢔꢏꢖ)ꢕꢕꢕꢕꢎ
z b
†
Annunciator
The symbols along the top and bottom of the display, shown in
the following figure, are called annunciators. Each one has a special
significance when it appears in the display.
picture 1–8
Getting Started
1–7
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HP 32SII Annunciator
Annunciator
Meaning
Chapter
1, 6
Upper Row:
TS
The
and
keys are
z ˜
z —
active for stepping through a list.
When in Fraction–display mode
5
(press
), only one of the
z Š
T
S
TS
" " or " " halves of the "
"'
annunciator will be turned on to
indicate whether the displayed
numerator is slightly less than or
slightly greater than its true value. If
ST
neither part of "
"' is on, the
exact value of the fraction is being
displayed.
Left shift is active.
Right shift is active.
1
1
¡
PRGM
Program–entry is active. Blinks while
program is running.
12
EQN
Equation–entry mode is active, or the
calculator is evaluating an expression
or executing an equation.
6
0 1 2 3
Indicates which flags are set (flags 4
through 11 have no annunciator.
13
4
RAD or GRAD
Radians or Grad angular mode is set.
DEC mode (default) has no
annunciator.
HEX OCT BIN
Indicates the active number base.
DEC (base 10, default) has no
annunciator.
10
1–8
Getting Started
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HP 32SII Annunciator (continued)
Annunciator
Meaning
Chapter
Lower Row:
The top–row keys on the calculator are
redefined according to the menu labels
displayed above menu pointers.
1
ž
,
There are more digits to the left or right.
1, 6
€
Use
to see the rest of a
{
decimal number; use the left and
right–scrolling keys (
,
) to see
< 6
the rest of an equation or binary
number.
Both these annunciators may appear
simultaneously in the display, indicating
that there are more characters to the left
and to the right. Press either of the
indicated menu keys (
or
) to
<
6
see the leading or trailing characters.
A..Z
The alphabetic keys are active.
3
1
£
Attention! Indicates a special condition
or an error.
Battery power is low.
A
¤
Keying in Numbers
You can key in a number that has up to 12 digits plus a 3–digit
exponent up to 499. If you try to key in a number larger than this, digit
entry halts and the
annunciator briefly appears.
£
If you make a mistake while keying in a number, press
to backspace
a
and delete the last digit, or press
to clear the whole number.
†
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Making Numbers Negative
The _ key changes the sign of a number.
ꢀ
ꢀ
To key in a negative number, type the number, then press
.
_
To change the sign of a number that was entered previously, just press
. (If the number has an exponent,
_
affects only the mantissa —
_
the non–exponent part of the number.)
Exponent of Ten
Exponents in the Display
–5
Numbers with exponents of ten (such as 4.2 × 10 are displayed with an ꢈ
preceding the exponent (such as ꢒ)ꢏꢕꢕꢕꢈ.ꢗ).
A number whose magnitude is too large or too small for the display format
will automatically be displayed in exponential form.
For example, in FIX 4 format for four decimal places, observe the effect of the
following keystrokes:
Keys:
Display:
Description:
.000062
Shows number being entered.
Rounds number to fit the display
format.
)ꢕꢕꢕꢕ ꢏ_
š
ꢕ)ꢕꢕꢕꢔꢎ
.000042
š
Automatically uses scientific
notation because otherwise no
significant digits would appear.
ꢒ)ꢏꢕꢕꢕꢈ.ꢗꢎ
Keying in Exponents of Ten
Use
(exponent) to key in numbers multiplied by powers of ten. For
`
–34
example, take Planck's constant, 6.6262 × 10
:
1. Key in the mantissa (the non–exponent part) of the number. If the mantissa
is negative, press
after keying in its digits.
_
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Keys:
Display:
6.6262
2. Press
`
) ꢏ ꢏ_
. Notice that the cursor moves behind the :
`
ꢈ
) ꢏ ꢏꢈ_ꢎ
3. Key in the exponent. (The largest possible exponent is 499.) If the
exponent is negative, press
the value of the exponent:
after you key in the E or after you key in
_
34
_
) ꢏ ꢏꢈ.ꢖꢒ_
34
For a power of ten without a multiplier, such as 10 , just press
34. The
`
calculator displays
.
ꢔꢈꢖꢒ
Other Exponent Functions
To calculate an exponent of ten (the base 10 antilogarithm), use
.
z (
To calculate the result of any number raised to a power (exponentiation), use
(see chapter 4).
0
Understanding Digit Entry
As you key in a number, the cursor (_) appears in the display. The cursor
shows you where the next digit will go; it therefore indicates that the number
is not complete.
Keys:
Display:
Description:
123
Digit entry not terminated: the number is
ꢔꢏꢖ_
not complete.
If you execute a function to calculate a result, the cursor disappears because
the number is complete — digit entry has been terminated.
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Digit entry is terminated.
Pressing š terminates digit entry. To separate two numbers, key in the
<
ꢔꢔ)ꢕꢓꢕꢗꢎ
first number, press
second number
to terminate digit, entry, and then key in the
š
123
A completed number.
š
ꢔꢏꢖ)ꢕꢕꢕꢕꢎ
ꢔꢏꢘ)ꢕꢕꢕꢕꢎ
4
Another completed number.
™
If digit entry is not terminated (if the cursor is present),
backspaces to
a
erase the last digit. If digit entry is terminated (no cursor),
acts like
a
and clears the entire number. Try it!
†
Range Number and OVERFLOW
–499
The smallest number available on the calculator is 1 × 10
. The largest
because
499
number is 9.99999999999 × 10
(displayed as
ꢔ)ꢕꢕꢕꢕꢈꢗꢕꢕ
of rounding).
ꢀ
ꢀ
If a calculation produces a result that exceeds the largest possible
499
number, 9.99999999999
10
is returned, and the warning
×
message
appears.
ꢑ#ꢈꢁꢋꢂꢑ$
If a calculation produces a result smaller that the smallest possible
number, zero is returned. No warning message appears.
Doing Arithmetic
All operands (numbers) must be present before you press a function key.
(When you press a function key, the calculator immediately executes the
function shown on that key.)
All calculations can be simplified into one–number functions and/or
two–number functions.
One–Number Functions
To use a one–number function (such as
,
3 < z :
.
, or
)
_
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1. Key in the number. ( You don't need to press
.)
š
2. Press the function key. (For a shifted function, press the appropriate
or
z
shift key first.)
{
For example, calculate 1/32 and
and change its sign.
Then square the last result
148.84
Keys:
Display:
Description:
32
Operand.
ꢖꢏ_
Reciprocal of 32.
3
ꢕ)ꢕꢖꢔꢖꢎ
148.84
z :
_
Square root of 148.84.
<
ꢔꢏ)ꢏꢕꢕꢕꢎ
ꢔꢒꢙ)ꢙꢒꢕꢕꢎ
.ꢔꢒꢙ)ꢙꢒꢕꢕꢎ
Square of 12.2.
Negation of 148.8400.
The one–number functions also include trigonometric, logarithmic,
hyperbolic, and parts–of–numbers functions, all of which are discussed in
chapter 4.
Two–Number Functions
To use a two–number function (such as
,
™ „ y p 0
,
.
,
or
{
.
S
1. Key in the first number.
2. Press
to separate the first number from the second.
š
3. Key in the second number. (Do not press
.)
š
4. Press the function key. (For a shifted function, press the appropriate shift
key first.)
Type in both cumbers (separate them by pressing
before pressing a function key.
by
š
Note
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For example:
To calculate:
Press:
Display:
123 + 3
12 – 3
12
12
12
12
3
š ™
ꢔꢗ)ꢕꢕꢕꢕꢎ
ꢓ)ꢕꢕꢕꢕꢎ
3
š „
12 3
3
š y
×
ꢖ )ꢕꢕꢎ
3
12
3
š 0
ꢔ8ꢘꢏꢙ)ꢕꢕꢕꢕꢎ
.ꢖꢘ)ꢗꢕꢕꢕꢎ
Percent change from 88 š 5 { S
to 5
The order of entry is important only for non–commutative functions such as
„,p, 0 or { S. If you type numbers in the wrong order, you
can still get the correct answer (without re–typing them) by pressing
to
Z
swap the order of the numbers on the stack. Then press the intended function
key. (This is explained in detail in chapter 2 under "Exchanging the X– and
Y–Registers in the Stack.")
Controlling the Display Format
Periods and Commas in Numbers
To exchange the periods and commas used for the decimal point (radix mark)
and digit separators in a number:
1. Press
to display the MODES menu.
z Ÿ
2. Specify the decimal point (radix mark) by pressing { } or { }.
)
8
For example, the number one million looks like:
ꢀ
if you press { } or
ꢔ8ꢕꢕꢕ8ꢕꢕꢕ8ꢕꢕꢕꢕ
)
ꢀ
if you press { }.
ꢔ)ꢕꢕꢕ)ꢕꢕꢕ8ꢕꢕꢕꢕ
8
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Number of Decimal Places
All numbers are stored with 12–digit precision, but you can select the number
of decimal places to be displayed by pressing
(the display menu).
z ž
During some complicated internal calculations, the calculator uses 15–digit
precision for intermediate results. The displayed number is rounded
according t the display format. The DISP menu gives you four options;
ꢋ% ꢃ ꢈꢄ ꢀꢂꢂ
Fixed–Decimal Format ({ })
ꢋ%
FIX format displays a number with up to 11 decimal places (11 digits to the
right of the " " or " " radix mark) if they fit. After the prompt
_ type in
)
8
ꢋꢊ%
the number of decimal places to be displayed. For 10 or 11 places, press
Œ
0 or
1.
Œ
For example, in the number
, the "7", "0", "8", and "9"
ꢔꢏꢖ8ꢒꢗ )ꢘꢕꢙꢓ
are the decimal digits you see when the calculator is set to FIX 4 display
mode.
Any number teat is too large or too small to display in the current
decimal–place setting will automatically be displayed in scientific
format.
Scientific Format ({ })
ꢃ
SCI format displays a number in scientific notation (one digit before the " "
)
or " " radix mark) with up to 11 decimal places (if they fit) and up to
8
three digits in the exponent. After the prompt,
_, type in the number
ꢃꢊ
of decimal places to be displayed. For 10 or 11 places, press
0 or
Œ
1. (The integer part of the number will always be less than 10.)
Œꢁ
For example, in the number
, the "2", "3", "4", and "6" are the
ꢔ)ꢏꢖꢒ ꢈꢗ
decimal digits you see when the calculator is set to SCI 4 display mode The
5
"5" following the "E" is the exponent of 10: 1.2346 × 10 .
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Engineering Format ({ })
ꢈꢄ
ENG format displays a number in a manner similar to scientific notation,
except that the exponent is a multiple of three (there can be up to three digits
before the " " or " " radix mark). This format is most useful for scientific and
)
8
3
engineering calculations that use units specified in multiples of I0 (such ass
micro–, milli–, and kilo–units.)
After the prompt,
_, type in the number of digits you want after the first
ꢈꢄꢆ
significant digit. For 10 or 11 places, press
0 or
1.
Œ
Œ
For example, in the number
, the "2", "3", "4", and "6" are
ꢔꢏꢖ)ꢒ ꢈꢖ
the significant digits after the first significant digit you see when the
calculator is set to ENG 4 display mode. The "3" following the "E" is the
3
(multiple of 3) exponent of 10: 123.46x 10 .
ALL Format ({
})
format displays a number as precisely as possible (12 digits maximum).
ꢀꢂꢂ
ALL
If all the digits don't fit in the display, the number is automatically displayed in
scientific format: 123,456.
SHOWing Full 12–Digit Precision
Changing the number of displayed decimal places affects what you see, but it
does not affect the internal representation of numbers. Any number stored
internally always has 12 digits.
For example, in the number 14.8745632019, you see only "14.8746"
when the display mode is set to FIX 4, but the last six digits ("632019") are
present internally in the calculator.
To temporarily display a number in full precision, press
. This
{
shows you the mantissa (but no exponent) of the number for as long as you
hold down
.
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Keys:
Display:
Description:
{
} 4
Displays four decimal places.
Four decimal places displayed.
Scientific format: two decimal
places and an exponent.
Engineering format.
z ž
ꢋ%
ꢎ
45
1.3
š
y
ꢗꢙ)ꢗꢕꢕꢕꢎ
ꢗ)ꢙꢗꢈꢔꢎ
{
ꢃ
} 2
z ž
{
ꢈꢄ
} 2
}
z ž
ꢗꢙ)ꢈꢕꢎ
ꢗꢙ)ꢗꢎ
z ž {
All significant digits; trailing
zeros dropped.
ꢀꢂꢂ
{
ꢋ%
} 4
Four decimal places, no
exponent.
z ž
ꢗꢙ)ꢗꢕꢕꢕꢎ
Reciprocal of 58.5.
3
ꢕ)ꢕꢔꢘꢔꢎ
(hold)
Shows full precision until you release
ꢔꢘꢕꢓꢒꢕꢔꢘꢕꢓꢒꢕꢎ
{
Fractions
T
he HP 32SII allows you to type in and display fractions, and to perform
orm
math operations on them. Fractions are real numbers of the f
a b/c
where a, b, and c are integers; 0 ≤ b ≤ c; and the denominator (c) must be
in the range 2 through 4095.
Entering Fractions
Fractions can be entered onto the stack at any time:
1. Key in the integer part of the number and press
. (The first
Œ
Œ
Œ
separates the integer part of the number from its fractional part.)
2. Key in the fraction numerator and press again. The second
Œ
separates the numerator from the denominator.
3. Key in the denominator, then press š or a function key to
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terminate digit entry. The number or result is formatted according to
the current display format.
The a b/c symbol under the
key is a reminder that the
key is used
Œ
Œ
twice for fraction entry.
3
For example, to enter the fractional number 12 / , press these keys:
8
Keys:
Display:
Description:
12
Enters the integer part of the number.
ꢔꢏ_ꢎ
The
key is interpreted in the normal
Œ
Œ
ꢔꢏ)_
manner.
3
Enters the numerator of the fraction (the
number is still displayed in decimal
form).
ꢔꢏ)ꢖ_ꢎ
Œ
The calculator interprets the second Œ
as a fraction and separates the
numerator from denominator.
Appends the denominator of the
fraction.
ꢔꢏ)ꢖ+_ꢎ
8
ꢔꢏ)ꢖ+ꢙ_ꢎ
Terminates digit entry; displays
the number in the current display format.
š
ꢔꢏ)ꢖꢘꢗꢕꢎ
3
If the number you enter has no integer part (for example, / ), just start the
8
number without an integer.
Keys:
Display:
Description:
3
Œ Œ
8
Enters no integer part. (3
8
Œ Œ
ꢕ ꢖ+ꢙꢎ
also works.)
Terminates digit entry; displays the
number in the current display format
(FIX 4).
š
ꢕ)ꢖꢘꢗꢕꢎ
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Displaying Fractions
Press zꢁŠ to switch between Fraction–display mode and the current
decimal display mode.
Keys:
Display:
Description:
12
3
Œ Œ
8
Displays characters as you key them in.
Terminates digit entry; displays the
number in the current display format.
Displays the number as a fraction.
ꢔꢏ ꢖ+ꢙꢎ
š
ꢔꢏ)ꢖꢘꢗꢕꢎ
z Š
ꢔꢏ ꢖ+ꢙꢎ
3
Now add 3/4 to the number in the X–register (12 / ):
8
Keys:
Display:
Description:
3
Œ Œ
4
Displays characters as, you key them
in.
ꢕ ꢖ+ꢒꢎ
Adds the numbers in the X– and
Y–registers; displays the result as a
fraction.
™
ꢔꢖ ꢔ+ꢙꢎ
z Š
Switches to current decimal display
format.
ꢔꢖ)ꢔꢏꢗꢕꢎ
Refer to chapter 5, "Fractions," for more information about using fractions.
Messages
The calculator responds to certain conditions or keystrokes by displaying a
message. The
symbol comes on to call your attention to the message.
£
ꢀ
ꢀ
To clear a message, press
or
.
†
a
To clear a message and perform another function, press any other key.
If no message appears but
that has no meaning in the current situation, such as ꢂ in Binary mode).
does, you have pressed an inactive key (a key
£
All displayed messages are explained in appendix E, "Messages."
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Calculator Memory
The HP 32SII has 384 bytes of memory in which you can store any
combination of data (variables, equations, or program lines). The memory
requirements of specific activities are given under "Managing Calculator
Memory" in appendix B.
Checking Available Memory
Pressing
displays the following menu:
zꢁXꢁ
ꢏꢔ )ꢕ #ꢀꢁ ꢅꢆꢇ
Where
is the number of bytes of memory available.
ꢏꢔ )ꢕ
Pressing the {
#ꢀꢁ
} menu key displays the catalog of variables (see
"Reviewing Variables in the VAR Catalog" in chapter 3). Pressing the {
menu key displays the catalog of programs.
}
ꢅꢆꢇ
1. To enter the catalog of variables, press {
#ꢀꢁ
} to enter the catalog of
programs, press { }.
ꢅꢆꢇ
2. To review the catalogs, press
3. To delete a variable or a program, press
or
z ˜ z —
z b
.
while viewing it in
its catalog.
4. To exit the catalog, press
.
†
Clearing All of Memory
Clearing all of memory erases all numbers, equations, and programs you've
stored. It does not affect mode and format settings. (To clear settings as
well as data, see "Clearing Memory" in appendix B.)
To clear all of memory:
1. Press z b {
}. You will then see the confirmation prompt
ꢀꢂꢂ
ꢃꢂꢁ
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{ } { }, which safeguards against the unintentional clearing of
ꢀꢂꢂ@ &
ꢄ
memory.
2. Press { } (yes).
&
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2
The Automatic Memory Stack
This chapter explains how calculations take place in the automatic memory
stack. You do not need to read and understand this material to use the
calculator, but understanding the material will greatly enhance your use of the
calculator, especially when programming.
In part 2, "Programming", you will learn how the stack can help you to
manipulate and organize data for programs.
What the Stack Is
Automatic storage of intermediate results is the reason that the HP 32SII easily
processes complex calculations, and does so without parentheses. The key to
automatic storage is the automatic, RPN memory stack.
HP's operating logic is based on an unambiguous, parentheses–free
mathematical logic known as "Polish Notation," developed by the Polish
Ł
logician Jan ukasiewicz (1878–1956).
While conventional algebraic notation places the operators between the
Ł
relevant numbers or variables, hukasiewicz's notation places them before
the numbers or variables. For optimal efficiency of the stack, we have
modified that notation to specify the operators after the numbers. Hence the
term Reverse Polish Notation, or RPN.
The stack consists of four storage locations, called registers, which are
"stacked" on top of each other. These registers—labeled X, Y, Z, and T–store
and manipulate four current numbers. The "oldest" number is stored in the T–
(top) register. The stack is the work area for calculations.
The Automatic Memory Stack
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T
Z
Y
X
0.0000
0.0000
0.0000
0.0000
“Oldest” number
Displayed
The most "recent" number is in the X–register: this is the number you see in the
display.
In programming, the slack is used to perform calculations, to temporarily store
intermediate results, to pass stored data (variables) among programs and
subroutines, to accept input, and to deliver output.
The X–Register Is in the Display
The X–register is what you see except when a menu, a message, or a
program line is being displayed. You might have noticed that several function
names include an x or y.
This is no coincidence: these letters refer to the X– and Y–registers. For
example,
raises ten to the power of the number in the X–register
z (
(the displayed number).
Clearing the X–Register
Pressing
{ } always clears the X–register to zero; it is also used
º
z b
to program this instruction. The
key, in contrast, is context–sensitive. It.
†
either clears or cancels the current display, depending on the situation: it acts
like
z b
{ } only when the X–register is displayed.
{ } when the X–register is displayed and digit entry is
º
also acts like
z b
a
º
terminated (no cursor present). It cancels other displays: menus, labeled
numbers, messages, equation entry, and program entry.
2–2
The Automatic Memory Stack
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Reviewing the stack
¶
R (Roll Down)
The (roll down) key lets you review the entire contents of the stack by
9
"rolling" the contents downward, one register at a time. You can see each
number when it enters the X–register.
Suppose the stack is filled with 1, 2, 3, 4 (press 1
š 4. Pressing 9 four times rolls the numbers all the way around and
2
3
š
š
back to where they started:
T
Z
Y
X
1
2
3
4
4
1
2
3
3
4
1
2
2
3
4
1
1
2
3
4
9
9
9
9
What was in the X–register rotates into the T–register, the contents of the
T–register rotate into the Z–register, etc. Notice that only the centents of the
registers are rolled — the registers themselves maintain their positions, and
only the X–register's contents are displayed.
µ
R (Roll Up)
The (roll up) key has a similar function to
except that it "rolls" the
{ 8
9
stack contents upward, one register at a time.
The contents of the X–register rotate into the Y–register; what was in the
T–register rotates into the X–register, and so on.
T
Z
Y
X
1
2
3
4
2
3
4
1
3
4
1
2
4
1
2
3
1
2
3
4
9
9
9
9
The Automatic Memory Stack
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Exchanging the X– and Y–Registers in the Stack
Another key that manipulates the stack contents is
(x exchange y). This
Z
key swaps the contents of the X– and Y–registers without affecting the rest of
the stack. Pressing
Y–register contents.
twice restores the original order of the X– and
Z
The
function is used primarily for two purposes:
Z
ꢀ
To view the contents of the Y–register and then return them to y (press
twice).
Z
Some functions yield two results: one in the X–register and one in the
Y–register. For example,
converts rectangular coordinates in
z q
the X– and Y–registers into polar coordinates in the X– and Y–registers.
ꢀ
To swap the order of numbers in a calculation.
For example, one way to calculate 9 ÷ (13 × 8):
Press 13 š 8 y 9 Z p
The keystrokes to calculate this expression from left–to–right are:
9
13
8
š y p
š
Note
Always make sure that there are no more than four numbers in
the stack at any given time – the contents of the T–register (the
top register) will be lost whenever a fifth number is entered.
Arithmetic–How the Stack Does It
The contents of the stack move up and down automatically as new numbers
enter the X–register (lifting the stack) and as operators combine two
numbers in the X– and Y–registers to produce one new number in the
X–register (dropping the stack).
Suppose the stack is filled with the numbers 1, 2, 3, and 4. See how
the stack drops and lifts its contents while calculating
2–4
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3 + 4 – 9
T
Z
Y
X
1
2
3
4
1
1
2
7
9
1
1
1
2
7
2
–2
™
ꢃꢁ
„
1
2
3
1. The stack "drops" its contents. The T– (top) register replicates its contents.
2. The stack "lifts" its contents. The T–register's contents are lost.
3. The stack drops.
ꢀ
Notice that when the stack lifts, it replaces the contents of the T– (top)
register with the contents of the Z–register, and that the former contents of
the T–register are lost. You can see, therefore, that the stack's memory is
limited to four numbers.
ꢀ
ꢀ
Because of the automatic movements of the stack, you do not need to
clear the X–register before doing a new calculation.
Most functions prepare the stack to lift its contents when the next number
enters the X–register. See appendix B for lists of functions that disable
stack lift.
How ENTER Works
You know that
separates two numbers keyed in one after the other. In
š
?
terms of the stack, how does it do this Suppose the stack is again filled with
1, 2, 3, and 4. Now enter and add two new numbers:
The Automatic Memory Stack
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5 + 6
1 lost
2 lost
T
Z
Y
X
1
2
3
4
2
3
4
5
3
4
5
5
3
4
5
6
3
3
4
11
ꢄꢁ
š
ꢅꢁ
™
1
2
3
4
1. Lifts the stack.
2. Lifts the stack and replicates the X–register.
3. Does not lift the stack.
4. Drops the stack n replicate the T–register.
replicates the contents of the X–register into the Y–register. The next
š
number you key in (or recall) writes over the copy of the first number left in
the X–register. The effect is simply to separate two sequentially entered
numbers.
You can use the replicating effect of
clear the stack quickly: press 0
š
. All stack registers now contain zero. Note,
š š š
however, that you don't need to clear the tech before doing calculations.
Using a Number Twice in a Row
You can use the replicating feature of
to other advantages. To add a
š
number to itself, press
š ™
Filling the to with a Constant
The re
plicating effect of
together with the replicating effect of stack
š
drop (from T into Z) allows you t fill the stack with a numeric constant for
calculations.
2–6
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Example:
Given bacterial culture with a constant growth rate of 50%, how large would
?
population of 100 be at the end 3 days
Replicates T–register
T
Z
Y
X
1.5
1.5
1.5
1.5
1.5
1.5
1.5
100
1.5
1.5
1.5
150
1.5
1.5
1.5
225
1.5
1.5
1.5
šꢁ
šꢁ
š
1.5
337.5
100
y
yꢁ
y
1
2
3
4
1
1. Fills the stack with the growth rate.
2. Keys in the initial population.
3. Calculates the population after 1 day.
4. Calculates the population after 2 days.
5. Calculates the population after 3 days.
How CLEAR x Works
Clearing the display (X–register) put zero in the X–register. The next number
you key in (or recall writes over this zero.
There are three ways to clear the contents of the X–register, that is, to clear x:
1. Press
†
2. Press
3. Press z b { } (Mainly used during program entry.)
a
º
Note these exceptions:
ꢀ
During program entry, a deletes the currently–displayed program line
and
cancels program entry.
†
ꢀ
ꢀ
During digit entry, a backspaces over the displayed number.
If the display shows a labeled number (such as
), pressing
ꢀ/ꢏ)ꢕꢕꢕꢕ
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or
cancel that display and shows the X–register.
†
a
ꢀ
ꢀ
When viewing an equation,
equation to allow for editing.
displays the cursor at the end the
a
During equation entry,
backspaces over the displayed equation,
a
one function at a time.
For example, if you intended to enter 1 and 3 but mistakenly entered 1 and 2,
this what you should do to correct your error:
T
Z
Y
X
1
1
1
2
1
0
1
3
ꢆ
1
š
ꢇ
†ꢁ
ꢂ
2
3
4
5
1
1. Lifts the stack
2. Lift the stack and replicates the X–register.
3. Overwrites the X–register.
4. Clears x by overwriting it with zero.
5. Overwrites x (replaces the zero.)
The LAST X Register
The LAST X register is a companion to the stack: it holds the number that was
in the X–register before the last numeric function was executed. (A numeric
function is an operation that produces a result from another number or
numbers, such as
X–register.
.) Pressing
returns this value into the
<
z Ž
This ability to retrieve the "last x" has two main uses:
1. Correcting errors.
2. Reusing a number in a calculation.
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See appendix B for a comprehensive list of the functions that save x in the
LAST X register.
Correcting Mistakes with LAST X
Wrong
e–Number Function
On
If you execute the wrong one–number function, use
to retrieve
z Ž
the number so you can execute the correct function. (Press † first if you
want to clear the incorrect result, from the stack.)
Since
and
don't cause the stack to drop, you can
{ P
{ S
recover from these functions in the same manner as from one–number
functions.
Example:
5
Suppose that you had just computed In 4.7839 × (3.879 × 10 ) and wanted
to find its square root, but pressed
by mistake. You don't have to start
*
over! To find the correct result, press
.
z Žꢁ<
Mistakes with a Two–number operation
If you make a mistake with a two–number operation, (
,
,
,
,
™ „ y p
or
), you can correct it by using
and inverse of the
0
.ꢁ
z Ž
two–number function (
or
,
or
,
or
).
„ꢁ ™ pꢁ y . 0ꢁ
1. Press
to recover the second number (x just before the
z Ž
operation).
2. Execute the inverse operation. This returns the number that was originally
first. The second number is still in the LAST X register. Then:
ꢀ
If you had used the wrong function, press
again to
z Ž
restore the original stack contents. Now execute the correct function.
If you had used the wrong second number, key in the correct one and
execute the function.
ꢀ
If you had used the wrong first number, keyin the correct first number, press
to recover the second number, and execute the function again.
z Ž
(Press
first if you want to clear the incorrect result from the stack.)
†
Example:
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Suppose you made a mistake while calculating
16 × 19 = 304.
There are three kinds of mistakes you could have made:
Wring
Calculation:
Mistake:
Correction:
16
19
Wrong function
š
„
z Ž ™ꢁ
z Ž y
15
16
19
18
Wrong first number 16
Wrong second
š
š
y
y
z Ž y
19
z Ž p
y
number
Reusing Numbers with LAST X
You can use z Ž to reuse a number (such as a constant) in a
calculation. Remember to enter the constant second, just before executing the
arithmetic operation, so that the constant is the last number in the X–register,
and therefore can be saved and retrieved with
z Ž
Example:
96.704+ 52.3947
Calculates
52.3947
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T
Z
Y
t
t
t
t
z
z
96.704
96.704
96.704
96.704
52.3947
z
š X
149.0987
52.3947
52.3947
™
™
LAST X
l
l
52.3947
T
t
t
Z
Y
z
t
z
149.0987
52.3947
z Ž X
2.8457
p
LAST X
52.3947
52.3947
Keys:
Display:
Description:
96.704 š
Enters first number.
ꢓ )ꢘꢕꢒꢎ
ꢔꢒꢓ)ꢕꢓꢙꢘꢎ
ꢗꢏ)ꢖꢓꢒꢘꢎ
52.3947
Intermediate result.
™
Brings back display from before
z Ž
.
™
Final result.
p
ꢏ)ꢙꢒꢗꢘꢎ
Example:
Two close stellar neighbors of Earth are Rigel Centaurus (4.3 light–years
away) and Sirius (8.7 light–years away). Use c, the speed of light (9.5
10 meters per year) to convert the distances from the Earth to these stars
into meters:
×
15
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15
To Rigel Centaurus: 4.3 yr (9.5 10 m/yr).
To Sirius: 8.7 yr × (9.5 × 10 m/yr).
×
×
15
Keys:
Display:
Description:
4.3
Light–years to Rigel Centaurus.
Speed of light, c.
š
ꢒ)ꢖꢕꢕꢕꢎ
9.5 ` 15
y
ꢓ)ꢗꢈꢔꢗꢎ
Meters to R. Centaurus.
Retrieves c.
ꢒ)ꢕꢙꢗꢕꢈꢔ ꢎ
ꢓ)ꢗꢕꢕꢕꢈꢔꢗꢎ
ꢙ)ꢏ ꢗꢕꢈꢔ ꢎ
8.7
z Ž
Meters to Sirius.
y
Chain Calculations
The
automatic lifting and dropping of the stack's contents let you retain
intermediate results without storing or reentering them, and without using
parentheses.
Work from the Parentheses Out
For example, solve (12 + 3) × 7.
If you were working out this problem on paper, you would first calculate the
intermediate result of (12 + 3) ...
(12 + 3) = 15
… then you would multiply the intermediate result by 7:
(15) × 7 = 105
Solve the problem in the same way on the HP 32SII, starting inside the
parentheses:
Keys:
Display:
Description:
12
3
š ™
Calculates the intermediate result first.
ꢔꢗ)ꢕꢕꢕꢕꢎ
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You don't need to press
to save this intermediate result before
š
proceeding; since it is a calculated result, it is saved automatically.
Keys:
Display:
Description:
Pressing the function key produces the
ꢔꢕꢗ)ꢕꢕꢕꢕꢎ
7
y
answer. This result can be used in
further calculations.
Now study the following examples. Remember that you need to press
š
only to separate ,sequentially–entered numbers, such as at the beginning
of a problem The operations themselves ( ™,„, etc.) separate
subsequent numbers and save intermediate results. The last result saved is the
first one retrieved as needed to carry out the calculation.
Calculate 2 ÷ (3 + 10):
Keys:
Display:
Description:
3
10
Calculates (3 + 10) first.
Puts 2 before 13 so the division is
correct: 2 ÷ 13.
š
™
ꢔꢖ)ꢕꢕꢕꢕꢎ
ꢕ)ꢔꢗꢖꢙꢎ
2 Z p
Calculate 4 ÷ [(14 + (7 × 3) – 2] :
Keys:
Display:
Description:
7
3
š y
Calculates (7 3).
×
ꢏꢔ)ꢕꢕꢕꢕꢎ
ꢖꢖ)ꢕꢕꢕꢕꢎ
ꢖꢖ)ꢕꢕꢕꢕꢎ
14
2
™ „
Calculates denominator.
Puts 4 before 33 in preparation for
division.
4 Z
p
Calculates 4 ÷ 33, the answer.
ꢕ)ꢔꢏꢔꢏꢎ
Problems that have multiple parentheses can be solved in the same manner
using the automatic storage of intermediate results. For example, to solve (3 +
4) (5 + 6) on paper, you would first calculate the quantity (3 + 4). Then you
×
would calculate (5 + 6). Finally, you would multiply the two intermediate
results to get the answer.
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Work through the problem the same way with the HP 32SII, except that you
don't have to write down intermediate answers—the calculator remembers
them for you.
Keys:
Display:
Description:
3
5
4
š ™
First adds (3+4)
ꢘ)ꢕꢕꢕꢕꢎ
ꢔꢔ)ꢕꢕꢕꢕꢎ
ꢘꢘ)ꢕꢕꢕꢕꢎ
6
š ™
Then adds (5+6)
Then multiplies the intermediate
answers together for the final
answer.
y
Exercises
Calculate:
(16.3805x5)
= 181.0000
0.05
Solution:
16.3805
5
š y <
.05
p
Calculate:
[(2+ 3)× (4+ 5)] + [(6+ 7)× (8+ 9) = 21.5743
Solution:
2
y < ™
3
4
5
6
7
8
9
š ™ š ™ y < š ™ š ™
Calculate:
(10 – 5) ÷ [(17 – 12) × 4] = 0.2500
Solution:
17
or
10
12
5
4
„ y
10
5
š „ Z p
š
17
12
4
„ y p
š „
š
2–14 The Automatic Memory Stack
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Order of Calculation
We recommend solving chain calculations by working from the innermost
parentheses outward. However, you can also choose to work problems in a
left–to–right order.
For example, you have already calculated:
4 ÷ [14 + (7 × 3) – 2]
by starting with the innermost parentheses (7 × 3) and working outward,
just as you would with pencil and paper. The keystrokes were 7
3
š
14
2
™ „ Z p
4
y
If you work the problem from left–to–right, press
14
4
7
š š y ™ „ p
3
2
.
š
This method takes one additional keystroke. Notice that the first intermediate
result is still the innermost parentheses (7 × 3). The advantage to working a
problem left–to–right is that you don't have to use
to reposition
Z
operands for nomcommutaiive functions (
and
).
„
p
However, the first method (starting with the innermost parentheses) is often
preferred because:
ꢀ
ꢀ
It takes fewer keystrokes.
It requires fewer registers in the stack.
Note
When using the left–to–right method, be sure that no more
than four intermediate numbers (or results) will be needed at
one time (the stack can hold no more than four numbers).
The above example, when solved left–to–right, needed all registers in the
stack at one point:
Keys:
Display:
Description:
4 š 14
š
Saves 4 and 14 as intermediate
numbers in the stack.
ꢎ
ꢔꢒ)ꢕꢕꢕꢕꢎ
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7
3
At this point the stack is full with
numbers for this calculation.
Intermediate result.
š
ꢖ_ꢎ
y
™
2
ꢏꢔ)ꢕꢕꢕꢕꢎ
ꢖꢗ)ꢕꢕꢕꢕꢎ
ꢖꢖ)ꢕꢕꢕꢕꢎ
ꢕ)ꢔꢏꢔꢏꢎ
Intermediate result.
Intermediate result.
„
Final result.
p
More Exercises
Practice using RPN by working through the following problems:
Calculate:
(14 + 12) × (18 – 12) ÷ (9 – 7) = 78.0000
A Solution:
14
12
18
12
9
„ y š „ p
7
š
™
š
Calculate:
2
23 – (13 × 9) + 1/7 = 412.1429
A Solution:
23 z : 13 š 9 y „ 7 3 ™
Calculate:
(5.4× 0.8)÷ (12.5− 0.73) = 0.5961
Solution:
5.4
or
.8
.7
3
š 0
12.5
š
y
Z „ pꢁ<
5.4 š .8 y 12.5 š .7 š 3 0 „ p <
Calculate:
8.33× (4− 5.2)÷ [(8.33− 7.46)× 0.32]
= 4.5728
4.3× (3.15− 2.75)− (1.71× 2.01)
A Solution:
2–16 The Automatic Memory Stack
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4
5.2
8.33
7.46
0.32
3.15
š
„
y z Ž
„
y p
š 2.75 „ 4.3 y 1.71 š 2.01 y „ p <ꢁ
The Automatic Memory Stack 2–17
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3
Storing Data into Variables
The HP 32II has 384 bytes of user memory: memory that you can use to store
numbers, equations, and program lines. Numbers are stored in locations
called variables, each named with a letter from A through Z. (You can choose
the letter to remind you of what is stored there, such as B for bank balance
and C for the speed of light.)
3-1 Picture
1. Cursor prompts for variable.
2. Indicates letter keys are active.
3. Letter keys.
Each white letter is associated with a key and a unique variable. The letter
keys are automatically active when needed. (The A..Z annunciator in the
display confirms this.)
Note that the variables, X, Y, Z and T are different storage locations from the
X–register, Y–register, Z–register, and T–register in the stack.
Storing and Recalling Numbers
Numbers are stored into and recalled from lettered variables with the H
(store) and
(recll) functions.
K
To store a copy of a displayed number (X–register) to a variable:
Press letter–key.
H
To recall a copy of a number from a variable to the display:
Press K letter–key.
Storing Data into Variables
3–1
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Example: Storing Numbers.
Store Avogadro's number (approximately 6.0225 × 10 ) in A.
23
Keys:
Display:
Description:
6.0225 ` 23
H
Avogadro's numbers.
)ꢕꢏꢏꢗꢈꢏꢖ_
Prompts for variable.
!ꢑ_
A (HOLD < key)
Displays function as long as key is
held down.
!ꢑ ꢀꢎ
(release)
Stores a copy of Avogadro's
numbers in A. This also terminates
digit entry (no cursor present)
Clears the number in the display.
Prompts for variable.
)ꢕꢏꢏꢗꢈꢏꢖꢎ
†
K
A
ꢕ)ꢕꢕꢕꢕꢎ
ꢁꢃꢂ_ꢎ
Copies Avogadro's numbers from A
the display.
)ꢕꢏꢏꢗꢈꢏꢖꢎ
Viewing a Variable without Recalling It
The
function shows you the contents of a variable without putting
{ ‰
that number in the X–register. The display is labeled for the variable, such as:
ꢀ/ꢔꢏꢖꢒ)ꢗ ꢘꢙꢎ
If the number is too large to fit completely in the display with its label, it is
rounded and the rightmost digits are dropped. (An exponent is displayed in
full.) To see the full mantissa, press
.
{
In Fraction–display mode (
), part of the integer may be dropped.
z Š
This will be indicated by "…" at the left end of the integer.
To see the full mantissa, press
the left of the radix ( or ).
. The integer part is the portion to
{
)
8
is most often used in programming, but it is useful anytime you
{ ‰
want to view a variable's value without affecting the contents of the stack.
3–2
Storing Data into Variables
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To cancel the VIEW display, press
or
once.
a
†
Reviewing Variables in the VAR Catalog
The
(memory) function provides information about memory:
z X
QQQ)Q #ꢀꢁ ꢅꢆꢇꢎ
where nnn.n is the number of bytes of available memory.
Pressing the {
Pressing the {
} menu key displays the catalog of variables.
} menu key displays the catalog of programs.
#ꢀꢁ
ꢅꢆꢇ
To review the values at any or all non–zero variables:
1. Press
2. Press
{VAR}.
or
z X
to move the list and display the desired
z ˜ z —
TS
variable. (Note the
annunciator, indicating that the left–shifted
˜
TS
and
keys are active, If Fraction–display mode is active,
does
—
not indicate accuracy.)
To see all the significant digits of a number displayed in the {
} catalog,
#ꢀꢁ
press
the < and 6 keys to see the rest.)
. (If it is a binary number with more than 12 digits, use
{
3. To copy a displayed variable from the catalog to the X–register, press
.
š
4. To clear a variable to zero, press
catalog.
while it is displayed in the
z b
5. Press † to cancel the catalog.
Clearing Variables
Variables' values are retained by Continuous Memory until you replace there
or clear them. Clearing a variable stores a zero there; a value of zero takes
no memory.
To clear a single variable:
Storing Data into Variables
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Store zero in it: Press 0
variable.
H
To clear selected variables:
1. Press } and use
{
or
z ˜ z —
to display the
z X
#ꢀꢁ
variable.
2. Press z b.
3. Press
to cancel the catalog.
†
To clear all variables at once:
}.
Press z b {
#ꢀꢁ
Arithmetic with Stored Variables
Storage arithmetic and recall arithmetic allow you to do calculations with a
number stored in a variable without recalling the variable into the stack. A
calculation uses one number from the X–register and one number from the
specified variable.
Storage Arithmetic
Storage arithmetic uses
,
,
, or
to do
H ™ H „ H y
H p
arithmetic in the variable itself and to store the result there. It uses the value in
the X–register and does riot affect the stack.
New value of variable = Previous value of variable {+, –, × , ÷} x.
For example, suppose you want to reduce the value in A(15) by the number in
the X–register (3, displayed). Press H „ A. Now A = 12, while 3 is still
in the display.
3–4
Storing Data into Variables
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Results: 15–3
thatis, A–x
A
A
15
12
T
Z
Y
X
T
Z
Y
X
t
t
z
y
3
z
y
3
H „ ꢈ
Recall Arithmetic
Recall arithmetic uses a
,
, or
to do arithmetic in
K ™ K y K p
the X–register using a recalled number and to leave the result in the display.
Only the X–register is affected.
New x = Previous x {+, –, ×, ÷ } Variable
For example, suppose you want to divide the number in the X–register (3,
displayed) by the value in A(12). Press K p A. Now x = 0.25, while 12
is still in A. Recall arithmetic saves memory in programs: using
A
K ™
(one instruction) uses half as much memory as
A,
(two instructions).
K
™
A
A
12
12
T
Z
Y
X
T
Z
Y
X
t
t
z
z
y
3
y
Results: 3÷12,
thatis, x÷A
0.25
K p ꢈ
Storing Data into Variables
3–5
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Example:
Suppose the variables D, E, and F contain the values 1, 2, and 3. Use storage
arithmetic to add 1 to each of those variables.
Keys:
Display:
Description:
1
2
3
1
D
Stores the assumed values into the
variable.
H
ꢔ)ꢕꢕꢕꢕꢎ
ꢏ)ꢕꢕꢕꢕꢎ
ꢖ)ꢕꢕꢕꢕꢎ
ꢎ
E
F
H
H
D
Add 1 to D, E, And F.
H ™
E
H ™ H
ꢎ
F
™
ꢔ)ꢕꢕꢕꢕꢎ
ꢍ/ꢏ)ꢕꢕꢕꢕꢎ
ꢈ/ꢖ)ꢕꢕꢕꢕꢎ
ꢋ/ꢒ)ꢕꢕꢕꢕꢎ
ꢔ)ꢕꢕꢕꢕꢎ
D
Displays the current value of D.
{ ‰
E
{ ‰
F
{ ‰
Clears the VIEW display; displays
X-register again.
@ꢁ
Suppose the variables D, E, and F contain the values 2, 3, and 4 from the last
example. Divide 3 by D, multiply it by E, and add F to the result.
Keys:
Display:
Description:
3
D
Calculates 3 D.
K p
÷
ꢔ)ꢗꢕꢕꢕꢎ
E
F
3 D E.
K y
K ™
÷
×
ꢒ)ꢗꢕꢕꢕꢎ
ꢙ)ꢗꢕꢕꢕꢎ
3 ÷ D × E + F
Exchanging x with Any Variable
The
key allows yon to exchange the contents of (the Displayed
{ Y
X –register with 1 contents of any variable. Executing this function does not
effect the Y–, Z–, or T–registers
3–6
Storing Data into Variables
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Example:
Keys:
Display:
Description:
12 H A
3
Stores 12 in variable A.
Display x.
ꢔꢏ)ꢕꢕꢕꢕꢎ
ꢖ_
{ Y A
Exchange contents of the X–register
and variable A.
ꢔꢏ)ꢕꢕꢕꢕꢎ
{ Y A
Exchange contents of the X–register
and variable A.
ꢖ)ꢕꢕꢕꢕꢎ
A
A
12
3
T
T
Z
Y
X
t
t
Z
z
y
3
z
Y
y
X
12
{ Y ꢈ
The Variable "i"
There is a 27th variables that you can access directly–the variable i. The Œ
key is labeled "i", and it means i whenever the A..Z annunciator is on.
Although it stores numbers as other variables do, i is special in that it can be
used to refer to other variables, including the statistics registers, using the (i)
function. This is a programming technique called indirect addressing that is
covered under "Indirectly Addressing variables and labels" in chapter 13.
Storing Data into Variables
3–7
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4
Real–Number Functions
This chapter covers most of the calculator's functions that perform
computations on real numbers, including some numeric functions used in
programs (such as ABS, the absolute–value function):
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
Exponential and logarithmic functions.
Power functions. ( 0 and .)
Trigonometric functions.
Hyperbolic functions.
Percentage functions.
Conversion functions for coordinates, angles, and units.
Probability functions.
Parts of numbers (number–altering functions).
Arithmetic functions and calculations were covered in chapters 1 and 2.
Advanced numeric operations (root–finding, integrating, complex numbers,
base conversions, and statistics) are described in later chapters.
All the numeric functions are on keys except for the probability and
parts–of–numbers functions.
The probability functions (
,
ꢃQ8T ꢅQ8T ꢍ
,
, and ) are in the PROB menu
ꢁ
(press { [PROB]).
The–parts–of numbers functions( , , and,
) are in PARTS menu (press
ꢊꢅ ꢋꢅ
ꢀꢌ
{ [PARS]).
Exponential and Logarithmic Functions
Put the number in the display, then execute the function — there is no need to
press
.
š
Real–Number Functions
4–1
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To Calculate:
Press:
Natural logarithm (base e)
Common logarithm (base 10)
Natural exponential
-
z +
*
Common exponential (antilogarithm)
z (
Power Functions
To calculate the square of a number x, key in x and press z :.
To calculate a power x of 10, key in x and press
.
z (
'To calculate a number y raised to a power x, key in y
x, then press
š
.(For y > 0, x can be any rational number; for y < 0, x must be are integer;
0
for y = 0, x must be positive.)
To Calculate:
Press:
Result:
2
15
10
15
6
z :
ꢏꢏꢗ)ꢕꢕꢕꢕꢎ
ꢔ8ꢕꢕꢕ8ꢕꢕꢕ)ꢕꢕꢕꢕꢎ
ꢖꢗ)ꢕꢕꢕꢕꢎ
6
z (
4
5
5
2
4
š 0
–1.4
2
1.4
š
_ 0
ꢕ)ꢖꢘꢙꢓꢎ
3
(–1.4)
1.4
3
_ š 0
.ꢏ)ꢘꢒꢒꢕꢎ
th
To calculate a root x of a number y (the x root of y), key in y
x, then
š
press
. For y<0, x must be an integer.
z .
To Calculate:
Press:
Result:
125
125
3
_ š z .
3
.ꢗ)ꢕꢕꢕꢕꢎ
ꢗ)ꢕꢕꢕꢕꢎ
−125
3
š z .
3 125
4–2
Real–Number Functions
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.37893
1.4
š
_ z .
ꢏ)ꢕꢕꢕꢕꢎ
−1.4 .37893
Trigonometry
Entering π
Press
to place the first 12 digits of π into the X–register.
{ M
(The number displayed depends on the display format.) Because π is a
function, it doesn't need to be separated from another number by š.
Note that calculator cannot exactly represent π, since π is an irrational
number.
Setting the Angular Mode
The angular rode specifies which unit of measure do assume for angles used
in trigonometric functions. The mode does not convert numbers already
present (see "Conversion Functions" later in this chapter)
360 degrees = 2π radians = 400 grads
To set, an angular mode, press z Ÿ. A menu will be displayed from
which you can select an option.
Option
Description
Annunciator
none
{
ꢍꢆ
}
Sets Degrees mode (DEG). Uses decimal
degrees, not degrees, minutes, and
seconds.
{
{
}
}
Sets Radians mode (RAD).
Sets Grads mode (GRAD).
RAD
ꢁꢍ
GRAD
ꢆꢁ
Real–Number Functions
4–3
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Trigonometric Functions
With x in the display:
To Calculate:
Press:
Sine of x.
N
Cosine of x.
Q
Tangent of x.
Arc sine of x.
Arc cosine of x.
Arc tangent of x.
T
z L
z O
z R
Note
Calculations with the irrational number π cannot be expressed
exactly by the 12–digit internal precision of the calculator. This
is particularly noticeable in trigonometry. For example, the
–13
calculated sin π (radians) is not zero but –2.0676 × 10 , a
very small number close to zero.
Example:
Show that cosine (5 7) radians and cosine 128.57° are equal (to four
÷
π
significant digits).
Keys:
Display:
Description:
{
ꢁꢍ
}
Sets Radians mode; RAD
annunciator on.
z Ÿ
ꢎ
5
Œ Œ š
7
5 ÷ 7 in decimal format.
ꢕ)ꢘꢔꢒꢖꢎ
.ꢕ) ꢏꢖꢗꢎ
.ꢕ) ꢏꢖꢗꢎ
Cos (5/7) .
{ M y Q
π
{
ꢍꢆ
}
Switches to Degrees mode (no
annunciator).
z Ÿ
128.57
Calculates cos 128.57°, which is
the same as cos (5/7)π.
Q
.ꢕ) ꢏꢖꢗꢎ
4–4
Real–Number Functions
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Programming Note:
Equations using inverse trigonometric functions to determine an angle , often
θ
look something like this:
= arctan (y x).
θ
/
If x = 0, then y/x is undefined, resulting in the error:
. For a
ꢍꢊ#ꢊꢍꢈ ꢌ& ꢕ
program, then, it would be more reliable to determine θ by a rectangular– to
polar conversion, which converts (x,y) to (r,θ). See "Coordinate Conversions"
later in this chapter.
Hyperbolic Functions
With x in the display:
To Calculate
Press:
Hyperbolic sine of x (SINH).
z 7 N
Hyperbolic cosine of x (COSH).
Hyperbolic tangent of x (TANH).
Hyperbolic arc sine of x (ASINH).
Hyperbolic arc cosine of x (ACOSH).
Hyperbolic arc tangent of x (ATANH).
z 7 Q
z 7 T
z 7 z L
z 7 z O
z 7 z R
Percentage Functions
The percentage functions are special (compared with
and
) because
y
p
they preserve the value of the base number (in the Y–register) when they
return the result of the percentage calculation (in the X–register). You can then
carry out subsequent calculations using both the base number and the result
without reentering the base number.
Real–Number Functions
4–5
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To Calculate
Press:
x% of y
y
x
š { P
Percentage change from y to x. (y≠ 0) y
x
š { S
Example:
Find the sales tax at 6% and the total cost of a $15.76 item.
Use FIX 2 display format so the costs are rounded appropriately.
Keys:
Display:
Description:
{
ꢋ%
} 2
Rounds display to two decimal
places.
z ž
ꢎ
15.76
6
š
ꢔꢗ)ꢘ ꢎ
ꢕ)ꢓꢗꢎ
ꢔ )ꢘꢔꢎ
Calculates 6% tax.
{ P
Total cost (base price + 6% tax).
™
Suppose that the $15.76 item cost $16.12 last year. What is the percentage
?
change from last year's price to this year's
Keys:
Display:
Description:
16.12
15.76
š
ꢔ )ꢔꢏꢎ
.ꢏ)ꢏꢖꢎ
This year's price dropped about
2.2% from last year's price.
Restores FIX 4 format.
{ S
{
ꢋ%
} 4
z ž
.ꢏ)ꢏꢖꢖꢖꢎ
Note
The order of the two numbers is important for the %CHG
function. The order affects whether the percentage change is
considered positive or negative.
4–6
Real–Number Functions
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Conversion Functions
There are four types of conversions: coordinate (polar/rectangular), angular
(degrees/radians), time (decimal/minutes–seconds), and unit (cm/in, °C/°F,
l/gal, Kg/lb).
Coordinate Conversions
ꢁ
, ꢁ
The function names for these conversions are y,x ,r and r y,x.
θ
θ
,
Polar coordinates (r θ) and rectangular coordinates (x,y) are measured as
shown in the illustration. The angle uses units set by the current angular
θ
mode. A calculated result for θ will be between –180° and 180°, between –π
and π radians, or between –200 and 200 grads.
x
r
y
θ
To convert between rectangular and polar coordinates:
1. Enter the coordinates (in rectangular or polar form) that you want to
convert. The order is y
2. Execute the conversion you want: press
x or θ
r.
z q
š
š
(rectangular–to–polar)
or
(polar–to–rectangular). The converted coordinates occupy
{ r
the X– and Y–registers.
3. The resulting display (the X–register) shows either r (polar result) or x
(rectangular result). Press
to see θ or y.
Z
Real–Number Functions
4–7
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y,x θ, r
y
x
θ
Y
X
r
θ
,
r y, x
Example: Polar to Rectangular Conversion.
In the following right triangles, find sides x and y in the triangle on the left,
and hypotenuse r and angle θ in the triangle on the right.
10
r
y
4
30o
θ
x
3
Keys:
Display:
Description:
{
ꢍꢆ
}
Sets Degrees mode.
z Ÿ
ꢎ
30
10
Calculates x.
š
{
ꢙ) ꢕꢖꢎ
r
Z
Displays y.
ꢗ)ꢕꢕꢕꢕꢎ
ꢗ)ꢕꢕꢕꢕꢎ
4 š 3 z q
Calculates hypotenuse (r).
Displays θ.
Z
ꢗꢖ)ꢔꢖꢕꢔꢎ
4–8
Real–Number Functions
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Example: Conversion with Vectors.
Engineer P.C. Bard has determined that in the RC circuit shown, the total
impedance is 77.8 ohms and voltage lags current by 36.5 º. What a .re the
?
values of resistance R and capacitive reactance X in the circuit
C
Use a vector diagram as shown, with impedance equal to the polar
magnitude, r, and voltage lag equal to the angle, θ, in degrees. When the
values are converted to rectangular coordinates, the x–value yields R, in ohms;
the y–value yields X ,in ohms.
C
R
θ
_
36.5o
R
X
c
77.8 ohms
C
Keys:
Display:
Description:
z Ÿ {
}
Sets Degrees mode.
ꢎ
ꢍꢆ
36.5
Enters θ, degrees of voltage lag.
.ꢖ )ꢗꢕꢕꢕꢎ
_ š
77.8
Enters r, ohms of total impedance.
ꢘꢘ)ꢙ_
Calculates x, ohms resistance, R.
{ r
Z
ꢏ)ꢗꢒꢕꢔꢎ
Displays y, ohms reactance, X .
.ꢒ )ꢏꢘꢘꢏꢎ
C
For more sophisticated operations with vectors (addition, subtraction, cross
product, and dot product), refer to the "Vector Operations" program in
chapter 15, "Mathematics Programs"
Time Conversions
Values for time (in hours, H) or angles (in degrees, D) can be converted
between a decimal–fraction form (H.h or D.d) and a minutes–seconds form
(H.MMSSss or D.MMSSss) using the
or
z s { t
keys.
Real–Number Functions
4–9
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To convert between decimal fractions and minutes–seconds:
1. Key in the time or angle (in decimal form or minutes–seconds form) that
you want to convert.
2. Press
or . The result is displayed.
{ t z s
Example: Converting Time Formats.
?
How many minutes and seconds are there in 1 ÷ 7 of an hour Use FIX 6
display format.
Keys:
Display:
Description:
{
} 6
} 4
Sets FIX 6 display format.
1 ÷ 7 as a decimal fraction.
Equals 8 minutes and 34.29
z ž
ꢋ%
ꢎ
1
Œ Œ
7
ꢕ ꢔ+ꢘꢎ
ꢕ)ꢕꢙꢖꢒꢏꢓꢎ
{ t
seconds.
Restores FIX 4 display format.
{
ꢋ%
z ž
ꢕ)ꢕꢙꢖꢒꢎ
Angle Conversions
When converting to radians, the number in the x–register is assumed to be
degrees; when converting to degrees, the number in the x–register is assumed
to be radians.
To convert an angle between degrees and radians:
1. Key in the angle (in decimal degrees or radians) that you want to convert.
2. Press
or . The result is displayed.
{ v z u
4–10 Real–Number Functions
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Unit conversions
€
€
The HP 32SII has eight unit–conversion functions on the keybord: kg, lb,
€
€
€
€
€ €
ºC, ºF, cm, in, l, gal.
To Convert:
To:
Press:
Displayed Results:
1 lb
kg
lb
1
1
(kilograms)
z }
{ |
ꢕ)ꢒꢗꢖ
1 kg
(pounds)
ꢏ)ꢏꢕꢒ
32 ºF
100 ºC
1 in
ºC
ºF
32
(°C)
z ~
ꢕ)ꢕꢕꢕꢕ
100
1
(°F)
{
ꢏꢔꢏ)ꢕꢕꢕꢕ
cm
in
(centimeters)
z €
ꢏ)ꢗꢒꢕꢕ
100 cm
1 gal
1 l
100
(inches)
{
z ‚
{ ƒ
ꢖꢓ)ꢖꢘꢕꢔ
l
1
1
(liters)
ꢖ)ꢘꢙꢗꢒ
gal
(gallons)
ꢕ)ꢏ ꢒꢏ
Probability Functions
Factorial
To calculate the factorial of a displayed positive integer x (o ≤ x ≤ 253), press
(the left–shifted
key).
z 1
3
Gamma
To calculate the gamma function of a noninteger x, Γ(x), key in (x – 1) and
press . The x! function calculates (x + 1). The value for x cannot be
{ 1
Γ
a negative integer.
Real–Number Functions 4–11
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Probability Menu
Press { [PROB] to see the PROB (probability) menu shown, in the following
table. It has functions to calculate combinations and permutations, to
generate seeds for random numbers, and to obtain random numbers from
those seeds.
PROB Menu
Menu Label
Description
{ , }
ꢃQ T
Combinations. Enter n first, then r (nonnegative
integers only). Calculates the number of possible sets
of n items taken r at a time. No item occurs more than
once in a set, and different orders of the same r items
are not counted separately.
{ , }
ꢅQ T
Permutations. Enter n first, then r (nonnegative
integers only). Calculates the number of possible
arrangements of n items taken r at a time. No item
occurs more than once in an arrangement, and
different orders of the same r items are counted
separately.
{SD}
{R}
Seed. Stores the number in x as a new seed for the
random number generator.
Random number generator. Generates a random
number in the range 0 ≤ x < 1 (The number is part of
a uniformly–distributed pseudo–random number
sequence. It passes the spectral test of D. Knuth,
Seminumerical Algotithims, vol. 2, London: Addison
Wesley, 1981.)
The RANDOM function (executed by pressing { }) uses a seed to generate a
ꢁ
random number. Each random number generated becomes the seed for the
next random number. Therefore, a sequence of random numbers can be
repeated by starting with the same seed. You can store a new seed with the
SEED function (executed by pressing { }). If memory is cleared, the seed is
ꢍ
reset to zero.
4–12 Real–Number Functions
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Example: Combinations of People.
A company employing 14 women and 10 men is forming a six–person safety
?
committee. How many different combinations of people are possible
Keys:
Display:
Description:
24
6
Twenty–four people grouped six
at a time.
š
_
[PROB]
Probability menu.
ꢃQ8T ꢅQ8T ꢍ ꢁꢎ
{
{ , }
ꢃQ T
Total number of combinations
ꢔꢖꢒ8ꢗꢓ )ꢕꢕꢕꢕꢎ
possible.
If employees are chosen at random, what is the probability that the committee
?
will contain six women To find the probability of an event, divide the number
of combinations for that event by the total number of combinations.
Keys:
Display:
Description:
Fourteen worriers grouped six
at a time.
Number of combinations of six
women on the committee.
14
6
š
_ꢎ
ꢖ8ꢕꢕꢖ)ꢕꢕꢕꢕꢎ
[PROB] { , }
ꢃQ T
{
Brings total number of
Z
p
ꢔꢖꢒ8ꢗꢓ )ꢕꢕꢕꢕꢎ
combinations back into the
X–register.
Divides combinations of
women by total combinations
to find probability that any one
combination would have all
warriors.
ꢕ)ꢕꢏꢏꢖꢎ
Real–Number Functions 4–13
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Parts of Numbers
The functions in the PARTS menu (
[PARTS]) shown in the following table
{
and the
function alter the number in the X–register in simple ways.
z I
These functions are primarily used in programming.
PARTS Menu
Menu Label
Description
{
{
{
}
ꢊꢅ
Integer part. Removes the fractional part of x and replaces
it with zeros. (For example, the integer part of 14.2300 is
14.000.)
}
ꢋꢅ
Fractional part. Removes the integer part of x and
replaces it with zeros. (For example, the fractional part of
14.2300 is 0.2300)
}
ꢀꢌ
Absolute value. Replaces x with its absolute value.
The RND function ( z I ) rounds x internally to the number of digits
specified by the display format. (The internal number is represented by 12
digits.) Refer to chapter 5 for the behavior of RND in Fraction–display mode.
Names of Function
You might have noticed that the name of a function appears in the display
when you press and hold the key to execute it. (The name remains displayed
for as long as you hold the key down.) For instance, while pressing
, the
<
display shows
. "SQRT" is the name of the function as it will appear in
ꢉꢁ!
program lines (and usually in equations also).
4–14 Real–Number Functions
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5
Fractions
"Fractions" in chapter 1 introduces the basics about entering, displaying, and
calculating with fractions:
ꢀ
To enter a fraction, press
twice—after the integer part, and between
Œ
3
the numerator and denominator. To enter 2 / , press 2
3
Œ Œ
8. To
8
5
enter / , press
5
Œ Œ
8 or 5
8.
Œ Œ
8
ꢀ
To turn Fraction–display mode on and off, press
. When you
z Š
turn off Fraction–display mode, the display goes back to the previous
display format. (FIX, SCI, ENG, and ALL also turn off Fraction–display
mode.)
ꢀ
Functions work the same with fractions as with decimal numbers—except
for RND, which is discussed later in this chapter.
This chapter gives more information about using and displaying fractions.
Entering Fractions
You can type almost any number as a fraction on the keyboard — including
an improper fraction (where the numerator is larger than the denominator).
However, the calculator displays
if you disregard these two restrictions.
£
ꢀ
ꢀ
The integer and numerator must not contain more than 12 digits total.
The denominator must not contain more than 4 digits.
Fractions
5–1
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Example:
Keys:
Display:
Description:
z Š
Turns on Fraction–display mode.
ꢎ
1.5
Enters 1.5; shown as a fraction.
š
ꢔ ꢔ+ꢏꢎ
ꢔ ꢖ+ꢒꢎ
ꢔ)ꢘꢗꢕꢕꢎ
ꢔ ꢖ+ꢒꢎ
3
1
3
Œ Œ š
4
Enters 1 / .
4
Displays x as a decimal number.
Displays x as a fraction.
z Š
z Š
If you didn't get the same results as the example, you may have accidentally
changed how fractions are displayed. (See "Changing the Fraction Display"
later in this chapter.)
The next topic includes more examples of valid and invalid input fractions.
You can type fractions only if the number base is 10 — the normal number
base. See chapter 10 for information about changing the number base.
Fractions in the Display
In Fraction–display mode, numbers are evaluated internally as decimal
numbers, then they're displayed using the most precise fractions allowed. In
addition, accuracy annunciators show the direction of any inaccuracy of the
fraction compared to its 12–digit decimal value. (Most statistics registers are
exceptions — they're always shown as decimal numbers.)
Display Rules
The fraction you see may differ from the one you enter. In its default condition,
the calculator displays a fractional number according to the following rules.
(To change the rules, see "Changing the Fraction Display" later in this
chapter.)
ꢀ
The number has an integer part and, if necessary, a proper fraction (the
numerator is less than the denominator).
5–2
Fractions
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ꢀ
ꢀ
The denominator is no greater than 4095.
The fraction is reduced as far as possible.
Examples:
These are examples of entered values and the resulting displays. For
S
T
comparison, the internal 12–digit values are also shown. The
annunciators in the last column are explained below.
and
Entered Value
Internal Value
2.37500000000
14.4687500000
4.50000000000
9.60000000000
Displayed Fraction
3
2 /
8
ꢏ ꢖ+ꢙꢎ
ꢔꢒ ꢔꢗ+ꢖꢏꢎ
ꢒ ꢔ+ꢏꢎ
ꢓ ꢖ+ꢗꢎ
15
14 /
32
54
/
12
18
6 /
5
34
T
/
2.83333333333
.183105468750
(Illegal entry)
12
ꢏ ꢗ+
15
S
/
ꢕ ꢘ+ꢖꢙꢏꢖ
8192
12345
12345678
/
3
£
£
3
16 /
(Illegal entry)
16384
Accuracy Indicators
S
T
The accuracy of a displayed fraction is indicated by the
and
annunciators at the top of the display. The calculator compares the value of
the fractional part of the internal 12–digit number with the value of the
displayed fraction:
ꢀ
If no indicator is lit, the fractional part of the internal 12–digit value
exactly matches the value of the displayed fraction.
ꢀ
T
If
is fit, the fractional part of the internal 12–digit value is slightly less
than the displayed fraction — the exact numerator is no more than 0.5
below the displayed numerator.
Fractions
5–3
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S
This diagram shows how the displayed fraction relates to nearby values —
means the exact numerator is "a little above" the displayed numerator, and
T
means the exact numerator is "a little below".
0 7/16
0 7/16
0 7/16
6
6.5
7
7.5
8
/
/
/
/
/
16
16
16
16
16
(0.40625)
(0.43750)
(0.46875)
This is especially important if you change the rules about how fractions are
displayed. (See "Changing the Fraction Display" later.) For example, if you
2
S
force all fractions to have 5 as the denominator, then / is displayed as
ꢕ
3
3.3333
because the exact fraction is approximately
/ , "a little above"
ꢖ+ꢗ
5
3
2
S
/ . Similarly, – / is displayed as
5 because the true numerator is
.ꢕ ꢖ+
5
3
"a little above" 3.
ST
If you press z X {
} to view the VAR catalog, the
annunciator
to move
#ꢀꢁ
doesn't indicate accuracy — it means you can use
and
—
˜
through the list of variables. The accuracy isn't shown.
Sometimes an annunciator is lit when you wouldn't expect it to be. For
2
S
example, if you enter 2 / , you see
, even though that's the exact
ꢏ ꢏ+ꢖ
3
number you entered. The calculator always compares the fractional part of
the internal value and the 12–digit value of just the fraction. If the internal
value has an integer part, its fractional part contains less than 12 digits–and it
can't exactly match a fraction that uses all 12 digits.
Longer Fractions
If the displayed fraction is too long to fit in the display, it's shown with ... at
the beginning. The fraction part always fits — the ... means the integer part
isn't shown completely. To see the integer part (and the decimal fraction),
proms and hold
(You can't scroll a fraction in the display.)
{
5–4
Fractions
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Example:
Keys:
Display:
Description:
14
14 *
Calculates e .
...ꢕꢒ ꢙꢙꢙ+ꢖꢔꢏꢗꢎ
Shows all decimal digits.
ꢔꢏꢕꢏ ꢕꢒ)ꢏꢙꢒꢔ ꢎ
{
H A
Stores value in A.
...ꢕꢒ ꢙꢙꢙ+ꢖꢔꢏꢗꢎ
A
Views A.
ꢀ/... ꢙꢙꢙ+ꢖꢔꢏꢗꢎ
{ ‰
† †
Clears x.
ꢕꢎ
Changing the Fraction Display
In its default condition, the calculator displays a fractional number according
to certain rules. (See "Display Rules" earlier in this chapter.) However, you
can change the rules according to how you want fractions displayed:
ꢀ
ꢀ
You can set the maximum denominator that's used.
You can select one of three fraction formats.
The next few topics show how to change the fraction display.
Setting the Maximum Denominator
For any fraction, the denominator is selected based on a value stored in the
calculator. If you think of fractions as a b/c, then /c corresponds to the value
that controls the denominator.
The /c value defines only the maximum denominator used in Fraction–display
mode — the specific denominator that's used is determined by the fraction
format (discussed in the next topic).
ꢀ
To set the /c value, press n {ꢁ‹, where n is the maximum
denominator you want. n can't exceed 4095. This also turns on Fraction–
display mode.
ꢀ
ꢀ
To recall the /c value to the X–register, press 1
.
{ ‹
To restore the default value or 4095, press 0
. (You also restore
{ ‹
Fractions
5–5
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the default if you use 4095 or greater.) This also turns on
Fraction–display mode.
The /c function uses the absolute value of the integer part of the number in the
X–register. It doesn't change the value in the LAST X register.
Choosing Fraction Format
The calculator has three fraction formats. Regardless of the format, the
displayed fractions are always the closest fractions within the rules for that
format.
ꢀ
Most precise fractions. Fractions have any denominator up to the /c
value, and they're reduced as much as possible. For example, if you're
studying math concepts with fractions, you might want any denominator
to be possible (/c value is 4095). This is the default fraction format.
ꢀ
Factors of denominator. Fractions have only denominators that are
factors of the /c value, and they're reduced as much as possible. For
example, if you're calculating stock prices, you might want to see
ꢗꢖ
and
(/c value is 8). Or if the /c value is 12, possible
ꢔ+ꢔꢒ
ꢖꢘ ꢘ+ꢙ
denominators are 2, 3, 4, 6, and 12.
ꢀ
Fixed denominator. Fractions always use the /c value as the
denominator—they're not reduced. For example, if you're working with
time measurements, you might want to see
(/c value is 60).
ꢔ ꢏꢗ+ ꢕ
To select a fraction format, you must change the states of two flags. Each flag
can be "set" or "clear," and in one case the state of flag 9 doesn't matter.
To Get This Fraction Format:
Change These Flags:
8
9
—
Most precise
Clear
Set
Factors of denominator
Fixed denominator
Clear
Set
Set
5–6
Fractions
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You can change flags 8 and 9 to set the fraction format using the steps listed
here. (Because flags are especially useful in program, their use us covered in
detail in chapter 13.)
1. Press
to get the flag menu.
{ x
2. To set a flag, press { } and type the flag number, such as 8.
ꢋ
To clear a flag, press { ) and type the flag number.
ꢃꢋ
To see if a flag is set, press {
} and type the flag number. Press
or
†
ꢋ @
to clear the
or
response.
a
&ꢈ
ꢄꢑ
Examples of Fraction Displays
The following table shows how the number 2.77 is displayed in the three
fraction formats for two /c values.
Fraction
Format
How 2.77 Is Displayed
/c= 4095 /c= 16
(2.7700)
(2.7692)
(2.7500)
S
Most precise
2 77/100
2 10/13
2 3/4
Factors of
denominator
(2.7699)
S
S
S
2 1051/1365
2 3153/4095
Fixed
denominator
(2.7699)
(2.7500)
S
2 12/16
Fractions
5–7
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The following table shows how different numbers are displayed in the three
fraction formats for a /c value of 16.
Fraction
Number Entered and Fraction Displayed
Format ꢀ
2
16
2
2.5
2 1/2
2 /
2.9999
2
/
3
25
2
2
2 2/3
3
T
2 7/11
S
S
Most precise
Factors of
denominator
2 1/2
T2 11/16
T3
S2 5/8
Fixed
denominator
2 0/16
2 8/16
2 11/16
T
2 16/16
T
2 10/16
S
ꢀ For a /F value of 16.
Example:
1
5
Suppose a stock has a current value of 48 / . If it goes down 2 / , what
4
8
?
?
would be its value What would then be 85 percent of that value
Keys:
Display:
Description:
{
} 8
} 9
Sets flag 8, clears flag 9 for
{ x
ꢎ
ꢋ
{
ꢃꢋ
"factors of denominator" format.
{ x
8 { ‹
1
Sets up fraction format for /
ꢎ
8
increments.
48 Œ 1 Œ 4 š
Enters the starting value.
ꢒꢙ ꢔ+ꢒꢎ
ꢒꢗ ꢗ+ꢙꢎ
Sꢖꢙ ꢖ+ꢒꢎ
2
5
Œ Œ Œ „
8
Subtracts the change.
85
Finds the 85–percent value to the
{ P
1
nearest / .
8
Rounding Fractions
If Fraction–display mode is active, the RND function converts the number in
the X–register to the closest decimal representation of the fraction. The
rounding is done according to the current /c value and the states of flags 8
5–8
Fractions
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and 9. The accuracy indicator turns off if the fraction matches the decimal
representation exactly. Otherwise, the accuracy indicator stays on, (See
"Accuracy Indicators" earlier in this chapter.)
In an equation or program, the RND function does fractional rounding if
Fraction–display mode is active.
Example:
3
Suppose you have a 56 / –inch space that you want to divide into six equal
4
sections. How wide is each section, assuming you can conveniently measure
1
?
?
/ –inch increments What's the cumulative roundoff error
16
Keys:
Display:
Description:
16
Sets up fraction format for
{ ‹
ꢎ
1
/ –inch increments. (Flags 8
16
and 9 should be the same as for
the previous example.)
56 Π3 Π4 H D
Stores the distance in D.
ꢎ
ꢗ ꢖ+ꢒꢎ
Sꢓ ꢘ+ꢔ ꢎ
6
The sections are a bit wider than
p
7
9 / inches.
16
Rounds the width to this value.
Width of six sections.
z I
ꢓ ꢘ+ꢔ ꢎ
ꢗ ꢗ+ꢙꢎ
.ꢕ ꢔ+ꢙꢎ
.ꢕ ꢔ+ꢙꢎ
.ꢕ)ꢔꢏꢗꢕꢎ
6
y
D
K „
The cumulative round off error.
Clears flag 8.
{
ꢃꢋ
} 8
{ x
z Š
Turns off Fraction–display mode.
Fractions in Equations
When you're typing an equation, you can't type a number as a fraction.
When an equation is displayed, all numeric values are shown as decimal
values–Fraction — display mode is ignored.
Fractions
5–9
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When you're evaluating an equation and you're prompted for variable
values, you may enter fractions — values are displayed using the current
display format.
See chapter 6 for information about working with equations.
Fractions in Programs
When you're typing a program, you can type a number as a fraction — but
it's converted to its decimal value. All numeric values in a program are shown
as decimal values — Fraction–display mode is ignored.
When you're running a program, displayed values are shown using
Fraction–display mode if it's active. If you're prompted for Values by INPUT
instructions, you may enter fractions, regardless of the display mode.
A program can control the fraction display using the /c function and by
setting and clearing flags 7, 8, and 9. Setting flag 7 turns on Fraction–display
mode — z Š isn't programmable. See "Flags" in chapter 13.
See chapters 12 and 13 for information about working with programs.
5–10 Fractions
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6
Entering and Evaluating
Equations
How You Can Use Equations
You can use equations on the HP 32SII in several way:
ꢀ
ꢀ
ꢀ
For specifying an equation to evaluate (this chapter).
For specifying an equation to solve for unknown values (chapter 7).
For specifying a function to integrate (chapter 8).
Example: Calculating with an Equation.
Suppose you frequently need to determine the volume of a straight section of
pipe. The equation is
2
V = .25 π d l
There d is the inside diameter of the pipe, and l is its length.
You could key in the calculation over and over, for example, .25 š
2.5
16
calculates the volume of 16 inches of
{ M y
1
z : y
y
2 / –inch diameter pipe (78.5398 cubic inches). However, by storing the
2
equation, you get the HP 32SII to "remember" the relationship between
diameter, length, and volume—so you can use it many times.
Put the calculator in Equation mode and type in the equation using the
following keystrokes:
Keys:
Display:
Description:
Selects Equation mode, or the
current shown by the EQN
annunciator.
{ G
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅꢎ
or the current equation
Entering and Evaluating Equations
6–1
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Begins a new equation, turning on
K
¾ꢎ
the " " equation–entry cursor.
¾
turns on the A..Z
K
annunciator so you can enter a
variable name.
V
V types and moves the
{ c
K
#/¾ꢎ
#
cursor to the right.
Digit entry uses the "_" digit–entry
cursor.
.25
#/ꢕ)ꢏꢗ_ꢎ
ends the number and restores
y { M y
y
#/ꢕ)ꢏꢗºπº¾ꢎ
the " " cursor.
¾
D
K 0
2
types .
0
/ꢕ)ꢏꢗºπºꢍ: ꢏ_ꢎ
:
L
scrolls o f the left side of the
y K
ꢕ)ꢏꢗºπºꢍ:ꢏºꢂ¾ꢎ
#/
display.
Terminates and displays the
š
#/ꢕ)ꢏꢗºπºꢍ:ꢏºꢎ
equation.
shows that part of the
€
equation doesn't fit in the display,
and above means you can
6
ž
press
to see characters in that
6
direction.
Shows the checksum and length for
{
ꢃꢚ/ꢗꢙꢖ
ꢕ)ꢏ )ꢕꢎ
the equation, so you can check
your keystrokes.
By comparing the checksum and length of your equation with those in the
example, you can verify that you've entered the equation properly. (See
"Verifying Equations" at the end of this chapter for more information.)
Evaluate the equation (to calculate V):
Keys:
Display:
Description:
Prompts for variables on the
š
ꢍ@valueꢎ
right–hand side of the equation.
6–2
Entering and Evaluating Equations
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Keys:
Display:
Description:
Prompts for D first; value is the
current value of D.
1
2 Π1 Π2
Enters 2 / inches as a fraction.
ꢍ@ ꢏ ꢔ+ꢏꢎ
2
Stores D, prompts for L; value is
current value of L.
f
ꢂ@valueꢎ
16
Stores L; calculates V in cubic inches
and stores the result in V.
f
#/ꢘꢙ)ꢗꢖꢓꢙꢎ
Summary of Equation Operations
All equations you create are saved in the equation list. This list is visible
whenever you activate Equation mode.
You use certain keys to perform operations involving equations. They're
described in more detail later.
Entering and Evaluating Equations
6–3
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Key
Operation
Enters and leaves Equation mode.
{ G
š
Evaluates the displayed equation. If the equation is an
assignment, evaluates the right–hand side and stores
the result in the variable on the left–hand side. If the
equation is an equality or expression, calculates its
value like
chapter.)
. (See "Types of Equations" later in this
W
W
Evaluates the displayed equation. Calculates its value,
replacing "=" with "–" if an "=" is present.
Solves the displayed equation for the unknown
variable you specify. (See chapter 7.)
{ œ
{ )
a
Integrates the displayed equation with respect, to the
variable you specify. (See chapter 8.)
Begins editing the displayed equation; subsequent
presses delete the rightmost function or variable.
Deletes the displayed equation from the equation list.
Steps up or down through the equation list.
z b
z —
or
z ˜
{
Shows the displayed equation's checksum (verification
value) and length (bytes of memory).
Leaves Equation mode.
†
You can also use equations in programs—this is discussed in chapter 12.
Entering Equations into the Equation List
The equation list is a collection of equations you enter. The list is saved in the
calculator's memory. Each equation you enter is automatically saved in the
equation list.
6–4
Entering and Evaluating Equations
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To enter an equation:
1. Make sure the calculator is in its normal operating mode, usually with a
number in the display. For example, you can't be viewing the catalog of
variables or programs.
2. Press
. The EQN annunciator shows that Equation mode is
{ G
active, and an entry from the equation list is displayed.
3. Start typing the equation. The previous display is replaced by the equation
you're entering — the previous equation isn't affected. If you make a
mistake, press
as required.
a
4. Press
to terminate the equation and see it in the display. The
š
equation is automatically saved in the equation list—right after the entry
that was displayed when you started typing. (If you press
equation is saved, but Equation mode is turned off.)
instead, the
†
You can make an equation as long as you want—you're limited only by the
amount of memory available.
Equations can contain variables, numbers, functions, and parentheses —
they're described in the following topics. The example that follows illustrates
these elements.
Variables in Equations
You can use any of the calculator's 28 variables in an equation: A through Z,
i, and (i). You can use each variable as many times as you want. (For
information about (i), see "Indirectly Addressing Variables and Labels" in
chapter 13.)
To enter a variable in an equation, press
variable (or
variable).
K
H
When you press
, the A..Z annunciator shows that you can press a
K
variable key to enter its name in the equation.
Number in Equations
You can enter any valid number in an equation except fractions and numbers
that aren't base 10 numbers. Numbers are always shown using ALL display
format, which displays up to 12 characters.
Entering and Evaluating Equations
6–5
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To enter a number in an equation, you can use the standard number–entry
keys, including , and . Press only after you type one or
,
Π_
`
_
more digits. Don't use
for subtraction.
_
When you start entering the number, the cursor changes from " " to "_" to
¾
show numeric entry. The cursor changes back when you press a nonnumeric
key.
Functions in Equations
You can enter many HP 32SII functions in an equation. A complete list is
given tinder "Equation Functions" later in this chapter. Appendix F,
"Operation Index," also gives this information.
When you enter an equation, you enter functions in about the same way you
put them in ordinary algebraic equations:
ꢀ
In an equation, certain functions are normally shown between its
arguments, such as "+" and "÷". For such infix operators, enter them in
an equation in the same order.
ꢀ
Other functions normally have one or more arguments after the function
name, such as "COS" and "LN". For such prefix functions, enter them in
an equation where the function occurs—the key you press puts a left
parenthesis after the function name so you can enter its arguments.
If the function has two or more arguments, press
key) to separate them.
(on the
o
f
If the function is followed by other operations, press
to complete
{ ]
the function arguments — otherwise, you don't have to add the trailing
")".
If the first key in an equation is a function from the top row of keys on the
calculator, and if the displayed equation has the annunciator turned on,
ž
you have to press
[SCRL] first to turn off the annunciator. (See
{
"Displaying and Selecting Equations" later in this chapter for more
information.)
6–6
Entering and Evaluating Equations
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Parentheses in Equations
You can include parentheses in equations to control the order in which
operations are performed. Press
and
to insert parentheses.
{ \
{ ]
(For more information, see "Operator Precedence" later in this chapter.)
Example: Entering an Equation.
Enter the equation r = 2 × c × cos (t – a).
Keys:
Display:
Description:
Shows the last equation used in the
#/ꢕ)ꢏꢗºπºꢍ:ꢏºꢎ
{ G
equation list.
K R { c
Starts a new equation with variable
R.
ꢁ/¾ꢎ
2
Enters a number, changing the
cursor to "_".
ꢁ/ ꢏ_ꢎ
C
y K y
Enters infix operators.
Enters a prefix function with a left
parenthesis.
ꢁ/ꢏºꢃº¾ꢎ
Q
ꢁ/ꢏºꢃºꢃꢑ 1¾ꢎ
T
K K
{ ]
A
Enters the argument and right
parenthesis. This final parenthesis
is optional.
ºꢃºꢃꢑ 1!.ꢀ2¾ꢎ
Terminates the equation and
displays it.
š
ꢁ/ꢏºꢃºꢃꢑ 1!.ꢎ
Shows its checksum and length.
{
†
ꢃꢚ/ꢗ ꢃꢔ ꢕꢔꢙ)ꢕꢎ
Leaves Equation mode.
ꢎ
Displaying and Selecting Equations
The equation list contains the equations you've entered. You can display the
equations and select one to work with.
Entering and Evaluating Equations
6–7
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To display equations:
1. Press . This activates Equation mode and turns on the EQN
{ G
annunciator. The display shows an entry from the equation list:
ꢀ
if there are no equations in the equation list or if the
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅ
equation pointer is at the top of the list.
ꢀ
The current equation (the last equation you viewed).
2. Press
or
to step through the equation list and view
z — z ˜
each equation. The list "wraps around" at the top and bottom.
ꢈꢉꢄ ꢂꢊ !
marks the "top" of the list.
!ꢑꢅ
To view a long equation:
1. Display the equation in the equation list, as described above. If it's more
than 12 characters long, only 12 characters are shown. The
€
annunciator indicates more characters to the right. The annunciator over
ž
means scrolling is turned on.
6
2. Press
to scroll the equation one character at a time, showing
6
characters to the right. Press < to show characters to the left.
and
€
turn off if there are no more characters to the left or right.
Press { [SCRL] to turn scrolling off and on. When scrolling is turned off, the
left end of the equation is displayed, the
annunciators are off, and the
ž
unshifted top–row keys perform their labeled functions. You must turn off
scrolling if you want to enter a new equation that starts with a top–row
function, such as LN.
To select an equation:
Display the equation in the equation list, as described above. The displayed
equation is the one that's used for all equation operations.
Example: Viewing an Equation.
View the last equation you entered.
Keys:
Display:
Description:
{ G
Displays the current equation in the
equation list.
ꢁ/ꢏºꢃºꢃꢑ 1!.ꢎ
Shows two more characters to the
ꢏºꢃºꢃꢑ 1!.ꢀ2ꢎ
6 4
6–8
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right.
<
†
Shows one character to the left.
Leaves Equation mode.
/ꢏºꢃºꢃꢑ 1!.ꢀꢎ
ꢎ
Editing and Clearing Equations
You can edit or clear an equation that you're typing. You can also edit or
clear equations saved in the equation list.
To edit an equation you're typing:
1. Press
repeatedly until you delete the unwanted number or function.
a
If you're typing a decimal number and the "_" digit–entry cursor is on,
deletes only the rightmost character. If you delete all characters in the
a
number, the calculator switches back to the " " equation–entry cursor.
¾
If the " " equation–entry cursor is on, pressing
deletes the entire
a
¾
rightmost number or function.
2. Retype the rest of the equation.
3. Press š (or †) to save the equation in the equation list.
To edit a saved equation:
1. Display the desired equation. (See "Displaying and Selecting Equations"
above.)
2. Press
(once only) to start editing the equation. The " "
a
¾
equation–entry cursor appears at the end of the equation. Nothing is
deleted from the equation.
3. Use
to edit the equation as described above.
a
4. Press
(or
) to save the edited equation in the equation list,
š
†
replacing the previous version.
To clear an equation you're typing:
Press
then press { }. The display goes back to the previous entry
z b
&
in the equation list.
Entering and Evaluating Equations
6–9
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To clear a saved equation:
1. Display the desired equation. (See "Displaying and Selecting Equations"
above.)
2. Press
. The display shows the previous entry in the equation
z b
list.
To clear all equations, clear them one at a time: scroll through the equation
list until you come to
b repeatedly as each equation is displayed until you see
, press
, then press
z —
z
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅ
ꢈꢉꢄ ꢂꢊ !
.
!ꢑꢅ
Example: Editing an Equation.
R
emove the optional right parenthesis in the equation from the previous
example.
Keys:
Display:
Description:
Shows the current equation in the
ꢁ/ꢏºꢃºꢃꢑ 1!.ꢎ
{ G
equation list.
Turns on Equation–entry mode and
a
ºꢃºꢃꢑ 1!.ꢀ2¾ꢎ
shows the " " cursor at the end of
¾
the equation.
a
Deletes the right parenthesis.
ꢏºꢃºꢃꢑ 1!.ꢀ¾ꢎ
Shows the end of edited equation in
/ꢏºꢃºꢃꢑ 1!.ꢀꢎ
š 6
6
the equation list.
Leaves Equation mode.
†
ꢎ
Types of Equations
The HP 32SII works with three types of equations:
ꢀ
Equalities. The equation contains an "=" and the left side contains
more than just a single variable. For example, x + y = r is an equality.
2
2
2
ꢀ
Assignments. The equation contains an "=" and the left side contains
just a single variable. For example, A = 0.5 × b × h is an assignment.
6–10 Entering and Evaluating Equations
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3
ꢀ
Expressions. The equation does not contain an "=". For example, x
+ 1 is an expression.
When you're calculating with an equation, you might use any type of
equation—although the type can affect how it's evaluated. When you're
solving a problem for an unknown variable, you'll probably use an equality
or assignment. When you're integrating a Function, you'll probably use an
expression.
Evaluating Equations
One of the most useful characteristics of equations is their ability to be
evaluated — to generate numeric values. This is what enables you to
calculate result from an equation. (It also enables you to solve and integrate
equations, as described in chapters 7 and 8).
Because many equations have two sides separated by "=", the basic value of
an equation is the difference between the values of the two sides. For this
calculation, "=" in an equation essentially treated as "_".
The value is a measure of lour well the equation balances.
The HP 32SII has two keys for evaluating equations:
and
. Their
W
š
actions differ only in how they evaluate assignment equations:
ꢀ
ꢀ
returns the value of the equation, regardless of the type: equation.
W
returns the value of the equation—unless it's an assignment–type
š
equation. For an assignment equation,
returns the value f the
š
right side only, and also "enters" that value into the variable on the left
side — it stores the value in the variable.
Entering and Evaluating Equations 6–11
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The following table shoves the two ways to evaluate equations.
Type of Equation
Result for š
Result for W
Equality: g (x) = f(x)
Example: x + y = r
g (x) – f(x)
2
2
2
2
2
2
x + y – r
f(x) ꢀ
0.5 × b × h ꢀ
y – f(x)
Assignment: y = f(x)
Example: A = 0.5 × b x h
Expression: f(x)
A – 0.5 × b × h
f(x)
3
Example: x + 1
3
x + 1
ꢀ
Also stores the result in the left–hand variable, A for example.
To evaluate an equation:
1. Display the desired equation. (See "Displaying and Selecting Equations"
above.)
2. Press
or
. The equation prompts for a value for each
W
š
variable needed. (If you've changed the number base, it's automatically
changed back to base 10.)
3. For each prompt, enter the desired value:
ꢀ
If the displayed value is good, press
.
f
ꢀ
If you want, a different value, type the value and press
. (Also see
f
"Responding to Equation Prompts" later in this chapter.)
The evaluation of an equation takes no values from the stack — it uses only
numbers in the equation and variable values. The value of the equation is
returned to the X–register. The LAST X register isn't affected.
Using ENTER for Evaluation
If an equation is displayed in the equation list, you can press š to
evaluate the equation. (If you're in the process of typing the equation,
pressing
only ends the equation—it doesn't evaluate it.)
š
6–12 Entering and Evaluating Equations
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ꢀ
ꢀ
If th
e equation is an assignment, only the right–hand side is evaluated.
The result is returned to the X–register and stored in the left–hand
variable, then the variable is VIEWed in the display. Essentially,
š
finds the value of the left–hand variable.
If the equation is an equality or expression, the entire equation is
evaluated — just as it is for
. The result is returned to the X–register.
W
Example: Evaluating an Equation with ENTER.
Use the equation from the beginning of this chapter to find the volume of a
35–mm diameter pipe that's 20 meters long.
Keys:
Display:
Description:
(
Displays the desired equation.
{ G z
— as required)
š
#/ꢕ)ꢏꢗºπºꢍ:ꢏºꢎ
Starts evaluating the assignment
ꢍ@ꢏ)ꢗꢕꢕꢕꢎ
equation so the value will be
stored in V. Prompts for variables
on the right–hand side of the
equation. Tile current value for D is
2.5000.
35
Stores D, prompts for L, whose
current value, 16.0000.
f
ꢂ@ꢔ )ꢕꢕꢕꢕꢎ
ꢎ
20
y f
1000
Stores L in millimeters; calculates V
in cubic: millimeters, stores the
š
#/ꢔꢓ8ꢏꢒꢏ8ꢏꢗꢗ)ꢕꢕꢎ
result in V, and displays V.
6
` p
Changes cubic millimeters to liters
ꢔꢓ)ꢏꢒꢏꢖꢎ
(but doesn't change V).
Entering and Evaluating Equations 6–13
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Using XEQ for Evaluation
If an equation is displayed in the equation list, you can press W to
evaluate the equation. The entire equation is evaluated, regardless of the type
of equation. The result is returned to the X–register.
Example: Evaluating an Equation with XEQ.
Use the results from the previous example to find out how much the volume of
the pipe changes if the diameter is changes to 35.5 millimeters.
Keys:
Display:
Description:
{ G
W
Displays the desired equation.
#/ꢕ)ꢏꢗºπºꢍ:ꢏºꢎ
Starts evaluating the equation to
¶@ꢔꢓ8ꢏꢒꢏ8ꢏꢗꢗ)ꢕꢕꢎ
find its value. Prompts for all
variables.
Keeps the same V, prompts for D.
store new D, Prompts for L.
f
ꢍ@ꢖꢗ)ꢕꢕꢕꢕꢎ
35.5
fꢁ
ꢂ@ꢏꢕ8ꢕꢕꢕ)ꢕꢕꢕꢕꢎ
Keeps the same L; calculates the
.ꢗꢗꢖ8ꢘꢕꢗ)ꢘꢕꢗꢔꢎ
f
value of the equation—the
imbalance between the left and
right sides.
6
` p
Changes cubic millimeters to liters.
.ꢕ)ꢗꢗꢖꢘꢎ
The value of the equation is the old volume (from V) minus the new volume
(calculated using the new D value) — so the old volume is smaller by the
amount shown.
Responding to Equation Prompts
When you evaluate an equation, you're prompted for a value for each
variable that's needed. The prompt gives the variable name and its current
value, such as
.
%@ꢏ)ꢗꢕꢕꢕ
6–14 Entering and Evaluating Equations
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ꢀ
ꢀ
To leave the number unchanged, just press
.
f
To change the number, type the new number and press
.This
f
new number writes over the old value in the X–register. You can enter a
number as a fraction if you want. If you need to calculate a number, use
normal keyboard calculations, then press f. For example, you can
press 2
5
š 0 f
.
ꢀ
ꢀ
To calculate with the displayed number, press š before
typing another number.
To cancel the prompt, press
. The current value for the variable
†
remains in the X–register. If you press
during digit entry, it clears the
†
number to zero. Press
again to cancel the prompt.
†
ꢀ
To display digits hidden by the prompt, press { .
Each prompt puts the variable value in the X–register and disables stack lift. If
you type a number at the prompt, it replaces the value in the X–register.
When you press
stack.
, stack lift is enabled, so the value is retained on the
f
The Syntax of Equations
Equations follow certain conventions that determine how they're evaluated:
ꢀ
ꢀ
ꢀ
How operators interact.
What functions are valid in equations.
How equations are checked for syntax errors.
Operator Precedence
Operators in an equation are processed in a certain order that makes the
evaluation logical and predictable:
Entering and Evaluating Equations 6–15
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Order
Operation
Example
Functions and Parentheses
1
2
3
4
5
6
ꢊꢄ1%-ꢔ2, 1%-ꢔ2ꢎ
.ꢀꢎ
Unary Minus (
)
_
Power (
)
0
%:ꢖꢎ
Multiply and Divide
Add and Subtract
Equality
%º&, ꢀªꢌꢎ
ꢅ-ꢉ, ꢀ.ꢌꢎ
ꢌ/ꢃꢎ
So, for example, all operations inside parentheses are performed before
operations outside the parentheses.
Examples:
Equations
Meaning
3
a × (b ) = c
ꢀºꢌ:ꢖ/ꢃꢎ
3
(a × b) = c
1ꢀºꢌ2:ꢖ/ꢃ
a + (b c) = 12
÷
ꢀ-ꢌªꢃ/ꢔꢏ
(a + b) ÷ c = 12
1ꢀ-ꢌ2ªꢃ/ꢔꢏꢎ
0ꢃꢐꢆ1!-ꢔꢏ ꢀ. 2:ꢏꢎ
2
[%CHG(t + 12), (a – 6)]
You can't use parentheses for implied multiplication. For example, the
expression p (1 – f) must be entered as , with the " " operator
ꢅº1ꢔ.ꢋ2
º
inserted between P and the left parenthesis.
6–16 Entering and Evaluating Equations
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Equation Function
The following table lists the functions that are valid in equations. Appendix F,
"Operation Index," also gives this information.
LN
LOG
IP
EXP
ALOG
RND
SQ
SQRT
x!
INV
SIN
SINH
FP
ABS
COS
COSH
TAN
TANH
ASIN
ASINH
ACOS
ACOSH
%CHG
ATAN
ATANH
XROOT
DEG
€
RAD
€
HR
KG
L
HMS
€
€
€
×
€
€
€
÷
Cn,r
Pn,r
LB
°C
€
°F
€
CM
€
IN
€
GAL
RANDOM
π
+
–
^
x
y
sx
x w
n
sy
σ x
σ y
ˆ
y
r
m
b
ˆ
x
2
2 2
Σx
Σy
Σx
Σx y
Σxy
For convenience, prefix–type functions, which require one or two arguments,
display a left parenthesis when you enter them.
The prefix functions that require two arguments are %CHG, XROOT, Cn,r and
Pn,r. Separate the two arguments with a space.
In an equation, the XROOT function takes its arguments in the opposite order
from RPN usage. For example, –8
3
to is equivalent to
š
.
.
%ꢁꢑꢑ!1ꢖ.ꢙ2
All other two–argument functions take their arguments in the Y, X order used
for RPN. For example, 28 š 4 { } is equivalent to
.
ꢃQ8T
ꢃQ8T1ꢏꢙ ꢒ2
For two–argument functions, be careful if the second argument is negative.
The second argument must not start with "subtraction" ( ). For a number,
„
use
. For a variable, use parentheses and
_
. These are valid equations:
„
Entering and Evaluating Equations 6–17
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0ꢃꢐꢆ1.% .ꢏ2ꢎ
0ꢃꢐꢆ1% 1.&22ꢎ
Six of the equation functions have names that differ from their equivalent RPN
operations:
RPN Operation
Equation function
2
x
SQ
EXP
x
e
x
10
ALOG
INV
1/x
X
y
x
X ROOT
^
y
Example: Perimeter of a Trapezoid.
The following equation calculates the perimeter of a trapezoid. This is how
the equation might appear in a book:
1
1
+
Perimeter = a + b + h (
)
sinθ sinφ
a
h
φ
θ
b
The following equation obeys the syntax rules for HP 32SII equations:
6–18 Entering and Evaluating Equations
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Parentheses used to group items
P=A+B+Hx(1
÷
SIN(T)+1 SIN(F))
÷
Single
letter
name
No implied
multiplication
Division is done
before addition
Th
e next equation also obeys the syntax rules. This equation uses the inverse
function, , instead of the fractional form,
.
ꢊꢄ#1 ꢊꢄ1!22
ꢔª ꢊꢄ1!2
Notice that the SIN function is "nested" inside the INV function. (INV is typed
by 3.)
ꢅ/ꢀ-ꢌ-ꢐº1ꢊꢄ#1 ꢊꢄ1!22-1ꢊꢄ#1 ꢊꢄ1ꢋ222ꢎ
Example: Area of a Polygon.
The equation for area of a regular polygon with n sides of length d is:
1
4
cos(π /n)
sin(π/n)
n d 2
Area =
d
2
π
/n
You can specify this equation as
Entering and Evaluating Equations 6–19
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π
ꢀ/ꢕ)ꢏꢗºꢄºꢍ:ꢏºꢃꢑ 1 ªꢄ2ª ꢊꢄ1 ªꢄ2ꢎ
π
Notice how the operators and functions combine to give the desired
equation.
You can enter the equation into the equation list using the following
keystrokes:
A
.25
N
D
2
{ G K {
y K y K 0 y Q
N
N
{ M p K { ] p N { M p K { ]
š
Syntax Errors
The calculator doesn't check the syntax of an equation until you evaluate the
equation and respond to all the prompts–only when a value is actually being
calculated. If an error is detected,
is displayed. You have to
ꢊꢄ#ꢀꢂꢊꢍ ꢈꢉꢄ
edit the equation to correct the error. (See "Editing and Clearing Equations"
earlier in this chapter.)
By not checking equation syntax until evaluation, the HP 32SII lets you create
"equations" that might actually be messages. This is especially useful in
programs, as described in chapter 12.
Verifying Equations
When you're viewing an equation — not while you're typing an equation —
you can press
to show you two things about the equation: the
{
equation's checksum and its length. Hold the
in the display.
key to keep the values
The checksum is a four–digit hexadecimal value that uniquely identifies this
equation. No other equation will have this value. If you enter the equation
incorrectly, it will not have this checksum. The length is the number of bytes of
calculator memory used by the equation.
6–20 Entering and Evaluating Equations
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The checksum and length allow you to verify that equations you type are
correct. The checksum and length of the equation you type in an example
should match the values shown in this manual.
Example: Checksum and Length of an Equation.
Find the checksum and length for the pipe–volume equation at the beginning
of this chapter.
Keys:
Display:
Description:
(
Displays the desired equation.
#/ꢕ)ꢏꢗºπºꢍ:ꢏºꢎ
{ G z
as required)
—ꢁ
(hold)
{
Display equation's checksum and
ꢕꢏ )ꢕꢎ
ꢃꢚ/ꢗꢙꢖ
length.
(release)
Redisplays the equation.
#/ꢕ)ꢏꢗºπºꢍ:ꢏºꢎ
Leaves Equation mode.
†
ꢎ
Entering and Evaluating Equations 6–21
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7
Solving Equations
In chapter 6 you saw how you can use
to find the value of the
š
left–hand variable in an assignment–type equation. Well, you can use SOLVE
to find the value of any variable in any type of equation.
For example, consider the equation
2
x – 3y = 10
If you know the value of y in this equation, then SOLVE can solve for the
unknown x. If you know the value of x, then SOLVE can solve for the unknown
y. This works for "word problems" just as well:
Markup × Cost = Price
If you know any two of these variables, then SOLVE can calculate the value of
the third.
When the equation has only one variable, or when known values are
supplied for all variables except one, then to solve for x is to find a root of the
equation. A root of an equation occurs where an equality or assignment
equation balances exactly, or where an expression equation equals zero.
(This is equivalent to the value of the equation being zero.)
Solving an Equation
To solve an equation for an unknown variable:
1. Press
and display the desired equation. If necessary, type the
{ G
equation as explained in chapter under "Entering Equations into the
Equation List."
2. Press
then press the key for the unknown variable. For
{ œ
example, press
X to solve for x. The equation then prompts
{ œ
Solving Equations
7–1
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for a value for every other variable in the equation.
3. For each prompt, enter the desired value;
ꢀ
If the displayed clue is the one you want, press
.
f
ꢀ
If you want a different clue, type or calculate the value and press
.
f
(For details, see "Responding to Equation Prompts" in chapter 6.)
You can half a running calculation b pressing or
†
.
f
When the root is found, it's stored in the unknown variable, and the variable
value is VIEWed in the display. In addition, the X–register contains the root,
the Y–register contains the previous estimate, and the Z–register contains the
value of the equation at the root (which should be zero).
For some complicated mathematical conditions, a definitive solution cannot
he found—and the calculator displays
. See "Verifying the
ꢄꢑ ꢁꢑꢑ! ꢋꢑ"ꢄꢍ
Result" later in this chapter, and "Interpreting results" and "When SOLVE
Cannot Find Root" in appendix C.
For certain equations it helps t provide one or two initial guesses for the
unknown variable before solving the equation. This can speed up the
calculation, direct the answer toward realistic solution, and find more than
one solution, if appropriate. See "Choosing Initial Guesses" later in this
chapter.
Example: Solving the Equation of Linear Motion.
The equation of motion for a free–falling object is:
1
2
d = v t + / g t
0
2
where d is the distance, v is the initial velocity, t is the time, and g is the
0
acceleration due to gravity.
Type in the equation:
Keys:
Display:
Description:
Clears memory.
z b
ꢎ
{ } { }
ꢀꢂꢂ &
Selects Equation mode.
{ G
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅꢎ
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or current equationꢎ
K D { c
Starts the equation.
ꢍ/#º!-¾ꢎ
V
K y K
T
™
.5
y K 0
G
y K
!-ꢕ)ꢗºꢆº!: ꢏ_ꢎ
T
2
Terminates the equation and
ꢍ/#º!-ꢕ)ꢗºꢆº!ꢎ
š
displays the left end.
Checksum end length.
{
ꢃꢚ/ ꢀꢓꢏ ꢕꢏꢓ)ꢕꢎ
g (acceleration due to gravity) is included as a variable so you can change it
2
2
for different units (98 m/s or 32.2 ft/s ).
Calculate hove ran meters an object falls in 5 seconds, starting from rest.
Since Equation mode is turned on and the desired equation is turn on and the
desired is already in the display, you can start solving for D:
Keys:
Display:
Description:
Prompts for unknown known
variable.
{ œ
ꢑꢂ#ꢈ_ꢎ
D
Selects D; prompts for V.
Stores 0 in V; prompts for T.
Stores 5 in T; prompts for G.
Stores 9.8 in G; prompts for D.
#@valueꢎ
0
f
!@valueꢎ
5
f
ꢆ@valueꢎ
9.8
f
ꢑꢂ#ꢊꢄꢆꢎ
ꢍ/ꢔꢏꢏ)ꢗꢕꢕꢕꢎ
Try another calculation using the same equation: how long does it take are
?
object to fall 500 meters from rest
Keys:
Display:
Description:
Displays the equation.
ꢍ/#º!-ꢕ)ꢗºꢆº!ꢎ
{ G
T
Solves for T; prompts for D.
{ œ
ꢍ@ꢔꢏꢏ)ꢗꢕꢕꢎ
#@ꢕ)ꢕꢕꢕꢕꢎ
ꢆ@ꢓ)ꢙꢕꢕꢕꢎ
500
Stores 500 in D; prompts for V.
Retains 0 in V; prompts for G.
f
f
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Retains 9.8 in G; prompts for T.
f
ꢑꢂ#ꢊꢄꢆꢎ
!/ꢔꢕ)ꢔꢕꢔꢗꢎ
Example: Solving the Ideal Gas Law Equation.
The Ideal Gas Law describes the relationship between pressure, volume,
temperature, and the amount (moles) of an ideal gas:
P × V = N × R × T
2
where P is pressure (in atmospheres or N/m ), V is volume (in liters), N is the
number of moles of gas, R is the universal gas constant (0.0821 liter–atm
mole–K or 8.314 J/mole–K), and T is temperature (Kelvins: K=°C + 273.1).
Enter the equation:
Keys:
Display:
Description:
P
{ G K y
Selects Equation mode and
starts the equation.
ꢅº¾ꢎ
V
K { cꢁ
K N yꢁ
ꢎ
ꢎ
R
K yꢁK
T
ꢅº#/ꢄºꢁº!¾ꢎ
ꢅº#/ꢄºꢁº!ꢎ
Terminates and displays the
equation.
š
Checksum and length.
ꢃꢚ/ꢔꢖꢈꢖ ꢕꢔꢗ)ꢕꢎ
{
A 2–liter bottle contains 0.005 moles of carbon dioxide gas at 24°C.
Assuming that the gas behaves as an ideal gas, calculate its pressure. Since
Equation mode is turned on and the desired equation is already in the display,
you can start solving for P:
Keys:
Display:
Description:
P
Solves for P; prompts for V.
{ œ
2 f
#@valueꢎ
ꢄ@valueꢎ
ꢁ@valueꢎ
!@valueꢎ
Stores 2 in V; prompts for N.
Stores .005 in N; prompts for R.
Stores .0821 in R; prompts for T.
.005
f
.0821 f
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24
Calculates T (Kelvins).
š
!@ꢏꢓꢘ)ꢔꢕꢕꢕꢎ
273.1™
Stores 297.1 in T; solves for P in
f
ꢑꢂ#Oꢄꢆꢎ
ꢅ/ꢕ)ꢕ ꢔꢕꢎ
atmospheres.
A 5–liter flask contains nitrogen gas. The pressure is 0.05 atmospheres when
the temperature is 18°C. Calculate the density of the gas (N × 28/V, where
28 is the molecular weight of nitrogen).
Keys:
Display:
Description:
Displays the equation.
{ G
ꢅº#/ꢄºꢁº!ꢎ
ꢅ@ꢑ)ꢑ ꢔꢕꢎ
#@ꢏ)ꢕꢕꢕꢕꢎ
ꢁ@ꢑ)ꢑꢙꢏꢔꢎ
!@ꢏꢓꢘ)ꢔꢕꢕꢕꢎ
ꢎ
N
Solves for N; prompts for P.
Stores .05 in P; prompts for V.
Stores 5 in V; prompts for H.
Retains previous R; prompts for T.
Calculates T (Kelvins).
{ œ
.05
f
5
f
f
18
šꢁ
273.1
™
!@ꢏꢓꢔ)ꢔꢕꢕꢕꢎ
ꢑꢂ#ꢊꢄꢆꢎ
ꢄ/ꢕ)ꢕꢔꢕꢗꢎ
ꢕ)ꢏꢓꢏꢓꢎ
Stores 291.1 in T; solves for N.
f
28
Calculates mass in grams, N × 28.
Calculates density in grams per
liter.
y
K V p
ꢕ)ꢕꢗꢙ ꢎ
Understanding and Controlling SOLVE
SOLVE uses an iterative (repetitive) procedure to solve for the unknown
variable. The procedure starts by evaluating the equation using two initial
guesses for the unknown variable. Based on the results with those two guesses,
SOLVE generates another, better guess. Through successive iterations, SOLVE
finds a value for the unknown that makes the value of the equation equal to
zero.
Solving Equations
7–5
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When SOLVE evaluates an equation, it does it the same way
does —
W
any "=" in the equation is treated as a " – " For example, the Ideal Gas Law
equation is evaluated as P V – (N R T). This ensures that an equality or
×
×
×
assignment equation balances at the root, and that an expression equation
equals zero at the root.
Some equations are more difficult to solve than others. In some cases, you
need to enter initial guesses in order to find a solution. (See "Choosing Initial
Guesses for SOLVE," below.) If SOLVE is unable to find a solution, the
calculator displays
.
ꢄꢑ ꢁꢑꢑ! ꢋꢄꢍ
See appendix C for more information about how SOLVE works.
Verifying the Result
After the SOLVE calculation ends, you can verify that the result is indeed a
solution of the equation by reviewing the values left in the stack:
ꢀ
The X–register (press
to clear the VIEWed variable) contains the
†
solution (root) for the unknown; that is, the value that makes the
evaluation of the equation equal to zero,
ꢀ
The Y–register (press
) contains the previous estimate for the root. This
9
number should be the same as the value in the X–register. If it is not, then
the root returned was only an approximation, and the values in the X–
and Y–registers bracket the root. These bracketing numbers should be
close together.
ꢀ
The Z– register (press
again) contains this value of the equation at
9
the root. For an exact root, this should be zero. If it is not zero, the root
given was only an approximation; this number should be close to zero.
If a calculation ends with the
, the calculator could not
ꢄꢑ ꢁꢑꢑ! ꢋꢄꢍ
converge on a root. (You can see the value in the X–register — the final
estimate of the root — by pressing † or a to clear the message.) The
values in the X– and Y–registers bracket the interval that was last searched to
find the root. The Z–register contains the value of the equation at the final
estimate of the root.
ꢀ
If the X– and Y–register values aren't close together, or the Z–register
value isn't close to zero, the estimate from the X–register probably isn't a
7–6
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root.
ꢀ
If the X– and Y–register values are close together, and the Z–register
value is close to zero, the estimate from the X–register may be an
approximation to a root.
Interrupting a SOLVE Calculation
To halt a calculation, press † or f. The current best estimate of the root
is in the unknown variable; use
stack.
to view it without disturbing the
{ ‰
Choosing Initial Guesses for SOLVE
The two initial guesses come from:
ꢀ
ꢀ
The number currently stored in the unknown variable.
The number in the X–register (the display).
These sources are used for guesses whether you enter guesses or not. If you
enter only one guess and store it in the variable, the second guess will be the
same value since the display also holds the number you just stored in the
variable. (If such is the case, the calculator changes one guess slightly so that
it has two different guesses.)
Entering your own guesses has the following advantages:
ꢀ
By narrowing the range of search, guesses can reduce the time to find a
solution.
ꢀ
If there is more than one mathematical solution, guesses can direct tote
SOLVE procedure to the desired answer or range of answers. For
example, the equation of linear motion
1
2
d = v t + / gt
0
2
can have two solutions for t. You can direct the answer to the only
meaningful one (t > 0) by entering appropriate guesses.
T
he example using this equation earlier in this chapter didn't require you
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to enter guesses before solving for T because in the first part of that
example you stored a value for T and solved for D. The value that was left
in T was a good (realistic) one, so it was used as a guess when solving
.
for T
ꢀ
If an equation does not allow certain values for the unknown, guesses
can prevent these values from occurring. For example,
y = t + log x
results in an error if x ≤ 0 (messages
or
).
ꢂꢑꢆ1ꢕ2
ꢂꢑꢆ1ꢄꢈꢆ2
In the following example, the equation has more than one root, but guesses
help find the desired root.
Example. Using Guesses to Find a Root.
Using a rectangular piece of sheet metal 40 cm by 80 cm, form an open–top
3
box having a volume of 7500 cm . You need to find the height of the box
(that is, the amount to be folded up along each of the four sides) that gives the
specified volume. A taller box is preferred to a shorter one.
H
_
40 2H
40
H
_
H
80 2H
H
80
7–8
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If H is the height, then the length of the box is (80 – 2H) and the width is (40 –
2H). The volume V is:
V = ( 80 – 2H ) (40 – 2H ) H
×
×
which you can simplify and enter as
V= ( 40 – H ) ( 20 – H ) 4 H
×
×
×
Type in the equation:
Keys:
Display:
Description:
{ Gꢁ
Selects Equation mode and
starts the equation.
ꢎ
V
K { c
#/¾ꢎ
40
{ \
„ꢁ
ꢎ
H
K { ]
#/1ꢒꢕ.ꢐ2¾ꢎ
„
ꢕ.ꢐ2º1ꢏꢕ.ꢐ2¾ꢎ
20
y { \
H
K { ]
4
y y K
H
º1ꢏꢕ.ꢐ2ºꢒºꢐ¾ꢎ
#/1ꢒꢕ.ꢐ2º1ꢏꢕꢎ
Terminates and displays the
equation.
š
Checksum and length.
ꢃꢚ/ꢕꢏꢀꢃ ꢕꢏꢘ)ꢕꢎ
{
It seems reasonable that either a tall, narrow box or a short, flat box could be
formed having the desired volume. Because the taller box is preferred, larger
initial estimates of the height are reasonable. However, heights greater than
20 cm are not physically possible because the metal sheet is only 40 cm wide.
Initial estimates of 10 and 20 cm are therefore appropriate.
Keys:
Display:
Description:
Leaves Equation mode.
Stores lower and upper limit
guesses.
†
ꢎ
10
H 20
H
ꢏꢕ_ꢎ
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Displays current equation.
Solves for H; prompts for V.
Stores 7500 in V; solves for H.
{ G
#/1ꢒꢕ.ꢐ2ºꢏꢕꢎ
#@valueꢎ
H
{ œ
7500
f
ꢐ/ꢔꢗ)ꢕꢕꢕꢕꢎ
Now check the quality of this solution — that is, whether it returned an exact
root — by looking at the value of the previous estimate of the root (in the
Y–register) and the value of the equation at the root (in the Z–register).
Keys:
Display:
Description:
This value from the Y–register is the
estimate made just prior to the final
result. Since it is the same as the
solution, the solution is an exact root.
This value from the Z–register shows
the equation equals zero at the root.
9
ꢔꢗ)ꢕꢕꢕꢕꢎ
9
ꢕ)ꢕꢕꢕꢕꢎ
The dimensions of the desired box are 50 × 10 × 15 cm. If you ignored the
upper limit on the height (20 cm) and used initial estimates of 30 and 40 cm,
you would obtain a height of 42.0256 cm — a root that is physically
meaningless. If you used small initial estimates such as 0 and 10 cm, you
would obtain a height of 2.9774 cm — producing an undesirably short, flat
box.
If you don't know what guesses to use, you can use a graph to help the
behavior of the equation. Evaluate your equation for several values of the
unknown. For each point on the graph, display the equation and press
— at the prompt for x enter the x–coordinate, and then obtain the
Wꢁ
corresponding value of the equation, the y–coordinate. For the problem
above, you would always set V = 7500 and vary the value of H to produce
different values for the equation. Remember that the value for this equation is
the difference between the left and right sides of the equation. The plot of the
value of this equation looks like this.
7–10 Solving Equations
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_
_
_
7500 (40 H) (20 H) 4H
20,000
H
_
10
50
_
10,000
For More Information
This chapter gives you instructions for solving for unknowns or roots over a
wide range of applications. Appendix C contains more detailed information
about how the algorithm for SOLVE works, how to interpret results, what
happens when no solution is found, and conditions that can cause incorrect
results.
Solving Equations 7–11
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8
Integrating Equations
Many problems in mathematics, science, and engineering require calculating
the definite integral of a function– If the function is denoted by f(x) and the
interval of integration is a to b, then the integral can be expressed
mathematically as
b
I = f(x)dx
∫
a
f (x)
I
x
a
b
The quantity I can be interpreted geometrically as the area of a region
bounded by the graph of the function f(x), the x–axis, and the limits x = a and
x = b (provided that f(x) is nonnegative throughout the interval of integration).
The operation
operation (∫ FN) integrates the current equation with
)
respect to a specified variable (∫
d_). The function may have more than
ꢋꢄ
one variable.
Integrating Equations
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works only with real numbers.
)
Integrating Equations ( ∫ FN)
To Integrating Equations:
To integrate an equation:
1. If the equation that defines the integrand's function isn't stored in the
equation list, key it in (see "Entering Equations Into the Equation List" in
chapter 6) and leave Equation mode. The equation usually contains just
an expression.
2. Enter the limits of integration: key in the lower limit and press
,
š
then key in the upper limit.
3. Display the equation: Press
and, if necessary, scroll through
{ G
the equation list (press
or
z — z ˜
) to display the desired
equation.
4. Select the variable of integration: Press
variable. This starts the
{ )
calculation.
uses far more memory than any other operation in the calculator. If
)
executing ) causes a
message, refer to appendix B.
ꢇꢈꢇꢑꢁ& ꢋ"ꢂꢂ
You can halt a running integration calculation by pressing
or
.
†
f
However, no information about the integration is available until the
calculation finishes normally
The display format setting affects the level of accuracy assumed for your
function and used for the result. The integration is more precise but takes
much longer in the {
ꢀꢂꢂ
} and higher { }, { }, and { } settings. The
ꢋ% ꢃ ꢈꢄ
uncertainty of the result ends up in the Y–register, pushing the limits of
integration up into the T– and Z–registers. For more information, see
"Accuracy of Integration" later in this chapter.
8–2
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To integrate the same equation with different information:
If you use the same limits of integration, press move them into the X–
9 9
and Y–registers. Then start at step 3 in the above list. If you want to use
different limits, begin at step 2.
To work another problem using a different equation, start over from step 1
with an equation that defines the integrated.
Example: Bessel Function.
The Bessel function of the first kind of order 0 can be expressed as
πcos(xsint)dt
1
π
J0 =
∫
0
Find the Bessel function for x–values of 2 and 3.
Enter the expression that defines the integrand's function:
cos (x sin t )
Keys:
Display:
Description:
{ALL}
Clears memory.
z b
{Y}
ꢎ
Selects Equation mode.
Types the equation.
{ G
Current equation or
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅꢎ
X
Q K
y N
K T
ꢃꢑ 1%¾ꢎ
ꢃꢑ 1%º ꢊꢄ1¾ꢎ
ꢃꢑ 1%º ꢊꢄ1!¾ꢎ
1%º ꢊꢄ1!22¾ꢎ
Right closing parentheses are
optional.
{ ] { ]
Terminates the expression and
displays its left end.
š
ꢃꢑ 1%º ꢊꢄ1!2ꢎ
Checksum and length.
ꢃꢚ/ꢋꢓꢖꢌ ꢕꢔꢏ)ꢕꢎ
{ꢁ
Integrating Equations
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Leaves Equation mode.
Now integrate this function with respect to t from zero to π ; x = 2.
†
ꢎ
Keys:
Display:
Description:
z Ÿ {
}
Selects Radians mode.
Enters the limits of integration
(lower limit first).
ꢎ
ꢁꢍ
0
š { M
ꢖ)ꢔꢒꢔ ꢎ
Displays the function.
Prompts for the variable of
integration.
{ G
{ )
ꢃꢑ 1%º ꢊꢄ1!2ꢎ
∫ ꢋꢄ G_ꢎ
T
Prompts for value of X.
x = 2. Starts integrating;
%@value
2
f
ꢎ
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
calculates result for
πf(t)
∫ /ꢕ)ꢘꢕꢖꢒꢎ
∫
0
The final result for
{ M p
ꢕ)ꢏꢏꢖꢓꢎ
J (2).
0
Now calculate J (3) with the same limits of integration. You must respecify the
0
limits of integration (0, ) since they were pushed off the stack by the
π
subsequent division by π.
Keys:
Display:
Description:
0
Enters the limits of integration
(lower limit first).
š { M
ꢖ)ꢔꢒꢔ ꢎ
Displays the current equation.
{ G
{ )
ꢃꢑ 1%º ꢊꢄ1!2ꢎ
Prompts for the variable of
∫ ꢋꢄ G_
integration.
T
Prompts for value of X.
x = 3. Starts integrating and
calculates the result for
%@ꢏ)ꢕꢕꢕꢕꢎ
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
∫ /.ꢕ)ꢙꢘꢕꢎ
3
f
πf(t)
.
∫
0
The final result for
J (3).
{ M p
–0.260
0
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Integrating Equations
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Example: Sine Integral.
Certain problems in communications theory (for example, pulse transmission
through idealized networks) require calculating an integral (sometimes called
the sine integral) of the form
t
sinx
x
S(t) = (
)dx
i
∫
0
Find Si (2).
Enter the expression that defines the integrand's function:
sinx
x
If the calculator attempted to evaluate this function at x = 0, the lower limit of
integration, an error (
) would result. However, the integration
ꢍꢊ#ꢊꢍꢈ ꢌ& ꢕ
algorithm normally does not evaluate functions at either limit of integration,
unless the endpoints of the interval of integration are extremely close together
or the number of sample points is extremely large.
Keys:
Display:
Description:
Selects Equation mode.
{ G
The current equation
or ꢈꢉꢄ ꢂꢊ ! !ꢑꢅꢎ
X
Starts the equation.
N K
{ ]
ꢊꢄ1%¾ꢎ
The closing right parenthesis is
required in this case.
ꢊꢄ1%2¾ꢎ
X
p K
š
{
†
ꢊꢄ1%2ª%¾ꢎ
ꢊꢄ1%2ª%ꢎ
ꢃꢚ/ꢒꢓꢔꢓ ꢕꢕꢓ)ꢕꢎ
ꢎ
Terminates the equation.
Checksum and length.
Leaves Equation mode.
Now integrate this function with respect to x (that is, X) from zero to 2 (t = 2).
Keys:
Ÿ {
Display:
Description:
}
Selects Radians mode.
ꢎ
ꢁꢍ
Integrating Equations
8–5
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0
2
Enters limits of integration (lower
first).
š
ꢏ_ꢎ
Displays the current equation.
Calculates the result for Si(2).
{ G
ꢊꢄ1%2ª%ꢎ
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
∫ /ꢔ) ꢕꢗꢒꢎ
X
{ )
Accuracy of Integration
Since the calculator cannot compute the value of an integral exactly, it
approximates it. The accuracy of this approximation depends on the
accuracy of the integrand's function itself, as calculated by your equation.
This is affected by round–off error in the calculator and the accuracy of the
empirical constants.
Integrals of functions with certain characteristics such as spikes or very rapid
oscillations might be calculated inaccurately, but the likelihood is very small.
The general characteristics of functions that can cause problems, as well as
techniques for dealing with them, are discussed in appendix D.
Specifying Accuracy
The display format's setting (FIX, SCI, ENG, or ALL) determines the precision
of the integration calculation, the greater the number of digits displayed, the
greater the precision of the calculated integral (and the greater the time
required to calculate it.). The fewer the number of digits displayed, the faster
the calculation, but the calculator will presume that the function is accurate to
only the number of digits specified in the display format.
To specify the accuracy of the integration, set the display format so that the
display shows no more than the number of digits that you consider accurate in
the integrand's values. This same level of accuracy and precision will be
reflected in the result of integration.
If Fraction–display mode is on (flag 7 set), the accuracy is specified by the
previous display format.
8–6
Integrating Equations
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Interpreting Accuracy
After calculating the integral, the calculator places the estimated uncertainty
of that integral's result in the Y–register. Press
uncertainty.
to view the value of the
Z
For example, if the integral Si(2) is 1.6054 0.0001, then 0.0001 is its
uncertainty.
Example: Specifying Accuracy.
With the display format set to SCI 2, calculate the integral in the expression
for Si(2) (from the previous example).
Keys:
Display:
Description:
{SC} 2
Sets scientific notation with two
decimal places, specifying that
the function is accurate to two
decimal places.
z ž
ꢔ) ꢔꢈꢕꢎ
Rolls down the limits of
integration frown the Z–and
T–registers into the X–and
Y–registers.
9 9
ꢏ)ꢕꢕꢈꢕꢎ
{ G
Displays the current Equation.
The integral approximated to two
decimal places.
ꢊꢄ1%2ª%ꢎ
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
∫ /ꢔ) ꢕꢈꢕꢎ
ꢔ)ꢕꢕꢈ.ꢖꢎ
X
{ )
The uncertainty of the
Z
approximation of the integral.
The integral is 1.61 0.00100. Since the uncertainty would not affect the
approximation until its third decimal place, you can consider all the displayed
digits in this approximation to be accurate.
If the uncertainty of an approximation is larger than what you choose to
tolerate, you can increase the number of digits in the display format and
repeat the integration (provided that f(x) is still calculated accurately to the
number of digits shown in the display), In general, the uncertainty of an
Integrating Equations
8–7
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integration calculation decreases by a factor of ten for each additional digit,
specified in the display format.
Example: Changing the Accuracy.
For the integral of Si(2) just calculated, specify that the result be accurate to
four decimal places instead of only two.
Keys:
Display:
Description:
{
ꢃ
} 4
Specifies accuracy to four
decimal places. The uncertainty
from the last example is still in the
display.
z ž
ꢔ)ꢕꢕꢕꢕꢈ.ꢖꢎ
Rolls down the limits of
integration from the Z– and
T–registers into the X– and
Y–registers.
9 9
ꢏ)ꢕꢕꢕꢕꢈꢕꢎ
{ G
Displays the current equation.
Calculates the result.
ꢊꢄ1%2ª%ꢎ
X
{ )
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
∫ /ꢔ) ꢕꢗꢒꢈꢕꢎ
Note that the uncertainty is about
Z
ꢔ)ꢕꢕꢕꢕꢈ.ꢗꢎ
1/100 as large as the
uncertainty of the SCI 2 result
calculated previously.
{
ꢋ%
} 4
Restores FIX 4 format.
z ž
ꢔ)ꢕꢕꢕꢕꢈ.ꢗꢎ
{ }
ꢍꢈꢆ ꢔ)ꢕꢕꢕꢕꢈ.ꢗꢎ
Restores Degrees mode.
z Ÿ
This uncertainty indicates that the result might be correct to only four decimal
places. In reality, this result is accurate to seven decimal places when
compared with the actual value of this integral. Since the uncertainty of a
result is calculated conservatively, the calculator's approximation in most
cases is more accurate than its uncertainty indicates.
8–8
Integrating Equations
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For More Information
This chapter gives you instructions for using integration in the HP 32SII over a
wide range of applications. Appendix D contains more detailed information
about how the algorithm for integration works, conditions that could cause
incorrect results, conditions that prolong calculation time, and obtaining the
current approximation to an integral.
Integrating Equations
8–9
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9
Operations with Comb Numbers
The HP 32SII can use complex numbers in the form
x
iy
+ .
It has operations for complex arithmetic (+, –, ×, ÷), complex trigonometry (sin,
z1z
z
2
,
cos, tan), and the mathematics functions –z, 1/z,
ln z, and e . (where
z
2
z and
1
are complex numbers).
To enter a complex number:
1. Type the imaginary part.
2. Press
.
š
3. Type the real part.
Complex numbers in the HP 32SII are handled by entering each part
(imaginary and real) of a complex number as a separate entry. To enter two
complex numbers, you enter four separate numbers. To do a complex
operation, press
before the operator. For example, to do
z F
(2 + i 4) + (3 + i 5),
press 4
2
š š š z F ™
5
3
.
The result is 5 + i 9. (Press Z to see the imaginary part.)
The Complex Stack
The complex stack is really the regular memory stack split into two double
registers for holding two complex numbers, z , + i z and z + i z :
1X
1y
2X
2y
Operations with Comb Numbers
9–1
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T
Z
Y
X
t
iy
1
Z1
Z2
z
y
x
x
1
iy
2
x
2
Real Stack
Complex Stack
Since the imaginary and real parts of a complex number are entered and
stored separately, you can easily work with or alter either part by itself.
y
1
x
1
y
2
x
2
Z1
Z2
Complex function
y
x
imaginary part
real part
Complex input
z or z and z
Complex result, z
1
2
Always enter the imaginary part (the y–part)of a number first. The real portion
of the result (z ) is displayed; press to view the imaginary portion (z ).
Z
x
y
(For two–number operations, the first complex number, z , is replicated in the
1
stack's Z and T registers.)
9–2
Operations with Comb Numbers
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Complex Operations
Use the complex operations as you do real operations, but precede the
operator with
.
z F
To do an operation with one complex number:
1. Enter the complex number z, composed of x + i y, by keying in y
š
x.
2. Select the complex function.
Functions for One Complex Number, z
To Calculate:
Press:
z
z F _
z F 3
z F -
z F *
z F N
z F Q
z F T
Change sign,–
z
Inverse, 1/
z
Natural log, ln
z
e
Natural antilog,
z
Sin
z
Cos
z
Tan
Operations with Comb Numbers
9–3
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To do an arithmetic operation with two complex numbers:
1. Enter the first complex number, z (composed of x + i y ), by keying in y
1
1
1
1
z1z
š x š. (For
2 , key in the base part, z , first.)
1
1
2. Enter the second complex number, z , by keying in y
x . (For
2
š
2
2
z1z
2 , key in the exponent, z , second.)
2
3. Select the arithmetic operation:
Arithmetic With Two Complex Numbers, z1 and z2
To Calculate:
Press:
Addition, z + z
z F ™
z F „
z F y
z F p
z F 0
1
2
Subtraction, z – z
1
2
Multiplication, z × z
1
2
Division, z ÷ z
1
2
z1z
2
Power function,
Examples:
Here are some examples of trigonometry and arithmetic with complex
numbers:
Evaluate sin (2 + i 3)
Keys:
Display:
Description:
3
2
Real part of result.
š
ꢎ
z F N
Z
ꢓ)ꢔꢗꢒꢗꢎ
.ꢒ)ꢔ ꢙꢓꢎ
Result is 9.1545 – i 4.1689.
Evaluate the expression
z
(z + z ),
÷
1
2
3
where z = 23 + i 13, z = –2 + i z = 4 – i
1
2
2
3
Since the stack can retain only two complex numbers at a time, perform the
calculation as
9–4
Operations with Comb Numbers
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z
1
[1 (z + z )]
×
÷
2
3
Keys:
Display:
Description:
1
3 _ š 4 zꢁ
2
Add z + z ; displays real
š _ šꢁ
ꢎ
2
3
part.
ꢎ
F ™
ꢏ)ꢕꢕꢕꢕꢎ
ꢕ)ꢏꢗꢕꢕꢎ
ꢎ
1 ÷ (z +z ).
z F 3
2
3
13
23
z ÷ (z +z ).
š
1
2
3
z F y
Z
ꢏ)ꢗꢕꢕꢕꢎ
ꢓ)ꢕꢕꢕꢕꢎ
Result is 2.5 + i 9.
Evaluate (4 – i 2/5) (3 – i 2/3). Do not use complex operations
when
part of a complex number.
calculating just one
Keys:
Display:
Description:
2
Œ Œ _ š
5
Enters imaginary part of first
complex number as a
fraction.
.ꢕ)ꢒꢕꢕꢕꢎ
4
Enters real part of first
complex number.
š
ꢒ)ꢕꢕꢕꢕꢎ
2
Œ Œ _ š
3
Enters imaginary part of
second complex number as a
fraction.
.ꢕ)
ꢘꢎ
3 z F y
Completes entry of second
number and then multiplies
ꢔꢔ)ꢘꢖꢖꢖꢎ
the two
complex
numbers.
Result is 11.7333 – i
3.8667.
Z
.ꢖ)ꢙ ꢘꢎ
−2 , where z = (1 + i ). Use
to evaluate z ;
–2
ez
Evaluate
enter –2 as –2 + i 0.
z F 0
Keys:
Display:
Description:
Operations with Comb Numbers
9–5
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1
0
1
š š
Intermediate result of
ꢎ
–2
2
ꢎ
(1 + i )
š _ zꢁ
ꢕ)ꢕꢕꢕꢕꢎ
F 0
z F *
Z
Real part of final results.
Final result is
ꢕ)ꢙꢘꢘ ꢎ
.ꢕ)ꢒꢘꢓꢒꢎ
0.8776 – i 0.4794.
Using Complex Number in Polar Notation
Many applications use real numbers in polar form or polar notation. These
forms use pairs of numbers, as do complex numbers, so you can do arithmetic
with these numbers by using the complex operations. Since the HP 32SII's
complex operations work on numbers in rectangular form, convert polar form
to rectangular form (using
before executing the complex
{ꢁr
operation, then convert the result back to polar form.
iθ
a + i b = r (cos θ + i sin θ) = re
= r ∠θ (Polar or phasor form)
imaginary
(a, b)
r
θ
real
Example: Vector Addition.
9–6
Operations with Comb Numbers
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Add the following three loads. You will first need to convert the polar
coordinates to rectangular coordinates.
y
62o
L
185 lb
2
170 lb
143o
L
1
x
L
3
100 lb
261 o
Keys:
Display:
Description:
{
ꢍꢆ
}
Sets Degrees mode.
z Ÿ
ꢎ
62
{ r
185
Enters L and converts it to
š
ꢎ
1
rectangular form.
ꢙ )ꢙꢗꢏꢏꢎ
143
r
170
Eaters and converts L .
š
{
.ꢔꢖꢗ)ꢘ ꢙꢕꢎ
2
Adds vectors.
z F ™
.ꢒꢙ)ꢓꢔꢗꢙꢎ
.ꢔꢗ) ꢒꢖꢒꢎ
261 š 100 {
r
Enters and converts L .
3
z F ™
z q
Adds L + L + L .
. ꢒ)ꢗꢗꢓꢏꢎ
ꢔꢘꢙ)ꢓꢖꢘꢏꢎ
I
2
3
Converts vector hack to polar
form; displays r.
Displays θ.
Z
ꢔꢔꢔ)ꢔꢒꢙꢓꢎ
Operations with Comb Numbers
9–7
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10
Base Conversions and Arithmetic
The BASE menu (
) lets you change the number base used for
z w
entering numbers and other operations (including programming). Changing
bases also converts the displayed number to the new base.
BASE Menu
Menu label
Description
{
}
Decimal mode. No annunciator. Converts numbers to
base 10. Numbers have integer and fractional parts.
ꢍꢈꢃ
{
ꢐ%
}
Hexadecimal mode. HEX annunciator on. Converts
numbers to base 16; uses integers only. The top–row
keys become digits
through
.
ꢈ
ꢉ
{
{
}
Octal mode. OCT annunciator on. Converts numbers
to base 8; uses integers only. The ꢊ, ꢃ, and
unshifted top–row keys are inactive.
ꢑꢃ
}
Binary mode. BIN annunciator on. Converts numbers
to base 2; uses integers only. Digit keys other than
ꢋ and ꢆ, and the unshifted top–row functions are
inactive. If a number is longer than 12 digits, then the
ꢌꢄ
outer top–row keys (
and
are active for
<
6
viewing windows. (See "Windows for Long Binary
Numbers" later in this chapter.)
Examples: Converting the Base of a Number.
The following keystrokes do various base conversions.
Convert 125.99 to hexadecimal, octal, and binary numbers.
10
Keys:
Display:
Description:
125.99
Converts just the integer part (125)
ꢘꢍꢎ
z
Base Conversions and Arithmetic 10–1
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{
}
of the decimal number to base 16
and displays this value.
Base 8.
w
ꢐ%
{
{
{
}
}
z w
z w
z w
ꢑꢃ
ꢔꢘꢗꢎ
Base 2.
ꢔꢔꢔꢔꢔꢕꢔꢎ
ꢌꢄ
}
Restores base 10; the original
ꢔꢏꢗ)ꢓꢓꢕꢕꢎ
ꢍꢈꢃ
decimal value has been preserved,
including its fractional part.
Convert 24FF to binary base. The binary number will be more than 12
16
digits (the maximum display) long.
Keys:
Display:
Description:
{HX}
Use the
ꢏꢒꢋꢋ_ꢎ
key to type "F".
z w
24FF
6
{
ꢌꢄ
}
The entire binary number does riot
ꢕꢔꢕꢕꢔꢔꢔꢔꢔꢔꢔꢔꢎ
z w
fit. The
annunciator indicates
that the number continues to the
left; the annunciator Points to
ž
<.
Displays the rest of the number.
<
6
ꢔꢕꢎ
The full number is
10010011111111 .
2
Displays the first 12 digits again.
ꢕꢔꢕꢕꢔꢔꢔꢔꢔꢔꢔꢔꢎ
{
}
Restores base 10.
ꢓ8ꢒꢘꢔ)ꢕꢕꢎ
z w
ꢍꢈꢃ
Arithmetic in Bases 2, 8, and 16
You can perform arithmetic operations using (
,
,
, and
) in any
p
™ „ y
base. The only function keys that are actually deactivated outside of Decimal
mode are and . However, you should realize
,
,
,
,
< * - 0 3
6
that most operations other than arithmetic will not produce meaningful results
since the fractional parts of numbers are truncated.
10–2 Base Conversions and Arithmetic
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Arithmetic in bases 2, 8, and 16 is in 2's complement form and uses integers
only:
ꢀ
If a number has a fractional part, only the integer part is used for an
arithmetic calculation.
ꢀ
The result of an operation is always an integer (any fractional portion is
truncated).
Whereas conversions change only the displayed number and not the number
in the X–register, arithmetic does alter the number in the X–register.
If the result of an operation cannot be represented in 36 bits, the display
shows
and then shows the largest positive or negative number
ꢑ#ꢈꢁꢋꢂꢑ$
possible.
Example:
Here are some examples of arithmetic in Hexadecimal, Octal, and Binary
modes:
?
12F + E9A
=
16
16
Keys:
Display:
Description:
{
}
Sets base 16; HEX
annunciator on.
Result.
z w
ꢐ%
ꢎ
12F
E9A
š
™
ꢋꢃꢓꢎ
?
7760 – 4326 =
8
8
{
ꢑꢃ
}
Sets base 8: OCT
ꢘꢘꢔꢔꢎ
z w
annunciator on. Converts
displayed number to octal.
Result.
7760
4326
š
„
ꢖꢒꢖꢏꢎ
?
100 – 5 =
8
8
100 š 5 p
Integer part of result.
ꢔꢒꢎ
?
5A0 + 1001100 =
16
2
z w { } 5A0
ꢐ%
Set base 16; HEX
ꢗꢀꢕ_ꢎ
Base Conversions and Arithmetic 10–3
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annunciator on.
Changes to base 2; BIN
z w { } 1001100
ꢌꢄ
ꢔꢕꢕꢔꢔꢕꢕ_ꢎ
annunciator on. This
terminates digit entry, so no
is needed between
š
the numbers.
Result in binary base.
ꢔꢕꢔꢔꢔꢔꢕꢔꢔꢕꢕꢎ
™
{
ꢐ%
}
Result in hexadecimal base.
ꢗꢈꢃꢎ
z w
z w {
}
Restores decimal base.
ꢔ8ꢗꢔ )ꢑꢑꢑꢑꢎ
ꢍꢈꢃ
The Representation of Numbers
Although the display of a number is converted when the base is changed, its
stored form is not modified, so decimal numbers are not truncated — until
they are used in arithmetic calculations.
When a number appears in hexadecimal, octal, or binary base, it is shown
as a right–justified integer with up to 36 bits (12 octal digits or 9 hexadecimal
digits). Leading zeros are riot displayed, but they are important because they
indicate a positive number. For example, the binary representation of 125
is displayed as:
10
11111101
which is the same as these 36 digits:
000000000000000000000000000001111101
Negative Numbers
The leftmost (most significant or "highest") bit of a number's binary
representation is the sign bit; it is set (1) for negative numbers. If there are
(undisplayed) leading zeros, then the sign bit is 0 (positive). A negative
number is the 2's complement of its positive binary number.
Keys:
Display:
Description:
10–4 Base Conversions and Arithmetic
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546
{
}
Enters a positive, decimal
ꢏꢏꢏꢎ
z w
ꢐ%
number; then converts it to
hexadecimal.
2's complement (sign
ꢋꢋꢋꢋꢋꢋꢍꢍꢈꢎ
_
changed).
{
ꢌꢄ
}
Binary version;
indicates
z w
< <
ꢔꢔꢕꢔꢔꢔꢕꢔꢔꢔꢔꢕꢎ
more digits.
Displays the leftmost
ꢔꢔꢔꢔꢔꢔꢔꢔꢔꢔꢔꢔꢎ
window; the number is
negative since the highest bit
is 1.
{
}
Negative decimal number.
z w
ꢍꢈꢃ
.ꢗꢒ )ꢕꢕꢕꢕꢎ
Range of Numbers
The 36-bit word size determines the range of numbers that can be
represented in hexadecimal (9 digits), octal (12 digits), and binary bases
(36 digits), and the range of decimal numbers (11 digits) that can be
converted to these other bases.
Range of Numbers for Base Conversions
Base
Positive Integer
of Largest Magnitude
Negative Integer
of Largest Magnitude
Hexadecimal 7FFFFFFFF
800000000
400000000000
Octal
377777777777
Binary
011111111111111111111 100000000000000000000
111111111111111
34,359,738,367
000000000000000
–34,359,738;368
Decimal
When you key in numbers, the calculator will not accept more than the
maximum number of digits for each base. For example, if you attempt to key
in a 10–digit hexadecimal number, digit entry halts and the £ annunciator
appears.
Base Conversions and Arithmetic 10–5
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If a number entered in decimal base is outside the range given above, then it
produces the message in the other base modes. Any operation
!ꢑꢑ ꢌꢊꢆ
causes an overflow condition, which substitutes the largest
using
!ꢑꢑ ꢌꢊꢆ
positive or negative number possible for the too–big number.
Windows for Long Binary Numbers
The longest binary number can have 36 digits–three times as many digits as
fit in the display. Each 12–digit display of a long number is called a window.
36 - bit number
Lowest window
(displayed)
Highest window
When a binary number is larger than the 12 digits, the
or
annunciator
€
(or both) appears, indicating in which direction the additional digits lie. Press
the indicated key ( < or 6 ) to view the obscured window.
10-7B Picture
SHOWing Partially Hidden Numbers
The
and
functions work with non–decimal numbers
{ ‰
z ˆ
as they do with decimal numbers. However, if the Bali octal or binary number
does not fit in the display, the leftmost digits are replaced with an ellipsis
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(
)))
ꢀ@
). Press
…label.
to view the digits obscured by the
… or
{
ꢀ/
Keys:
Display:
Description:
{
ꢑꢃ
}
Enters a large octal number.
z w
ꢏꢖꢒꢗ ꢘꢔꢏꢖꢒꢗ_ꢎ
123456712345
A
H
ꢔꢏꢖꢒꢗ ꢘꢔꢏꢖꢒꢗꢎ
ꢀ/...ꢒꢗ ꢘꢔꢏꢖꢒꢗꢎ
ꢔꢏꢖꢒꢗ ꢘꢔꢏꢖꢒꢗ
A
Drops leftmost three digit's.
Shows all digits.
{ ‰
(hold)
{
{
}
Restores Decimal mode.
ꢔꢔ8ꢏꢔꢓ8ꢒꢘꢖ8 ꢖꢘ)ꢕꢎ
z w
ꢍꢈꢃ
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11
Statistical Operations
The statistics menus in the HP 32SII provide functions to statistically analyze a
set of one– or two–variable data:
ꢀ
Mean, sample and population standard deviations.
ˆ
ˆ
ꢀ
ꢀ
ꢀ
Linear regression and linear estimation (x and y).
Weighted mean (x weighted by y).
2
2
A Summation statistics: n, Σx, Σy, Σx , Σy , and Σxy.
s ,
σ
SUMS
x , y
L.R.
r
sx
sy
σ
x
σ
y
x
y
m
b
x
2
2
n
x
y x y xy
y
x w
Entering Statistical Data
One– and two–variable statistical data are entered (or deleted) in similar
fashion using the 6 (or z 4 ) key. Data values are accumulated as
summation statistics in six statistic's registers (28 through 33), whose names
ꢏ
ꢏ
are displayed ire the SUMS menu. (Press { 5 and see
.
Qº¸º ¸ º¸
Note
Always clear the statistics registers before entering a new set of
statistical data (press { } ).
z b Σ
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Entering One–Variable Data
1. Press
{Σ} to clear existing statistical data.
z b
2. Key in each x–value and press
.
6
3. The display shows n, the number of statistical data values now
accumulated.
Pressing
actually enters two variables into the statistics registers because
6
the value already in the Y–register is accumulated as the y–value. For this
reason, the calculator will perform linear regression and show you values
based on y even when you have entered only x–data — or even if you have
entered an unequal number of x–and y–values. No error occurs, but the
results are obviously not meaningful.
To recall a value to the display immediately after it has been entered, press
.
z Ž
Entering Two–Variable Data
When your data consist of two variables, x is the independent variable and y
is the dependent variable. Remember to enter an (x, y) pair in reverse order (y
x) so that y ends up in the Y–register and X in the X–register.
š
1. Press
{ } to clear existing statistical data.
z b Σ
2. Key in the y–value first and press
.
š
3. Key in the corresponding x–value and press 6.
4. The display shows n, the number of statistical data pairs you have
accumulated.
5. Continue entering x, y–pairs. n is updated with each entry.
To recall an x–value to the display immediately after it has been entered,
press
.
z Ž
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Correcting Errors in Data Entry
If you make a mistake when entering statistical data, delete the incorrect data
and add the correct data. Even if only one value of an x, y–pair is incorrect,
you must delete and reenter both values.
To correct statistical data:
1. Reenter the incorrect data, but instead of pressing 6, press z 4.
This deletes the value(s) and decrements n.
2. Enter the correct value(s) using
.
6
If the incorrect values were the ones just entered, press
to
z Ž
retrieve them, then press
to delete them. (The incorrect y–value was
z 4
still in the Y–register, and its T–value was saved in the LAST X register.)
Example:
Key in the x, y–values on the left, these make the corrections shown on the
right:
Initial x, y
Corrected x, y
20,4
20,5
40,6
400,6
Keys:
Display:
Description:
{ }
´
Clears existing statistical
data.
z b
ꢎ
4
6
20
Enters the first new data pair.
Display shows n, the number
of data pairs yon entered.
Brings back last x–value. Last
y is still in Y–register. (Press
š
š
6
ꢔ)ꢕꢕꢕꢕꢎ
ꢏ)ꢕꢕꢕꢕꢎ
400
6
z Ž
ꢒꢕꢕ)ꢕꢕꢕꢕꢎ
twice to check y.)
Z
Deletes the last data pair.
Reenters the last data pair.
z 4
ꢔ)ꢕꢕꢕꢕꢎ
ꢏ)ꢕꢕꢕꢕꢎ
6
40
š
6
Statistical Operations 11–3
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4
5
20
20
Deletes the first data pair.
Reenters the first data pair.
There is still a. total of two
data pairs in the statistics
registers.
š
š
z 4
6
ꢔ)ꢕꢕꢕꢕꢎ
ꢏ)ꢕꢕꢕꢕꢎ
Statistical Calculations
Once you have entered your data, you can use the functions in the statistics
menus.
Statistics Menus
Menu
Key
Description
L.R.
The linear–regression menu: linear
{ ,
ˆ ˆ
estimation { } { } and curve–fitting { }
T
º ¸
{ } { }. See ''Linear Regression'' later in
P E
this chapter.
y
x,
z /
z 2
The mean menu: { } { } {
}. See
º ¸ º·
"Mean" below.
s,σ
The standard–deviation menu: { } {
}
Uº U¸
{σ } {σ }. See "Sample Standard
º
¸
Deviation" and "Population Standard
Deviation" later in this chapter.
ꢏ
ꢏ
SUMS
The summation menu: { } { } { } { } { }
º¸
z 5
Q º ¸ º
¸
{
}. See "Summation Statistics" later in
this chapter.
Mean
Mean is the arithmetic average of a group of numbers.
ꢀ
ꢀ
ꢀ
Press { / { } for the mean of the x–values.
º
Press
Press
{ } for the mean of the y–values.
{ /
{ /
¸
{
} for the weighted mean of the x–values using the
º·
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y–values as weights or frequencies. The weights can be integers or
non–integers.
Example: Mean (One Variable).
Production supervisor May Kitt wants to determine the average time that a
certain process takes. She randomly picks six people, observes each one as
he or she carries out the process, and records the time required (in minutes):
15.5
12.5
9.25
12.0
10.0
8.5
Calculate the mean of the times. (Treat all data as x–values.)
Keys:
Display:
Description:
z b { }
´
Clears the statistics registers.
Enters the first time.
ꢎ
15.5
6
ꢔ)ꢕꢕꢕꢕꢎ
ꢎ
9.25 6 10 6 12.5
Enters the remaining data;
six data points accumulated.
Calculates the mean time to
complete the process.
12
8.5
6
6
6
)ꢕꢕꢕꢕꢎ
ꢔꢔ)ꢏꢓꢔꢘꢎ
{ }
º
{ /
Example: Weighted Mean (Two Variables).
A manufacturing company purchases a certain part four times a year. Last
year's purchases were:
Price per Part (x)
$4.25 $4.60 $4.70 S4.10
800 900 1000
Number of Parts (y) 250
Find the average: price (weighted for the purchase quantity) for this part.
Remember to enter y, the weight (frequency), before x, the price.
Keys:
Display:
Description:
{ }
´
Clears the statistics registers.
z b
ꢎ
250
800
900
4.25
4.6
Enters data; displays n.
š
š
š
6ꢁ
ꢔ)ꢕꢕꢕꢕꢎ
ꢏ)ꢕꢕꢕꢕꢎ
ꢖ)ꢕꢕꢕꢕꢎ
6ꢁ
4.7
6
Statistical Operations 11–5
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1000
4.1
}
Four data pairs accumulated.
Calculates the mean price
weighted for the quantity
purchased.
šꢁ
6
ꢒ)ꢕꢕꢕꢕꢎ
ꢒ)ꢒꢖꢔꢒꢎ
{
{ /
º·
Sample Standard Deviation
Sample standard deviation is a measure of how dispersed the data values are
about the mean. standard deviation assumes the data is a sampling of a
larger, complete set of data, and is calculated using n – 1 as a divisor.
ꢀ
ꢀ
Press
Press
{
{
} for the standard deviation of x–values.
} for the standard deviation of y–values.
{ 2
{ 2
Uº
U¸
The {σ } and {σ } keys in this menu are described in the next section,
º
¸
"Population Standard Deviation."
Example: Sample Standard Deviation.
Using the same process–times as in the above "mean" example, May Kitt
now wants to determine the standard deviation time (s ) of the process:
x
9.25
12.0
15.5
12.5
10.0
8.5
Calculate the standard deviation of the times. (Treat all the data as x–values.)
Keys:
Display:
Description:
{ }
Clears the statistics registers.
Enters the first time.
z b
ꢎ
´
15.5
9.25
6
6
6
12
ꢔ)ꢕꢕꢕꢕꢎ
ꢎ
10
12.5
Enters the remaining data; six
data points entered.
6
8.5
6
6
)ꢕꢕꢕꢕꢎ
ꢏ)ꢗꢙꢕꢙꢎ
{
}
Calculates the standard deviation
time.
{ 2
%
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Population Standard Deviation
Population standard deviation is a measure of how dispersed the data values
are about the mean. Population standard deviation assumes the data
constitutes the complete set of data, and is calculated using n as a divisor.
ꢀ
Press
{ } for the population standard deviation of the
{ 2 σ
º
x–values.
ꢀ
Press
{ } for the population standard deviation of the
{ 2 σ
¸
y–values.
Example: Population Standard Deviation.
Grandma Tinkle has four grown sons with heights of 170, 173, 174, and
180 cm. Find the population standard deviation of their heights.
Keys:
Display:
Description:
{ }
Clears the statistics registers.
Enters data.
z b
´
ꢎ
170
6
173
6
174
6
6
ꢎ
180
ꢏ)ꢕꢕꢕꢕꢎ
ꢒ)ꢕꢕꢕꢕꢎ
ꢖ) ꢖꢔꢗꢎ
{ }
Four data points accumulated.
Calculates the population
standard deviation.
{ 2 σ
º
Linear regression
Linear regression, L.R. (also called linear estimation) is a statistical method for
finding a straight line that best fits a set of x,y–data.
Note
To avoid a
message, enter your data before
!ꢀ! ꢈꢁꢁꢑꢁ
executing any of the functions in the L.R. menu.
Statistical Operations 11–7
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L.R. (Linear Regression) Menu
Description
Menu Label
ˆ
{ }
Estimates (predicts) x for a given hypothetical value of
y, based on the line calculated to fit the data.
º
ˆ
{ }
Estimates (predicts) y for a given hypothetical value of
x, based on the line calculated to fit the data.
¸
{ }
T
Correlation coefficient for the (x, y) data. The
correlation coefficient is a. number in the range –1
through +1 that measures how closely the calculated
line fits the data.
{ }
P
Slope of the calculated line.
{ }
E
y–intercept of the calculated line.
ꢀ
ꢀ
To find an estimated value for x (or y), key in a given hypothetical value
ˆ
ˆ
for y (or x), then press
{ } (or
{ }).
{ ,
{ ,
º
¸
To find the values that define the line that best fits your data, press {
followed by { }, { }, or { }.
,
T
P
E
Example: Curve Fitting.
The yield of a new variety of rice depends on its rate of fertilization with
nitrogen. For the following data, determine the linear relationship: the
correlation coefficient, the slope, and the y–intercept.
X, Nitrogen Applied
(kg per hectare)
0.00
20.00 40.00 60.00 80.00
4.63
5.78 6.61 7.21 7.78
Y, Grain Yield
(metric tons per hectare)
Keys:
Display:
Description:
{ }
z b
´
Clears all, previous statistical
ꢎ
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data.
4.63 š 0 6
Enters data; displays n.
ꢔ)ꢕꢕꢕꢕꢎ
ꢏ)ꢕꢕꢕꢕꢎ
ꢖ)ꢕꢕꢕꢕꢎ
ꢒ)ꢕꢕꢕꢕꢎ
ꢗ)ꢕꢕꢕꢕꢎ
5.78
6.61
7.21
7.78
20
40
60
80
š
š
š
š
6
6
6
6
Five data pairs entered.
Displays linear–regression
menu.
ˆ ˆ
º ¸
{ ,
T P E
{ }
T
Correction coefficient; data
closely approximate a straight
line.
ꢕ)ꢓꢙꢙꢕꢎ
{ }
Slope of the line.
y–intercept.
z ,
P
ꢕ)ꢕꢖꢙꢘꢎ
ꢒ)ꢙꢗ ꢕꢎ
z , { }
E
Statistical Operations 11–9
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y
8.50
7.50
6.50
5.50
4.50
X
(70, y)
r = 0.9880
m = 0.0387
b = 4.8560
20
x
0
40
60
80
?
What if 70 kg of nitrogen fertilizer were applied to the rice field Predict the
grain yield based on the above statistics.
Keys:
Display:
Description:
70
Enters hypothetical x–value.
The predicted yield in tons per
hectare.
ꢘꢕ_ꢎ
ꢘ)ꢗ ꢔꢗꢎ
ˆ
{ }
{ ,
¸
Limitations on Precision of Data
Since the calculator uses finite precision (12 to 15 digits), it follows that there
are limitations to calculations due to rounding. Here are two examples:
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Normalizing Close, Large Numbers
The calculator might be unable to correctly calculate the standard deviation
and linear regression for a variable whose data values differ by a relatively
small amount. To avoid this, normalize the data by entering each value as the
difference from one central value (such as the mean). For normalized
x–values, this difference must then be added back to the calculation of x
ˆ
y
and
, and
and b roust also be adjusted. For example, if your x–values
were 7776999, 7777000, and 7777001, you should enter the data as –1,
. For b, add back
ˆ
x
0, and 1; then add 7777000 back to x and
7777000 × m. To calculate , be sure to supply an x–value that is less
ˆ
x
ˆ
y
7777000.
Similar inaccuracies can result if your x and y values have greatly different
magnitudes. Again, scaling the data can avoid this problem.
Effect of Deleted Data
Executing z 4 does not delete any rounding errors that might have been
generated in the statistics registers by the original data values. This difference
is not serious unless the incorrect data have a magnitude that is enormous
compared with the correct data; in such a case, it would be wise to clear and
reenter all the data.
Summation Values and the Statistics Registers
The statistics registers are six unique locations in memory that store the
accumulation of the six summation values.
Summation Statistics
Pressing
gives you access to the contents of the statistics
{ 5
registers:
ꢀ
ꢀ
ꢀ
Press { } to recall the number of accumulated data sets.
Q
Press { } to recall the sum of the x–values.
º
Press { } to recall the sum of the y–values.
¸
Statistical Operations 11–11
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ꢏ
ꢏ
ꢀ
Press { }, { }, and { } to recall the sums of the squares and the sum
º¸
º
¸
of the products of the x and y — values that are of interest when
performing other statistical calculations in addition to those provided by
the calculator.
If you've entered statistical data, you can see the contents of the statistics
registers. Press }, then use and to view
{
z X
z —
z ˜
#ꢀꢁ
the statistics registers.
Example: Viewing the Statistics Registers.
Use to store data pairs (1,2) and (3,4) in the statistics registers. Then
6
view the stored statistical values.
Keys:
Display:
Description:
z b { }
´
Clears the statistics registers.
Stores the first data pair (1,2).
Stores the second data pair (3,4).
Displays VAR catalog and views
ꢎ
2
1
š 6
ꢔ)ꢕꢕꢕꢕꢎ
ꢏ)ꢕꢕꢕꢕ
4 š 3 6
{
z X
#ꢀꢁ
}
´º¸/ꢔꢒ)ꢕꢕꢕꢕ
Σxy register.
´¸ꢏ/ꢏꢕ)ꢕꢕꢕꢕ
´ºꢏ/ꢔꢕ)ꢕꢕꢕꢕ
´¸/ )ꢕꢕꢕꢕ
´º/ꢒ)ꢕꢕꢕꢕ
2
Views y register.
z —
z —
z —
z —
z —
†
Σ
2
Views Σx register.
Views Σy register.
Views Σx register.
Views n register.
Q/ꢏ)ꢕꢕꢕꢕꢎ
Leaves VAR, catalog.
2.0000
The Statistics Registers in Calculator Memory
The memory space (48 bytes) for the statistics registers is automatically
allocated (if it doesn't already exist) when you press
registers are deleted and the memory deallocated when you execute z
or
. The
6
4
{ }.
b
´
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If not enough calculator memory is available to hold the statistics registers
when you first press (or ), the calculator displays
.
6
4
ꢇꢈꢇꢑꢁ& ꢋ"ꢂꢂ
You will rived to clear variables, equations, or programs (or a combination)
to make room for the statistics registers before you can enter statistical data.
Refer to "Managing Calculator Memory" in appendix B.
Access to the Statistics Registers
The statistics register assignments in the HP 32SII are shown in the following
table.
Statistics Registers
Register
Number
28
Description
Number of accumulated data pairs.
Sum of accumulated x–values.
n
Σx
29
y
Σx
y
30
31
32
Sum of accumulated y–values.
Sum of squares of accumulated x–values.
Sum of squares of accumulated y–values.
Σ
2
2
Σ
Σxy
33
Sum of products of accumulated x–and
y–values.
You can load a statistics register with a summation by storing the numb r (28
through 33) of the register you want in i (number and then storing
H ‘
the summation (value H ’. Similarly, you can press { ‰ ’ to
view a register value–the display is labeled with the register name. The SUMS
menu contains functions for recalling the register values. See "Indirectly
Addressing Variables and Labels" in chapter 13 for more information.
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Part 2
Programming
Statistics Programs
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12
Simple Programming
Part 1 of this manual introduced you to functions and operations that you can
use manually, that is, by pressing a key for each individual operation. And
you saw how you can use equations to repeat calculations without doing all
of the keystrokes each time.
In part 2, you'll learn how you can use programs for repetitive
calculations —–calculations that may involve more input or output control or
more intricate logic. A program lets you repeat operations and calculations in
the precise manner you want.
In this chapter you will learn how to program a series of operations. In the
next chapter, "Programming Techniques," you will learn about subroutines
and conditional instructions.
Example: A Simple Program.
To find the area of a circle with a radius of 5, you would use the
2
formula A = π r and press
5
z : { M y
to get the result for this circle, 78.5398.
?
But what if you wanted to find the area of many different circles
Rather than repeat the given keystrokes each time (varying only the "5" for
the different radii), you can put the repeatable keystrokes into a program:
ꢕꢕꢔ ºꢏꢎ
π
ꢕꢕꢏ ꢎ
ꢕꢕꢖ ºꢎ
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This very simple program assumes that the value for the radius is in the X–
register (the display) when the program starts to run. It computes the area and
leaves it in the X–register.
To enter this program into program memory, do the following:
Keys:
Display:
Description:
{ALL} {Y}
Clears memory.
z b
z d
ꢎ
Activates Program–entry mode
(PRGM annunciator on).
Resets program pointer to PRGM
TOP.
ꢎ
z U Œ Œ
ꢅꢁꢆꢇ !ꢑꢅꢎ
2
ꢕꢕꢔ ºꢏ
ꢕꢕꢏ π
ꢕꢕꢖ º
z :
{ M
y
(Radius)
2
Area = x
π
Exits Program–entry mode.
z d
Try running this program to find the area of a circle with a radius of 5:
Keys:
Display:
Description:
This sets the program to its
beginning.
z U Œ Œ
ꢎ
5
The answer!
f
ꢘꢙ)ꢗꢖꢓꢙꢎ
We will continue using the above program for the area of a circle to illustrate
programming concepts and methods.
Designing a Program
The following topics show what instructions you can put in a program. What
you put in a program affects how it appears when you view it and how it
works when you run it.
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Program Boundaries (LBL and RTN)
If you want more than one program stored in program memory, then a
program needs a label to mark its beginning (such as
) and a
ꢀꢕꢔ ꢂꢌꢂ ꢀ
return to mark its end (such as
).
ꢀꢕꢗ ꢁ!ꢄ
Notice–that the line numbers acquire an to match their label.
ꢀ
Program Labels
Programs and segments of programs (called routines) should start with a label.
To record a label, press:
letter–key
z “
The label is a single letter from A through Z. The letter keys are used as they
are for variables (as discussed in chapter 3). You cannot assign the same
label more than once (this causes the message
label can use the same letter that a variable uses.
), but a
ꢍ"ꢅꢂꢊꢃꢀ!)ꢂꢌꢂ
It is possible to have one program (the top one) in memory without any label.
However, adjacent programs need a label between them to keep them
distinct.
Program Line Numbers
Line numbers are preceded by the letter for the label, such as
.
ꢀꢕꢔ
If one label's routine has more than 99 lines, then the line number appears
with a decimal point instead of the leftmost number, such as for line
ꢀ)ꢕꢔ
101 in label A. For more than 199 lines, the line number uses a comma, such
as for line 201.
ꢀ8ꢕꢔ
Program Returns
Programs and subroutines should end with a return instruction. The keystrokes
are:
{ ”
When a program finishes running, the last RTN instruction returns the
program pointer to
, the top of program memory.
ꢅꢁꢆꢇ !ꢑꢅ
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Using RPN and Equations in Programs
You can calculate in programs the same ways you calculate on the.
keyboard:
ꢀ
Using RPN operations (which work with the stack, as explained in
chapter 2).
ꢀ
Using equations (as explained in chapter 6).
The previous example used a series of RPN operations to calculate the area of
the circle. Instead, you could have used an equation in the program. (An
example follows later in this chapter.) Many programs are a. combination of
RPN and equations, using the strengths of both.
Strengths of RPN Operations
Use less memory.
Strengths of Equations
Easier to write and read.
Can automatically prompt.
Execute a bit faster.
When a program executes a line containing an equation, the equation is
evaluated in the same way that evaluates an equation in the equation
W
list. For program evaluation, "=" in an equation is essentially treated as "–".
(There's no programmable equivalent to š for an assignment
equation—other than writing the equation as an expression, then using STO
to store the value in a variable.)
For both types of calculations, you can include RPN instructions to control
input, output, and program flow.
Data Input and Output
For programs that need more than one input or return more than one output,
you can decide how you want the program to enter and return information.
For input, you can prompt for a variable with the INPUT instruction, you can
get an equation to prompt for its variables, or you can take values entered in
advance onto the stack.
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For output, you can display a variable with the VIEW instruction, you can
display a message derived from an equation, or you can leave unmarked
values on the stack.
These are covered later in this chapter tinder "Entering and Displaying Data."
Entering a Program
Pressing
toggles the calculator into and out of Program–entry
z d
mode — turns the PRGM annunciator on and off. Keystrokes in
Program–entry mode are stored as program lines in memory. Each instruction
or number occupies one program line, and there is no limit (other than
available memory) on the number of lines in a program.
To enter a program into memory:
1. Press
2. Press
to activate Program–entry mode.
z d
to display
. This sets the program
z U Œ Œ
ꢅꢁꢆꢇ !ꢑꢅ
pointer to a known spot, before any other programs. As you enter
program lines, they are inserted before all other program lines.
If you don't need any other programs that might be in memory, clear
program memory by pressing
{
}. To confirm that you
z b
ꢅꢆꢇ
want all programs deleted, press { } after the message
3. Give the program a label—a single letter, A through Z. Press
.
z “
&
ꢃꢂ ꢅꢆꢇ@ & ꢄ
letter. Choose a letter that will remind you of the program, such as "A" for
"area."
If the message
is displayed, use a different letter. You
ꢍ"ꢅꢂꢊꢃꢀ!)ꢂꢌꢂ
can clear the existing program instead—press
{
}, use
z — or z ˜ to find the label, and press z b and
z X
ꢅꢆꢇ
.
†
4. To record calculator operations as program instructions, press the same
keys you would to do an operation manually. Remember that many
functions don't appear on the keyboard but must be accessed using menus.
To enter an equation in a program line, see the instructions below.
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5. End the program with a return instruction, which sets the program pointer
back to
after the program runs. Press
.
{ ”
ꢅꢁꢆꢇ !ꢑꢅ
6. Press
(or
) to cancel program entry.
†
z d
Numbers in program lines are stored as precisely as you entered them, and
they're displayed using ALL or SCI format. (If a long number is shortened in
the display, press
to view all digits.)
{
To enter an equation in a program line:
1. Press to activate Equation–entry mode, The EQN annunciator
{ G
turns on.
2. Enter the equation as you would in the equation list. See chapter 6 for
details. Use a to correct errors as you type.
3. Press
to terminate the equation and display its left end. (The
š
equation does not become part of the equation list.)
After you've entered an equation, you can press
to see its
{
checksum and length. Hold the
key to keep the values in the display.
For a long equation, the
and annunciators show that scrolling is active
€
ž
for this program line. You can use
and
to scroll the display. Press
6
<
[SCRL] to turn off
and to use the top–row keys to enter program
{
ž
instructions
Keys That Clear
Note these special conditions during program entry:
ꢀ
ꢀ
always cancels program entry. It never clears a number to zero.
†
If the program line doesn't contain an equation, a deletes the current
program line. It backspaces if a digit is being entered ("_" cursor
present).
ꢀ
If the program line contains an equation,
begins editing the
a
equation. It deletes the rightmost function or variable if an equation is
being entered (" " cursor present).
¾
ꢀ
ꢀ
{
} deletes a program lime if it contains an equation.
z b
ꢈꢉꢄ
To program a function to clear the K–register, use
{ }.
z b
º
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Function Names in Programs
Then name of function that is used in a program line is not necessarily the
same as the function's name on its key, in its menu, or in an equation. The
name that is used in a program is usually a fuller abbreviation than that which
can fit on a key or in a menu. This fuller name appears briefly in the display
whenever you execute a function — as long as you hold down the key, the
name is displayed.
Example: Entering a Labeled Program.
The following keystrokes delete the previous program for the area of a circle
and enter a new one that includes a label and a return instruction. If you make
a mistake during entry, press
reenter the line correctly.
to delete the current program line, then
a
Keys:
Display:
Description:
Activates Program–entry mode
(PRGM on).
z d
ꢎ
{
} { }
ꢅꢆꢇ & ꢅꢁꢆꢇ !ꢑꢅꢎ
Clears all of program memory.
Labels this program routine A
(for "area").
z d
z “ A
ꢀꢕꢔ ꢂꢌꢂ ꢀꢎ
ꢀꢕꢏ ºꢏꢎ
ꢀꢕꢖ πꢎ
ꢀꢕꢒ ºꢎ
ꢀꢕꢗ ꢁ!ꢄꢎ
}
Enters the three program lines.
z :
{ M
y
Ends the program.
{ ”
{
Displays label A and the length
ꢕꢕꢘ)ꢗꢎ
z X
ꢅꢆꢇ
ꢂꢌꢂ ꢀ
of the program in bytes.
Checksum and length of
{
† †
ꢃꢚ/ꢈꢕꢏꢃ ꢕꢕꢘ)ꢗꢎ
program.
Cancels program entry
(PRGM annunciator off).
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A different checksum means the program was not entered exactly as given
here.
Example: Entering a Program with an Equation.
The following program calculates the area of a circle using an equation,
rather than using RPN operation like the previous program.
Keys:
Display:
Description:
z d z
U Œ Œ
Activates Program–entry mode;
sets pointer to top of memory.
Labels this program routine E (for
"equation").
ꢅꢁꢆꢇ !ꢑꢅꢎ
E
z “
ꢈꢕꢔ ꢂꢌꢂ ꢈꢎ
R
Stores radius in variable R.
Selects Equation–entry mode;
enters the equation; returns to
Program–entry mode.
H
ꢈꢕꢏ !ꢑ ꢁꢎ
ꢎ
{ G { Mꢁ
R
0 š
y K
ꢎ
2
ꢈꢕꢖ πºꢁ:ꢏꢎ
Checksum and length of
equation.
{
ꢃꢚ/ꢈꢖꢋꢍ
ꢕꢕꢓ)ꢕꢎ
Ends the program.
{ ”
ꢈꢕꢒ ꢁ!ꢄꢎ
z X {
}
Displays label E and the length of
the program in bytes.
ꢂꢌꢂ ꢈ
ꢕꢔꢖ)ꢗꢎ
ꢅꢆꢇ
{
† †
Cancels program entry.
ꢕꢔꢖ)ꢗꢎ
ꢃꢚ/ꢔꢖꢗꢏ
ꢎ
Running a Program
To run or execute a program, program entry cannot be active (no
program–line numbers displayed; PRGM off). Pressing
will cancel
†
Program–entry mode.
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Executing a Program (XEQ)
Press W label to execute the program labeled with that letter. If there is
only one program in memory, you can also execute it by pressing
ΠΠf (run/stop). The PRGM annunciator blinks on and off while the
z U
program is running.
If necessary, enter the data before executing the program.
Example:
Run the programs labeled A and E to find the areas of three different circles
with radii of 5, 2.5, and 2 . Remember to enter the radius before
π
executing .A or E.
Keys:
Display:
Description:
5
A
Enters the radius, then starts
program A. The resulting area is
displayed.
W
ꢁ"ꢄꢄꢊꢄꢆꢎ
ꢘꢙ)ꢗꢖꢓꢙꢎ
2.5
2
E
Calculates area of the second
circle using program E.
Calculates area of the third
circle.
W
ꢔꢓ) ꢖꢗꢕꢎ
ꢔꢏꢒ)ꢕꢏꢗꢔꢎ
A
{ M y W
Testing a Program
If you know there is an error in a program, but are not sure where the error is,
then a good way to test the program is by stepwise execution. It is also a
good idea to test a long or complicated program before relying on it. By
stepping through its execution, one line at a time, you can see the result after
each program line is executed, so you can verify the progress of known data
whose correct results are also known.
1. As for regular execution, make sure program entry is not active (PRGM
annunciator off).
2. Press
label to set the program pointer to the start of the
z U
program (that is, at its LBL instruction). The
instruction moves the
ꢆ!ꢑ
program pointer without starting execution. (If the program is the first or
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only program, you can press
beginning.)
to move to its
z U Œ Œ
3. Press and hold
. This displays the current program line. When
z ˜
you release
, the line is executed. The result of that execution is then
˜
displayed (it is in the X–register).
To move to the preceding line, you can press
occurs.
. No execution
z —
4. The program pointer moves to the next line. Repeat step 3 until you find an
error (an incorrect result occurs) or reach the end of the program.
If Program–entry mode is active, then
or
simply changes
z ˜ z —
the programs pointer, without executing lines. Holding down an arrow key
during program entry makes the lines roll by automatically.
Example: Testing a Program.
Step through the execution of the program labeled A. Use a radius of 5 for
the test data. Check that Program–entry mode is not active before you start:
Keys:
Display:
Description:
5
A
Moves program counter to label A.
z U
ꢗ)ꢕꢕꢕꢕꢎ
ꢀꢕꢔ ꢂꢌꢂ ꢀꢎ
ꢗ)ꢕꢕꢕꢕꢎ
ꢀꢕꢏ ºꢏꢎ
(hold)
(hold)
(hold)
(hold)
(hold)
z ˜
(release)
z ˜
(release)
z ˜
(release)
z ˜
(release)
z ˜
(release)
Squares input.
ꢏꢗ)ꢕꢕꢕꢕꢎ
π
ꢀꢕꢖ ꢎ
Value of π.
ꢖ)ꢔꢒꢔ ꢎ
ꢀꢕꢒ ºꢎ
25π.
ꢘꢙ)ꢗꢖꢓꢙꢎ
ꢀꢕꢗ ꢁ!ꢄꢎ
ꢘꢙ)ꢗꢖꢓꢙꢎ
End of program. Result is correct.
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Entering and Displaying Data
The calculator's variables are used to store data input, intermediate results,
and final results. (Variables, as explained in chapter 3, are identified by a
letter from A through Z or i, but the variable names have nothing to do with
program labels.)
In a program, you can get data in these ways:
ꢀ
From an INPUT instruction, which prompts for the value of a variable.
(This is the most handy technique.)
ꢀ
From the stack. (You can use STO to store the value in a variable for later
use.)
ꢀ
ꢀ
From variables that already have values stored.
From automatic equation prompting (if enabled by flag 11 set).
(This is also handy if you're using equations.)
In a program, you can display information in these ways:
ꢀ
ꢀ
ꢀ
ꢀ
With a VIEW instruction, which shows the name and value of a variable.
(This is the most handy technique.)
On the stack—only the value in the X–register is visible. (You can use PSE
for a 1–second look at the X–register.)
In a displayed equation (if enabled by flag 10 set). (The "equation" is
usually a message, not a true equation.)
Some of these input and output techniques are described in the following
topics.
Using INPUT for Entering Data
The INPUT instruction (
Variable ) stops a running program and
a ˆ
displays a prompt for the given variable. This display includes the existing
value for the variable, such as
ꢁ@ꢕ)ꢕꢕꢕꢕꢎ
where
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"R" is the variable's name,
?
" " is the prompt for information, and
0.0000 is the current value stored in the variable.
Press
(run/stop) to resume the program. The value you keyed in then
f
writes over the contents of the X–register and is stored in the given variable. If
you have not changed the displayed value, then that value is retained in the
X–register.
The area–of–a–circle program with an INPUT instruction looks like this:
ꢀꢕꢔ ꢂꢌꢂ ꢀꢎ
ꢀꢕꢏ ꢊꢄꢅ"! ꢁꢎ
ꢀꢕꢖ ºꢏꢎ
π
ꢀꢕꢒ ꢎ
ꢀꢕꢗ ºꢎ
ꢀꢕ ꢁ!ꢄꢎ
To use the INPUT function in a program:
1. Decide which data values you will need, and assign them names.
(In the area–of–a–circle example, the only input needed is the radius,
which we can assign to R.)
2. In the beginning of the program, insert an INPUT instruction for each
variable whose value you will need. Later in the program, when you write
the part of the calculation that needs a given value, insert a
instruction to bring that value back into the stack.
variable
K
Since the INPUT instruction also leaves the value you just entered in the
X–register, you don't have to recall the variable at a later time — you
could INPUT it and use it when you need it. You might be able to save
some memory space this way. However, in a long program it is simpler to
just input all your data up front, and then recall individual variables as you
need them.
Remember also that the user of the program can do calculations while the
program is stopped, waiting for input. This can alter the contents of the
stack, which might affect the next calculation to be done by the program.
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Thus the program should not assume that the X–, Y–, and Z–registers'
contents will be the same before and after the INPUT instruction. If you
collect, all the data in the beginning and then recall then when needed for
calculation, then this prevents the stack's contents from being altered just,
before a calculation.
For example, see the "Coordinate Transformations" program in chapter 15.
Routine D collects all the necessary input for the variables M, N, and T (lines
D02 through D04) that define the x and y coordinates and angle of a new
θ
system.
To respond to a prompt:
Mien you run the program, it will stop at each INPUT and prompt you for that
variable, such as
. The value displayed (and the contents of the
ꢁ@ꢕ)ꢕꢕꢕꢕ
X–register) will be the current contents of R.
ꢀ
ꢀ
To leave the number unchanged, just press
.
f
To change the number, type the new number and press
, This
f
new number writes over the old value in the X–register. You can enter a
number as a fraction if you want. If you need to calculate a number, use
normal keyboard calculations, then press
. For example, you can
f
press 2
5
š 0 f
.
ꢀ
ꢀ
To calculate with the displayed number, press
typing another number.
before
š
To cancel the INPUT prompt, press
variable remains in the X–register. If you press f to resume the
. The current value for the
†
program, the canceled INPUT prompt is repeated. If you press
†
during digit entry, it clears the number to zero. Press
the INPUT prompt.
again to cancel
†
ꢀ
To display digits hidden by the prompt, press
it is a binary number with more than 12 digits, use the and < and
. (If
{
keys to see the rest.)
6
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Using VIEW for Displaying Data
The programmed VIEW instruction { ‰ variable stops a running
program and displays and identifies the contents of the given variable, such
as
ꢀ/ꢘꢙ)ꢗꢖꢓꢙꢎ
This is a display only, and does not copy the number to the X–register. If
Fraction–display mode is active, the value is displayed as a fraction.
ꢀ
ꢀ
Pressing
copies this number to the X–register.
š
If the number is wider than 10 characters, pressing
{
displays the entire number. (If it is a binary number with more than 12
digits, use the < and 6 keys to see the rest.)
ꢀ
ꢀ
Pressing
X–register.
Pressing
(or
) erases the VIEW display and shows the
†
a
clears the contents of the displayed variable.
z b
Press f to continue the program,
If you don't want the program to stop, see "Displaying Information without
Stopping" below.
For example, see the program for "Normal and Inverse–Normal
Distributions" in chapter 16. Lines T15 and T16 at, the end of the T routine
display the result for X. Note also that this VIEW instruction in this program is
preceded by a RCL instruction. The RCL instruction is not necessary, but it is
convenient because it brings the VIEWed variable to the X–register, making it
available for manual calculations. (Pressing
while viewing a VIEW
š
display would have the same effect.) The other application programs in
chapters 15 through 17 also ensure that the VIEWed variable is in the
X–register as well — except for the "Polynomial Root Finder" program.
Using Equations to Display Messages
Equations aren't checked for valid syntax until they're evaluated. This means
you can enter almost any sequence of characters into a program as an
equation — you enter it just as you enter any equation. On any program line,
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press
to start the equation. Press number and math keys to get
{ G
numbers and symbols. Press
equation.
before each letter. Press
to end the
K
š
If flag 10 is set, equations are displayed instead of being evaluated. This
means you can display any message you enter as are equation. (Flags are
discussed in detail in chapter 13.)
When the message is displayed, the program stops—.–press f to resume
execution. If the displayed message is longer than 12 characters, the
and
€
annunciators turn on when the message is displayed. You can then use
ž
6
and
to scroll the display. You can press
[SCRL] to turn off
<
{
ž
and make the top–row keys perform their normal functions.
If you don't want the program to stop, see "Displaying Information without
Stopping" below.
Example: INPUT, VIEW, and Messages in a Program.
Write an equation to find the surface area and volume of a cylinder given its
radius and height. Label the program C (for cylinder), and use the variables S
(surface area), V (volume), R (radius), and H (height). Use these formulas:
2
V = πR H
2
S = 2π R + 2π RH = 2π R ( R + H )
Keys:
Display:
Description:
Program, entry; sets pointer to top
of memory.
z d z
U Œ Œ
z “ C
ꢎ
ꢅꢁꢆꢇ !ꢑꢅꢎ
Labels program.
ꢃꢕꢔ ꢂꢌꢂ ꢃꢎ
ꢃꢕꢏ ꢊꢄꢅ"! ꢁꢎ
ꢃꢑꢖ ꢊꢄꢅ"! ꢐꢎ
R
Labels program.
z ˆ
H
Instructions to prompt for radius
and height.
z ˆ
{ G {
Calculates the volume.
ꢎ
ꢎ
ꢎ
R
0 2 y K H
M y K
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Keys:
Display:
Description:
š
ꢃꢕꢒ πºꢁ:ꢏºꢐꢎ
Checksum and length of equation.
ꢃꢚ/ꢓꢔꢓꢒ ꢕꢔꢏ)ꢕꢎ
{
V
Store the volume in V.
H
ꢃꢕꢗ !ꢑ #ꢎ
ꢎ
2
yꢁ{ M
Calculates the surface area.
{ G
ꢎ
R
yꢁK y
{ \ K
ꢎ
R
ꢎ
H
™ K {
ꢎ
] š
{
ꢃꢕ ꢏºπºꢁºꢁ1ꢎ
Checksum and length of equation.
ꢃꢚ/ꢀꢓꢔꢔ ꢕꢔꢙ)ꢕꢎ
S
Stores the surface area in S.
H
ꢃꢕꢘ !ꢑ ꢎ
ꢎ
{
{ x
ꢋ
}
Sets flag 10 to display equations.
0
Œ
ꢃꢕꢙ ꢋ ꢔꢕꢎ
ꢎ
{ G K
Displays message in equations.
V
o ™
O
L
K K
ꢎ
ꢎ
A
o K
ꢎ
K R K E
ꢎ
A
K š
ꢃꢕꢓ #ꢑꢂ - ꢀꢁꢎ
{ }
ꢃꢋ ꢎ
Clears flag 10.
{ x
0
Œ
ꢃꢔꢕ ꢃꢋ ꢔꢕꢎ
V
S
Displays volume.
{ ‰
{ ‰
{ ”
z X
ꢃꢔꢔ #ꢊꢈ$ #ꢎ
ꢃꢔꢏ #ꢊꢈ$ ꢎ
ꢃꢔꢖ ꢁ!ꢄꢎ
Displays surface area.
Ends program.
{
}
Displays label C and the length of
the program in bytes.
ꢂꢌꢂ ꢃ
ꢑ ꢔ)ꢗꢎ
ꢅꢆꢇ
{
† †
Checksum and length of program.
ꢃꢚ/ ꢕꢒꢘ
ꢎ
ꢑ ꢔ)ꢗꢎ
Cancels program entry.
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1
Now find the volume and surface area–of a cylinder with a radius of 2 /
2
cm and a height of 8 cm.
Keys:
Display:
Description:
C
Starts executing C; prompts for R.
W
ꢁ@valueꢎ
(It displays whatever value
happens to be in R.)
1
2
f
1
2
Enters 2 / as a fraction. Prompts
Œ Œ
ꢐ@valueꢎ
2
for H.
8
Message displayed.
f
#ꢑꢂ - ꢀꢁꢈꢀꢎ
#/ꢔꢗꢘ)ꢕꢘꢓ ꢎ
/ꢔ ꢒ)ꢓꢖꢖ ꢎ
3
Volume in cm .
f
f
2
Surface area in cm .
Displaying Information without Stopping
Normally, a program stops when it displays a variable with VIEW or displays
an equation message. You normally have to press f to resume execution.
If you want, you can make the program continue while the information is
displayed. If the next program line — after a VIEW instruction or a viewed
equation — contains a PSE (pause) instruction, the information is displayed
and execution continues after a 1–second pause. In this case, no scrolling or
keyboard input is allowed.
The display is cleared by other display operations, and by the RND operation
if flag 7 is set (rounding to a fraction).
Press
to enter PSE in a program.
{ e
The VIEW and PSE lines–or the equation and PSE lines — are treated as one
operation when you execute a program one line at a time.
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Stopping or Interrupting a Program
Programming a Stop or Pause (STOP, PSE)
ꢀ
Pressing f (run/stop) during program entry inserts a STOP instruction.
This will halt a running program until you resume it by pressing
f
from the keyboard. You can use STOP rather than RTN in order to end a
program without returning the program pointer to the top of memory.
ꢀ
Pressing
during program entry inserts a PSE (pause) instruction.
{ e
This will suspend a running program and display the contents of the
X–register for about 1 second — with the following exception. If PSE
immediately follows a VIEW instruction or an equation that's displayed
(flag 10 set), the variable or equation is displayed instead — and the
display remains after the 1–second pause.
Interrupting a Running Program
You can interrupt a running program at any time by pressing
The program completes its current instruction before stopping. Press
or
.
f
f
†
(run/stop) to resume the program.
If you interrupt a program and then press
,
, or
,
W z U
{ ”
you cannot resume the program with
. Reexecute the program instead
f
(
label).
W
Error Stops
If an error occurs in the course of a running program, program execution halts
and an error message appears in the display. (There is a list of messages and
conditions in appendix E.)
To see the line in the program containing the error–causing instruction, Press
. The program will have stopped at that point, (For instance, it
z d
might be a÷ instruction, which caused an illegal division by zero.)
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Editing Program
You can modify a program in program memory by inserting, deleting, and
editing program lines. If a program line contains an equation, you can edit
the equation—if any other program line requires even a minor change, you
must delete the old line and insert a new one.
To delete a program line:
1. Select the relevant program or routine (
label), activate
z U
program entry (
), and press
or
) to
z d
z ˜ z —
locate the program line that must be changed. Hold the arrow key down to
continue scrolling. (If you know the line number you want, pressing
z
label nn moves the program pointer there.)
U Œ
2. Delete the line you want to change—if it contains an equation, press
z
{
}; otherwise, press
. The pointer then moves to the
a
b
ꢈꢉꢄ
preceding line. (If you are deleting more than one consecutive program
line, start with the last line in the group.)
3. Key in the new instruction, if any. This replaces the one you deleted.
4. Exit program entry
or
).
†
d
To insert a program line:
1. Locate and display the program line that is before the spot where you
would like to insert a line.
2. Key in the new instruction; it is inserted after the currently displayed line.
For example, if you wanted to insert a new line between lines A04 and A05 of
a program, you would first display line A04, then key in the instruction or
instructions. Subsequent program lines, starting with the original line A05, are
moved down and renumbered accordingly.
To edit an equation in a program line:
1. Locate and display the program line containing the equation.
2. Press
. This turns on the " " editing cursor, but does riot delete
a
¾
anything in the equation.
3. Press
as required to delete the function or number you want to change,
a
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then enter the desired corrections.
4. Press
to end the equation.
š
Program Memory
Viewing Program Memory
Pressing
toggles the calculator into and out of program entry
z d
(PRGM annunciator on, program lines displayed). When Program–entry
mode is active, the contents of program memory are displayed.
Program memory starts at
. The list of program lines is circular, so
ꢅꢁꢆꢇ !ꢑꢅ
you can wrap the program pointer froze the bottom to the top and reverse.
While program entry is active, there are three ways to change the program
pointer (the displayed line):
ꢀ
Use the arrow keys,
last line wraps the pointer around to
and
. Pressing
, while pressing z
ꢅꢁꢆꢇ !ꢑꢅ
at the
z ˜
z —
z ˜
at
wraps the pointer around to the last program line.
—
ꢅꢁꢆꢇ !ꢑꢅ
To move more than one line at a time ("scrolling"), continue to hold the
or key.
˜
—
ꢀ
ꢀ
Press U ΠΠto move the program pointer to
.
ꢅꢁꢆꢇ !ꢑꢅ
Press
100.
label nn to move to a labeled line number less than
U Œ
If Program–entry mode is riot active (if no program lines are displayed), you
can also move the program pointer by pressing label.
z U
Canceling Program–entry mode does not change the position of the program
pointer.
Memory Usage
Each program line uses a certain amount of memory:
ꢀ
Numbers use 9.5 bytes, except for integer numbers from 0 through 254,
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which use only 1.5 bytes.
ꢀ
ꢀ
All other instructions use 1.5 bytes.
Equations use 1.5 bytes, plus 1.5 bytes for each function, plus 9.5 or 1.5
bytes for each number. Each "(" and each ")" uses 1.5 bytes except "("
for prefix functions.
If during program entry you encounter the message
, then
ꢇꢈꢇꢑꢁ& ꢋ"ꢂꢂ
there is not enough room in program memory for the line you just tried to enter.
You can make more room available by clearing programs or other data. See
"Clearing One or More Programs" below, or "Managing calculator Memory"
in appendix B.
The Catalog of Programs (MEM)
The catalog of programs is a list of all program labels with the number of
bytes of memory used by each label and the lines associated with it. Press
{
} to display the catalog, and press
or
z ˜ z —
z X
ꢅꢆꢇ
to move within the list. You can use this catalog to:
ꢀ
ꢀ
ꢀ
ꢀ
ꢀ
Review the labels in program memory and the memory cost of each
labeled program or routine.
Execute a labeled program. (Press
or
while the label is
W
f
displayed.)
Display a labeled program. (Press z d while the label is
displayed.)
Delete specific programs. (Press
displayed.)
while the label is
z b
See the checksum associated with a given program segment. (Press
{
.)
The catalog shows you how many bytes of memory each labeled program
segment uses. The programs are identified by program label:
ꢂꢌꢂ ꢃ ꢕ ꢔ)ꢗ
where 61.5 is the number of bytes used by the program.
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Clearing One or More Programs
To clear a specific program from memory
1. Press
{
} and display (using
and
)
z X
z ˜
z —
ꢅꢆꢇ
the label of the program.
2. Press
.
z b
3. Press † to cancel the catalog or a to back out.
To clear all programs from memory:
1. Press
2. Press
to display program lines (PRGM annunciator on).
z d
z b
{
} to clear program memory.
ꢅꢆꢇ
3. The message
4. Press
prompts you for confirmation. Press { }.
to cancel program entry.
ꢃꢂ ꢅꢆꢇ @ & ꢄ
&
z d
Clearing all of memory (
{
}) also clears all programs.
z b
ꢀꢂꢂ
The Checksum
The
checksum is a unique hexadecimal value given to each program label
and its associated lines (until the next label). This number is useful for
comparison with a known checksum for an existing program that you have
keyed into program memory. If the known checksum and the one shown by
e lines
your calculator are the same, then you have correctly entered all th
the program, To see your checksum:
of
1. Press z X {
} for the catalog of program labels.
ꢅꢆꢇ
2. Display the appropriate label by using the arrow keys, if necessary.
3. Press and hold to display value length.
{
ꢃꢚ/
For example, to see the checksum for the current program (the "cylinder"
program):
Keys:
Display:
Description:
{
}
Displays label C, which
takes 61.5 bytes.
z X
ꢂꢌꢂ ꢃ ꢕ ꢔ)ꢗꢎ
ꢅꢆꢇ
Checksum and length.
{
ꢃꢚ/ ꢕꢒꢘ ꢕ ꢔ)ꢗꢎ
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(hold)
If your checksum does not match this number, then you have not entered this
program correctly.
You will see that all of the application programs provided in chapters 15
through 17 include checksum values with each labeled routine so that you
can verify the accuracy of your program entry.
In addition, each equation in a program has a checksum. See "To enter an
equation in a program line" earlier in this chapter.
Nonprogrammable Functions
The following functions of the HP 32 II are not programmable:
{
{
}
}
z b
z U Œ Œ
ꢅꢆꢇ
label nn
z b
z U Œ
z X
ꢀꢂꢂ
a
,
z ˜ z —
z d
{
{ G
z Š
Programming with BASE
You can program instructions to change the base mode using
.
z w
These settings work in programs just as they do as functions executed from the
keyboard. This allows you to write programs that accept numbers in any of
the four bases, do arithmetic in any base, and display results in any base.
When writing programs that use numbers in a base other than 10, set the
base mode both as the current setting for the calculator and in the program
(as an instruction).
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Selecting a Base Mode in a Program
Insert a BIN, OCT, or HEX instruction into the beginning of the program. You
should usually include a DEC instruction at the end of the program so that the
calculator's setting will revert, to Decimal mode when the program is done.
An instruction in a program to change the base mode will determine bow
input is interpreted and how output looks during and after program execution,
but it does not affect the program lines as you enter them.
Equation evaluation, SOLVE, and ∫ FN automatically set Decimal mode.
Numbers Entered in Program Lines
Before starting program entry, set the base mode. The current setting for the
base mode determines the base of the numbers that are entered into program
lines. The display of these numbers changes when you change the base
mode.
Program line numbers always appear in base 10.
An annunciator tells you which base is the current setting. Compare the
program lines below in the left and right columns. All non–decimal numbers
are right justified in the calculator's display. Notice how the number 13
appears as "D" in Hexadecimal mode.
Decimal mode set:
:
:
Hexadecimal mode set:
:
:ꢎ
PRGM
PRGM
ꢀꢕꢓ ꢐꢈ%
PRGM
ꢀꢔꢕ
HEX
HEX
ꢀꢕꢓ ꢐꢈ%
PRGM
ꢀꢔꢕ ꢔꢖ
ꢍꢎ
:
:
:
:
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Polynomial Expressions and Horner's Method
Some expressions, such as polynomials, use the same variable several times
for their solution. For example, the expression
4
3
2
Ax + Bx + Cx + Dx + E
uses the variable x four different times. A program to calculate such an
expression using RPN operations could repeatedly recall a stored copy of x
from a variable. A shorter RPN programming method, however, would be to
use a stack which has been filled with the constant (see "Filling the Stack with
a Constant" in chapter 2).
Rorer's Method is a useful means of rearranging polynomial expressions to
cut calculation steps and calculation time. It is especially expedient with
SOLVE and FN, two relatively complex operations that use subroutines.
∫
This method involves rewriting a polynomial expression in a nested fashion to
eliminate exponents greater than 1:
4
3
2
Ax + 13x + Cx +D x + E
3
2
(Ax + Bx + Cx + D ) x + E
2
((Ax + Bx + C ) x + D )x + E
(((Ax + B )x + C ) x + D )x + E
Example:
4
3
Write a program using RPN operations for 5x + 2x as (((5x + 2)x)x)x, then
evaluate it for x = 7.
Keys:
Display:
Description:
z d z
U Œ Œ
ꢎ
ꢅꢁꢆꢇ !ꢑꢅꢎ
ꢅꢕꢔ ꢂꢌꢂ ꢅꢎ
ꢅꢕꢏ ꢊꢄꢅ"! %ꢎ
ꢅꢕꢖ ꢈꢄ!ꢈꢁꢎ
P
z “
X
Fills the stack with x.
z ˆ
š
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š
š
5
ꢅꢕꢒ ꢈꢄ!ꢈꢁꢎ
ꢅꢕꢗ ꢈꢄ!ꢈꢁꢎ
ꢅꢕ ꢗꢎ
5x.
y
ꢅꢕꢘ ºꢎ
2
ꢅꢕꢙ ꢏꢎ
5x + 2.
™
ꢅꢕꢓ -ꢎ
(5x + 2)x.
y
ꢅꢔꢕ ºꢎ
2
(5x + 2)x .
y
ꢅꢔꢔ ºꢎ
3
(5x + 2)x .
y
ꢅꢔꢏ ºꢎ
{ ”
z X
ꢅꢔꢖ ꢁ!ꢄꢎ
{
}
ꢅꢆꢇ ꢂꢌꢂ ꢅ ꢕꢔꢓ)ꢗꢎ
Displays label P, which takes 19.5
bytes.
Checksum and length.
z
† †
ꢃꢚ/ꢘꢋꢌꢒ ꢕꢔꢓ)ꢗꢎ
Cancels program entry.
ꢎ
Now evaluate this polynomial x = 7.
Keys:
Display:
Description:
P
Prompts for x.
Result.
W
%@valueꢎ
7
f
ꢔꢏ8 ꢓꢔ)ꢕꢕꢕꢕꢎ
A more general form of this program for any equation
(((Ax + B) + C) + D) + E would be:
×
×
×
ꢅꢕꢔ ꢂꢌꢂ ꢅꢎ
ꢅꢕꢏ ꢊꢄꢅ"! ꢀꢎ
ꢅꢕꢖ ꢊꢄꢅ"! ꢌꢎ
ꢅꢕꢒ ꢊꢄꢅ"! ꢃꢎ
ꢅꢕꢗ ꢊꢄꢅ"! ꢍꢎ
ꢅꢕ ꢊꢄꢅ"! ꢈꢎ
ꢅꢕꢘ ꢊꢄꢅ"! %ꢎ
ꢅꢕꢙ ꢈꢄ!ꢈꢁꢎ
ꢅꢕꢓ ꢈꢄ!ꢈꢁꢎ
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ꢅꢔꢕ ꢈꢄ!ꢈꢁꢎ
ꢅꢔꢔ ꢁꢃꢂº ꢀꢎ
ꢅꢔꢏ ꢁꢃꢂ- ꢌꢎ
ꢅꢔꢖ ºꢎ
ꢅꢔꢒ ꢁꢃꢂ- ꢃꢎ
ꢅꢔꢗ ºꢎ
ꢅꢔ ꢁꢃꢂ- ꢍꢎ
ꢅꢔꢘ ºꢎ
ꢅꢔꢙ ꢁꢃꢂ- ꢈꢎ
ꢅꢔꢓ ꢁ!ꢄꢎ
Checksum and length: E93F 028.5
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13
Programming Techniques
Chapter 12 covered the basics of programming. This chapter explores more
sophisticated but useful techniques:
ꢀ
Using subroutines to simplify programs by separating and labeling
portions of the program that are dedicated to particular tasks. The use of
subroutines also shortens a program that must perform a series of steps
more than once.
ꢀ
ꢀ
ꢀ
Using conditional instructions (comparisons and flags) to determine
which instructions or subroutines should be used,
Using loops with counters to execute a set of instructions a certain
number of times.
Using indirect addressing to access different variables using the same
program instruction.
Routines in Programs
A program is composed of one or more routines. A routine is a functional unit
that accomplishes something specific, Complicated programs need routines
to group and separate tasks. This makes a program easier to write, read,
understand, and alter.
For example, look at the program for "Normal and Inverse–Normal
Distributions" in chapter 16. Routine S "initializes" the program by collecting
the input for the mean and standard deviation. Routine D sets a limit of
integration, executes routine Q, and displays the result, Routine Q integrates
the function defined in routine F and finishes the probability calculation of
Q(x).
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A routine typically starts with a label (LBL) and ends with an instruction that
alters or stops program execution, such as RTN, GTO, or STOP, or perhaps
another label.
Calling Subroutines (XEQ, RTN)
A subroutine is a routine that is called from (executed by) another routine and
returns to that same routine when the subroutine is finished. The subroutine
must start with a LBL and end with a RTN. A subroutine is itself a routine, and
it can call other subroutines.
ꢀ
XEQ must branch to a label (LBL) for the subroutine. (It cannot branch to a
line number.)
ꢀ
At the very next RTN encountered, program execution returns to the line
after the originating XBQ.
For example, routine Q in the "Normal and Inverse–Normal Distributions"
program in chapter 16 is a subroutine (to calculate Q(x)) that is called from
routine D by line
. Routine Q ends with a RTN instruction that
ꢍꢕꢖ %ꢈꢉ ꢉ
sends program execution back to routine D (to store and display the result) at
line D04. See the flow diagrams below.
The flow diagrams in this chapter use this notation:
1
Program execution branches from this line to
€
ꢀꢕꢗ ꢆ!ꢑ ꢌ
ꢌꢕꢔ ꢂꢌꢂ ꢌ
1
the line marked
("from 1").
1
Program execution branches from a line
1
marked
("to 1") to this line.
€
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Starts here.
ꢍꢕꢔ ꢂꢌꢂ ꢍ
ꢍꢕꢏ ꢊꢄꢅ"! %
ꢍꢕꢖ %ꢈꢉ ꢉ
ꢍꢕꢒ !ꢑ ꢉ
ꢍꢕꢗ #ꢊꢈ$ ꢉ
ꢍꢕ ꢆ!ꢑ ꢍ
1
2
Calls subroutine Q.
Return here.
€
Starts D again.
1
2
Starts subroutine.
€
ꢉꢕꢔ ꢂꢌꢂ ꢉ
.
.
.
Returns to routines D.
ꢉꢔ ꢁ!ꢄ
Nested Subroutines
A subroutine can call another subroutine, and that subroutine can call yet
another subroutine. This "nesting" of subroutines—the calling of a subroutine
within another subroutine—is limited to a stack of subroutines seven levels
deep (not counting the topmost program level). The operation of nested
subroutines is as shown below:
MAIN program
(top level)
End of program
Attempting to execute a subroutine nested more than seven levels deep
causes an
error.
%ꢈꢉ ꢑ#ꢈꢁꢋꢂꢑ$
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Example: A Nested Subroutine.
The following subroutine, labeled S, calculates the value of the expression
a2 + b2 + c2 + d2
as part of a larger calculation in a larger program. The subroutine calls upon
another subroutine (a nested subroutine), labeled Q, to do the repetitive
squaring and addition. This saves memory by keeping the program shorter
than it would be without the subroutine.
ꢕꢔ ꢂꢌꢂ
Starts subroutine here.
ꢕꢏ ꢊꢄꢅ"! ꢀ
Enters A.
ꢕꢖ ꢊꢄꢅ"! ꢌ
Enters B.
ꢕꢒ ꢊꢄꢅ"! ꢃ
Enters C.
ꢕꢗ ꢊꢄꢅ"! ꢍ
Enters D.
ꢕ ꢁꢃꢂ ꢍ
Recalls the data.
ꢕꢘ ꢁꢃꢂ ꢃ
ꢕꢙ ꢁꢃꢂ ꢌ
ꢕꢓ ꢁꢃꢂ ꢀ
ꢔꢕ ºꢏ
2
1
3
5
A .
€
€
€
ꢔꢔ %ꢈꢉ ꢉ
ꢔꢏ %ꢈꢉ ꢉ
ꢔꢖ %ꢈꢉ ꢉ
ꢔꢒ ꢉꢁ!
ꢔꢗ ꢁ!ꢄ
2
2
2
4
6
A + B .
€
€
€
2
2
2
A + B + C
A2 + B2 + C2 + D2
Returns to main routine.
135 Nested subroutine
ꢉꢕꢔ ꢂꢌꢂ ꢉ
ꢉꢕꢏ º65¸
ꢉꢕꢖ ºꢏ
2
Adds x .
ꢉꢕꢒ -
246
Returns to subroutine S.
ꢉꢕꢗ ꢁ!ꢄ
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Branching (GTO)
As we have seen with subroutines, it is often desirable to transfer execution to
a part of the program other than the next line. This is called branching.
Unconditional branching uses the GTO (go to) instruction to branch to a
program label. It is not possible to branch to a specific line number during a
program.
A Programmed GTO Instruction
The GTO label instruction (press
label) transfers the execution of a
z U
running program to the program line containing that label, wherever it may
be. The program continues running from the new location, and never
automatically returns to its point of origination, so GTO is not used for
subroutines.
For example, consider the "Curve Fitting" program in chapter 16, The
ꢆ!ꢑ '
instruction branches execution from any one of three independent initializing
routines to LBL Z, the routine that is the common entry point into the heart of
the program:
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ꢕꢔ ꢂꢌꢂ
Can start here.
Branches to Z.
Can start here.
.
.
.
1
1
€
ꢕꢗ ꢆ!ꢑ '
ꢂꢕꢔ ꢂꢌꢂ ꢂ
.
.
.
Branches to Z.
Can start here.
€
ꢂꢕꢗ ꢆ!ꢑ '
ꢈꢕꢔ ꢂꢌꢂ ꢈ
.
.
.
Branches to Z.
Branch to here.
1
1
€
ꢈꢕꢗ ꢆ!ꢑ '
'ꢕꢔ ꢂꢌꢂ '
.
.
.
Using GTO from the Keyboard
You
can use z U to move the program pointer to a specified label or
line number without starting program execution.
ꢀ
ꢀ
To
:
.
z U Œ Œ
ꢅꢁꢆꢇ !ꢑꢅ
To a line number:
label nn (nn < 100). For example,
z U Œ
z
A05.
U Œ
ꢀ
To a label:
label —but only if program entry is not active (no
z U
program lines displayed; PRGM off). For example,
A.
z U
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Conditional Instructions
Another way to alter the sequence of program execution is by a conditional
test, a true/false test that compares two numbers and skips the next program
instruction if the proposition is false.
For instance, if a conditional instruction on line A05 is
(that is, is x
º/ꢕ@
?
equal to zero ), then the program compares the contents of the X–register
with zero. If the X–register does contain zero, then the program goes on to the
next line. If the X–register does not contain zero, then the program skips the
next line, thereby branching to line A07. This rule is commonly known as "Do
if true."
ꢀꢕꢔ ꢂꢌꢂ ꢀ
.
.
.
Do next if true.
2
2
Skip next if false.
€
ꢀꢕꢗ º/ꢕ@
ꢀꢕ ꢆ!ꢑ ꢌ
ꢀꢕꢘ ꢂꢄ
1
1
€
ꢀꢕꢙ !ꢑ ꢀ
.
.
.
ꢌꢕꢔ ꢂꢌꢂ ꢌ
.
.
.
ꢎ
The above example points out a common technique used with conditional
tests: the line immediately after the test (which is only executed in the "true"
case) is a branch to another label. So the net effect of the test is to branch to a
different routine under certain circumstances.
There are three categories of conditional instructions:
ꢀ
Comparison tests. These compare the X– and Y–registers, or the
X–register and zero.
ꢀ
ꢀ
Flag tests. These check the status of flags, which can be either set or clear.
Loop counters. These are usually used to loop a specified number of
times.
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Tests of Comparison (x?y, x?0)
There are 12 comparisons available for programming. Pressing
or
z l
displays a. menu for one of the two categories of tests:
{ n
ꢀ
ꢀ
?
x y for tests comparing x and y.
?
x 0 for tests comparing x and 0.
Remember that x refers to the number in the X–register, and y refers to the
number in the Y–register. These do not compare the variables X and Y.
Select the category of comparison, then press the menu key for the
conditional instruction you want.
The Test Menus
?
x y
?
x 0
?
?
?
{ } for x y
{ } for x 0
≠
≠
≠
≠
?
{≤} for x≤y
{≤} for x≤0
?
?
?
{ } for x 0
{ } for x y
<
<
<
/
?
{>} for x>y
{>} for x>0
{ } for x 0
?
?
{ } for x y
≥
≥
≥
≥
?
?
{ } for x=0
/
{ } for x=y
/
If you execute a conditional test from the keyboard, the calculator will display
or
.
&ꢈ
ꢄꢑ
Example:
The "Normal and Inverse–Normal Distributions" program in chapter 16 uses
?
the x y conditional in routine T:
<
Program Lines:
Description
.
.
.
Calculates the correction for X
.
!ꢕꢓ ª
guess
Adds the correction to yield a new X
.
guess
!ꢔꢕ !ꢑ- %
!ꢔꢔ ꢀꢌ
!ꢔꢏ ꢕ)ꢕꢕꢕꢕꢔ
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<
!ꢔꢖ º ¸@
!ꢔꢒ ꢆ!ꢑ !
Tests to see if the correction is significant.
Goes back to start of loop if correction is
significant. Continues if correction is not
significant.
!ꢔꢗ ꢁꢃꢂ %
Displays the calculated value of X.
!ꢔ #ꢊꢈ$ %
Line T09 calculates the correction for X . Line T13 compares the absolute
guess
value of the calculated correction with 0.0001. If the value is less than
0.0001 ("Do If True"), the program executes line T14; if the value is equal to
or greater than 0.0001, the program skips to line T15.
Flags
A flag is an indicator of status. It is either set (true) or clear (false). Testing a
flag is another conditional test that follows the "Do if true" rule: program
execution proceeds directly if the tested flag is set, and skips one line if the
flag is clear.
Meanings of Flags
The HP 32SII has 12 flags, numbered 0 through 11. All flags can be set.,
cleared, and tested from the keyboard or by a program instruction. The
default state of all 12 flags is clear. The three–key memory clearing operation
described in appendix B clears all flags. Flags are not affected by z
{ } { }.
ꢀꢂꢂ &
b
ꢀ
Flags 0, 1, 2, 3, and 4 have no preassigned meanings. That is, their
states will mean whatever you define it to mean in a given program. (See
the example below.)
ꢀ
Flag 5, when set, will interrupt a program when an overflow occurs
within the program, displaying
and
. An overflow occurs
£
ꢑ#ꢈꢁꢋꢂꢑ$
when a result exceeds the largest number that the calculator can handle.
The largest possible number is substituted for the overflow result. If flag 5
is clear, a program with an overflow is not interrupted, though
is displayed briefly when the program eventually stops.
ꢑ#ꢈꢁꢋꢂꢑ$
ꢀ
Flag 6 is automatically set by the calculator any time an overflow occurs
(although you can also set flag 6 yourself). It has no effect, but can be
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tested.
Flags 5 and 6 allow you to control overflow conditions that occur during
a program. Setting flag 5 stops a program at the line just after the line
that caused the overflow. By testing flag 6 in a program, you can alter
the program's flow or change a result anytime an overflow occurs.
ꢀ
Flags 7, 8, and 9 control the display of fractions. Flag 7 can also be
controlled from the keyboard, When Fraction–display mode is toggled
on or off by pressing
, flag 7 is set or cleared as well.
z Š
Flag
Status
Fraction–Control Flags
8
7
9
Clear
(Default)
Fraction display
off; display real
numbers in the
current display
format.
Fraction
Reduce
denominators
not greater than
the /c value.
fractions to
smallest form.
Set
Fraction display
on; display real
numbers as
Fraction
No reduction of
fractions. (Used
only if flag 8 is
set.)
denominators
are factors of
the /c Value.
fractions.
ꢀ
Flag 10 controls program execution of equations:
When flag 10 is clear (the default state), equations in running programs
are evaluated and the result put on the stack.
When flag 10 is set, equations in running programs are displayed as
messages, causing them to behave like a VIEW statement:
1. Program execution halts.
2. The program pointer moves to the next program line.
3. The equation is displayed without affecting the stack. You can clear
the display by pressing
that key's function.
or
. Pressing any other key executes
†
a
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4. If the next program line is a PSE instruction, execution continues after
a 1–second pause.
The status of flag 10 is controlled only by execution of the SF and CF
operations from the keyboard, or by SF and CF, statements in programs.
ꢀ
Flag 11 controls prompting when executing equations in a program —
it doesn't affect automatic prompting during keyboard execution:
When flag 11 is clear (the default state), evaluation, SOLVE, and FN of
∫
equations in programs proceed without interruption. The current value of
each variable in the equation is automatically recalled each time the
variable is encountered. INPUT prompting is not affected.
When flag 11 is set, each variable is prompted for wheat it is first
encountered in the equation. A prompt for a variable occurs only once,
regardless of the number of times the variable appears in the equation.
When solving, no prompt occurs for the unknown; when integrating, no
prompt occurs for the variable of integration. Prompts halt execution.
Pressing
resumes the calculation using the value for the variable
f
you keyed in, or the displayed (current) value of the variable if
your sole response to the prompt.
is
f
Flag 11 is automatically cleared after evaluation, SOLVE, or
∫ FN of an equation in a program. The status of flag 11 is also controlled
by execution of the SF and CF operations from the keyboard, or by SF
and CF statements in programs.
Annunciators for Set Flags
Flags 0, 1, 2, and 3 have annunciators in the display that turn on when the
corresponding flag is set. The presence or absence of 0, 1, 2, or 3 lets you
know at any time whether any of these four flags is set or not. However, there
is no such indication for the status of flags 4 through 11. These status of these
?
flags can be determined by executing the FS Instruction from the keyboard.
(See "Using Flags" below.)
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Using Flags
Pressing { x displays the FLAGS menu: { } { } {
}
ꢋ ꢃꢋ ꢋ @
After selecting the function you want, you will be prompted for the flag
number (0–11). For example, press { x { } 0 to set flag 0; press
ꢋ
{
ꢋ
}
to set flag 10; press
{
ꢋ
}
1 to set
{ x
Œ
{ x
Œ
flag 11.
FLAGS Menu
Menu Key
Description
{
{
{
} n
} n
Set flag. Set flag n.
ꢋ
Clear flag. Clears flag n.
ꢃꢋ
?
} n
Is flag set Tests the status of flag n.
ꢋ @
A flag test is a conditional test that affects program execution just as the
?
comparison tests do. The FS n instruction tests whether the given flag is set. If
it is, then the next line in the program is executed. If it is not, then the next line
is skipped. This is the "Do if True" rule, illustrated under "Conditional
Instructions" earlier in this chapter.
If you test a flag from the keyboard, the calculator will display "
&ꢈ
" or
"
ꢄꢑ
".
It is good practice in a program to make sure that any conditions you will be
testing start out in a known state. Current flag settings depend on how they
have been left by earlier programs that have been run. You should not
assume that any given flag is clear, for instance, and that it will be set only if
something in the program sets it. You should make sure of this by clearing the
flag before the condition arises that might set it. See the example below.
Example: Using Flags.
The "Curve Fitting" program in chapter 16 uses flags 0 and 1 to determine
whether to take the natural logarithm of the X– and Y–inputs:
ꢀ
Lines S03 and S04 clear both of these flags so that lines W07 and W11
(in the input loop routine) do not take the natural logarithms of the X–
and Y–inputs for a Straight–line model curve.
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ꢀ
ꢀ
ꢀ
Line L03 sets flag 0 so that line W07 takes the natural log of the X–input
for a Logarithmic–model curve.
Line E04 sets flag 1 so that line W11 takes the natural log of the Y–input
for an Exponential–model curve.
Lines P03 and P04 set both flags so that lines W07 and W11 take the
natural logarithms of both the X– and Y–inputs for a Power–model curve.
Note that lines S03, S04, L04, and E03 clear flags 0 and 1 to ensure that
they will be set only as required for the four curve models.
Program Lines:
Description:
.
.
.
.
.
.
Clears flag 0, the indicator for In X.
ꢕꢖ ꢃꢋ ꢕ
Clears flag 1, the indicator for In Y.
ꢕꢒ ꢃꢋ ꢔ
.
.
.
ꢎ
Sets flag 0, the indicator for In X.
Clears flag 1, the indicator for In Y.
ꢂꢕꢖ ꢋ ꢕ
ꢂꢕꢒ ꢃꢋ ꢔ
.
.
.
.
.
.
Clears flag 0, the indicator for In X.
ꢈꢕꢖ ꢃꢋ ꢕ
Sets flag 1, the indicator for In Y.
ꢈꢕꢒ ꢋ ꢔ
.
.
.
.
.
.
Sets flag 0, the indicator for ln X.
ꢅꢕꢖ ꢋ ꢕ
Sets flag 1, the indicator for In Y.
ꢅꢕꢒ ꢋ ꢔ
.
.
.
.
.
.
If flag 0 is set ...
$ꢕ ꢋ @ ꢕ
... takes the natural log of the X–input.
If flag 1 is set ...
$ꢕꢘ ꢂꢄ
$ꢔꢕ ꢋ @ ꢔ
$ꢔꢔ ꢂꢄ
... takes the natural log of the Y–input.
Programming Techniques 13–13
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Example: Controlling the Fraction Display.
The following program lets you exercise the calculator's fraction–display
capability. The program prompts for and uses your inputs for a fractional
number and a denominator (the /c value). The program also contains
examples of how the three fraction–display flags (7, 8, and 9) and the
"message–display" flag (10) are used.
Messages in this program are listed a MESSAGE and are entered as
equations:
1. Set Equation–entry mode by pressing
(the EQN annunciator
{ G
turns on).
2. Press
letter for each alpha character in the message; press
K
o
(the
key) for each space character.
f
3. Press
to insert the message in the current program line and end
š
Equation–entry mode.
Program Lines:
ꢋꢕꢔꢎꢂꢌꢂ ꢋ
Description:
Begins the fraction program.
Clears three fraction flags.
ꢋꢕꢏꢎꢃꢋ ꢘ
ꢋꢕꢖꢎꢃꢋ ꢙ
ꢋꢕꢒꢎꢃꢋ ꢓꢎ
ꢋꢕꢗꢎ ꢋ ꢔꢕ
Displays messages.
Selects decimal base.
Prompts for a number.
Prompts for denominator (2 – 4095).
Displays message, then shows the decimal
number.
ꢋꢕ ꢎꢍꢈꢃꢎ
ꢋꢕꢘꢎꢊꢄꢅ"! #
ꢋꢕꢙꢎꢊꢄꢅ"! ꢍ
ꢋꢕꢓꢎꢁꢃꢂ #
ꢋꢔꢕꢎꢍꢈꢃꢊꢇꢀꢂꢎ
ꢋꢔꢔꢎꢅ ꢈꢎ
ꢋꢔꢏꢎ !ꢑꢅꢎ
ꢋꢔꢖꢎꢁꢃꢂ ꢍ
ꢋꢔꢒꢎ+Fꢎ
Sets /c value and sets flag 7.
ꢋꢔꢗꢎꢇꢑ ! ꢅꢁꢈꢃꢊ ꢈ
Displays message, then shows the fraction.
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Program Lines:
Description:
ꢋꢔ ꢎꢅ ꢈ
ꢋꢔꢘꢎ !ꢑꢅ
ꢋꢔꢙꢎ ꢋ ꢙ
Sets flag 8.
ꢋꢔꢓꢎꢋꢀꢃ!ꢑꢁ ꢍꢈꢄꢑꢇ
ꢋꢏꢕꢎꢅ ꢈꢎ
ꢋꢏꢔꢎ !ꢑꢅꢎ
ꢋꢏꢏꢎ ꢋ ꢓ
Displays message, then shows the fraction.
Sets flag 9.
ꢋꢏꢖꢎꢋꢊ%ꢈꢍ ꢍꢈꢄꢑꢇ
ꢋꢏꢒꢎꢅ ꢈꢎ
ꢋꢏꢗꢎ !ꢑꢅꢎ
ꢋꢏ ꢎꢆ!ꢑ ꢋ
Displays message, then shows the fraction.
Goes to beginning of program.
Checksum and length: 10C3 102.0
Use the above program to see the different forms of fraction display:
Keys:
Display:
Description:
F
Executes label F; prompts for a
fractional number (V).
W
2.53
16
#@valueꢎ
Stores 2.53 in V; prompts for
denominator (D).
f
ꢍ@valueꢎ
Stores 16 as the /c value. Displays
message, then the decimal
number.
f
ꢍꢈꢃꢊꢇꢀꢂꢎ
ꢏ)ꢗꢖꢕꢕꢎ
Message indicates the fraction
f
f
ꢇꢑ ! ꢅꢁꢈꢃꢊ ꢈꢎ
format (denominator is no greater
than 16), then shows the fraction.
dꢏ ꢙ+ꢔꢗꢎ
d
indicates that the numerator is
"a little below" 8..
Message indicates the fraction
ꢋꢀꢃ!ꢑꢁ ꢍꢈꢄꢑꢇꢎ
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Keys:
Display:
Description:
format (denominator is factor of
16), then shows the fraction.
Message indicates the fraction
cꢏ ꢔ+ꢏꢎ
f
ꢋꢊ%ꢈꢍ ꢍꢈꢄꢑꢇꢎ
format (denominator is 16), then
cꢏ ꢙ+ꢔ ꢎ
shows the fraction.
Stops the program and clears flag
10
f † {
ꢏ)ꢗꢖꢕꢕꢎ
{
ꢃꢋ
}
0
x
Œ
Loops
Branching backwards — that is, to a label in a previous line — makes it
possible to execute part of a program more than once. This is called looping.
ꢍꢕꢔ ꢂꢌꢂ ꢍ
ꢍꢕꢏ ꢊꢄꢅ"! ꢇ
ꢍꢕꢖ ꢊꢄꢅ"! ꢄ
ꢍꢕꢒ ꢊꢄꢅ"! !ꢎ
ꢍꢕꢗ ꢆ!ꢑ ꢍ
This routine (taken from the "Coordinate Transformations" program on page
15–31 in chapter 15) is an example of an infinite loop. It is used to collect the
initial data prior to the coordinate transformation. After entering the three
values, it is up to the user to manually interrupt this loop by selecting the
transformation to be performed (pressing
N for the old–to–new system
W
or
O for the new–to–old system).
W
Conditional Loops (GTO)
When you want to perform an operation until a certain condition is met, but
you don't know how many times the loop needs to repeat itself, you can
create a loop with a conditional test and a GTO instruction.
For example, the following routine uses a loop to diminish a value A by a
constant amount B until the resulting A is less than or equal to B.
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Program lines:
Description:
ꢀꢕꢔ ꢂꢌꢂ ꢀ
ꢀꢕꢏ ꢊꢄꢅ"! ꢀ
ꢀꢕꢖ ꢊꢄꢅ"! ꢌ
Checksum and length: 6157 004.5
ꢕꢔ ꢂꢌꢂ
It is easier to recall A than to remember where it is in the
stack.
ꢕꢏ ꢁꢃꢂ ꢀ
Calculates A – B.
Replaces old A with new result.
Recalls constant for comparison.
ꢕꢖ ꢁꢃꢂ. ꢌ
ꢕꢒ !ꢑ ꢀ
ꢕꢗ ꢁꢃꢂ ꢌ
ꢕ º6¸@
ꢕꢘ ꢆ!ꢑ
ꢕꢙ #ꢊꢈ$ ꢀ
ꢕꢓ ꢁ!ꢄ
?
Is B < new A
Yes: loops to repeat subtraction.
No: displays new A.
Checksum and length: 5FE1 013.5
Loops With Counters (DSE, ISG)
When you want to execute a loop a specific number of times, use the
z
(increment; skip if greater than). or
(decrement; skip if less
k
{ m
than or equal to) conditional function keys. Each time a loop function is
executed in a program, it automatically decrements or increments a counter
value stored in a variable. It compares the current counter value to a final
counter value, then continues or exits the loop depending on the result.
For a count–down loop, use { m variable
For a count–up loop, use
variable
z k
These functions accomplish the same thing as a FOR–NEXT loop in BASIC:
variable = initial–value final–value increment
ꢋꢑꢁ
!ꢑ
!ꢈꢅ
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.
.
.
variable
ꢄꢈ%!
A DSE instruction is like a FOR–NEXT loop with a negative increment.
After pressing a shifted key for ISG or DSE ( or
), you
z k { m
will be prompted for a variable that will contain the loop–control number
(described below).
The Loop–Control Number
The specified variable should contain a loop–control number ccccccc.fffii,
where:
ꢀ
ꢀ
ꢀ
ccccccc is the current counter value (1 to 12 digits). This value changes
with loop execution.
fff is the final counter value (must be three digits). This value does not
change as the loop runs.
ii is the interval for incrementing and decrementing (must be two digits or
unspecified). This value does not change. An unspecified value for ii is
assumed to be 01 (increment/decrement by 1).
Given the loop–control number ccccccc.fffii, DSE decrements ccccccc to
ccccccc — ii, compares the new ccccccc with fff, and makes program
execution skip the next program line if this ccccccc ≤ fff.
Given the loop–control number ccccccc.fffii, ISG increments ccccccc to
ccccccc + ii, compares the new cccccccc with fff, and makes program
execution skip the next program line if this ccccccc > fff.
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1
1
1
1
€
€
$ꢕꢔ ꢂꢌꢂ $
.
.
.
2
2
€
$ꢕꢓ ꢍ ꢈ ꢀ
$ꢔꢕ ꢆ!ꢑ $
$ꢔꢔ %ꢈꢉ %
If current value >
final value,
If current value ≤
final value, exit
loop.
.
.
.
continue loop.
$ꢕꢔ ꢂꢌꢂ $
.
.
.
2
2
€
$ꢕꢓ ꢊ ꢆ ꢀ
$ꢔꢕ ꢆ!ꢑ $
$ꢔꢔ %ꢈꢉ %
If current value ≤
final value,
continue loop.
If current value >
final value, exit
loop.
.
.
.
For example, the loop–control number 0.050 for ISG means: start counting at
zero, count up to 50, and increase the number by 1 each loop.
The following program uses ISG to loop 10 times. The loop counter
(0000001.01000) is stored in the variable Z. Leading and trailing zeros can
be left off.
ꢂꢕꢔ ꢂꢌꢂꢎ
ꢂꢕꢏ ꢔ)ꢕꢔ
ꢂꢕꢖ !ꢕ '
ꢇꢕꢔ ꢂꢌꢂ ꢇꢎ
ꢇꢕꢏ ꢊ ꢆ '
ꢇꢕꢖ ꢆ!ꢑ
ꢇꢕꢒ ꢁ!ꢄ
Press
Z to see that the loop–control number is now 11.0100.
{ ‰
ꢎ
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Indirectly Addressing Variables and Labels
Indirect addressing is a technique used in advanced programming to specify
a variable or label without specifying beforehand exactly which one. This is
determined when the program runs, so it depends on the intermediate results
(or input) of the program.
Indirect addressing uses two different keys:
(with
) and
(with
‘
Œ
’
).
f
The variable I has nothing to do with
or the variable i. These keys are
’
active for many functions that take A through Z as variables or labels.
ꢀ
ꢀ
i is a variable whose contents can refer to another variable or label. It
holds a number just like any other variable (A through Z).
is a programming function that directs, "Use the number in i to
’
determine which variable or label to address."
This is an indirect address. (A through Z are direct addresses.)
Both
and
are used together to create an indirect address. (See the
‘
’
examples below.)
By itself, i is just another variable.
By itself, is either undefined (no number in i) or uncontrolled (using
’
whatever number happens to be left over in i).
The Variable "i"
Your can store, recall, and manipulate the contents of i just as you car, the
contents of other variables. You can even solve for i and integrate using i .
The functions listed below can use variable "i".
STO i
RCL i
STO +,–, × ,÷ i
RCL +,–, , i
× ÷
INPUT i
VIEW i
∫ FN d i
SOLVE i
DSE i
ISG i
x < > i
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The Indirect Address, (i)
Many functions that use A through Z (as variables or labels) can use ’ to
refer to A through Z (variables or labels) or statistics registers indirectly. The
function ’ uses the value in variable i to determine which variable, label, or
register to address. The following table shows how.
If i contains:
Then (i) will address:
1
variable A or label A
.
.
.
.
.
.
26
variable Z or label Z
27
variable i
28
n register
29
x register
Σ
Σ
30
y register
2
31
x register
Σ
2
32
33
y register
Σ
Σxy register
≥34 or ≤–34 or 0
error:
ꢊꢄ#ꢀꢂꢊꢍ 6L5
Only the absolute value of the integer portion of the number in i is used for
addressing.
The INPUT(i) and VIEW(i) operations label the display with the name of the
indirectly–addressed variable or register.
The SUMS menu enables you to recall values from the statistics registers.
However, you must use indirect addressing to do other operations, such as
STO, VIEW, and INPUT.
The functions listed below can use (i) as an address. For GTO, XEQ, and FN=,
(i) refers to a label; for all other functions (i) refers to a variable or register.
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STO(i)
RCL(i)
STO +, –,× ,÷, (i)
RCL +, –,× ,÷, (i)
XEQ(i)
GTO(i)
X<>(i)
INPUT(i)
VIEW(i)
DSE(i)
ISG (i)
SOLVE(i)
∫ FN d(i)
FN=(i)
Program Control with (i)
Since the contents of i can change each time a program runs–or even in
different parts of the same program — a program instruction such as
can branch to a different label at different times. This maintains
ꢆ!ꢑ6L5
flexibility by leaving open (until the program runs) exactly which variable or
program label will be needed. (See the first example below.)
Indirect addressing is very useful for counting and controlling loops. The
variable i serves as an index, holding the address of the variable that
contains the loop–control number for the functions DSE and ISG. (See the
second example below.)
Example: Choosing Subroutines With (i).
The "Curve Fitting" program in chapter 16 uses indirect addressing to
determine which model to use to compute estimated values for x and y.
(Different subroutines compute x and y for the different models.) Notice that i
is stored and then indirectly addressed in widely separated parts of the
program.
The first four routines (S, L, E, P) of the program specify the curve–fitting model
that will be used and assign a number (1, 2, 3, 4) to each of these models.
This number is then stored during routine Z, the common entry point for all
models:
'ꢕꢖ !ꢑ L
Routine Y uses i to call the appropriate subroutine (by model) to calculate the
x– and y–estimates. Line Y03 calls the subroutine to compute y:
&ꢕꢖ %ꢈꢉ1L2
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and line Y08 calls a different subroutine to compute
increased by 6:
after i has been
ˆ
x
&ꢕ
&ꢕꢘ !ꢑ- L
&ꢕꢙ %ꢈꢉ1L2
If i hold:
Then XEQ(i) calls:
To:
ˆ
y
1
LBL A
Compute
model.
Compute
model.
for straight–line
ˆ
y
2
3
LBL B
LBL C
for logarithmic
ˆ
y
Compute for exponential
model.
Compute
Compute
model.
ˆ
y
4
7
LBL D
LBL G
for power model.
for straight–line
ˆ
x
8
9
LBL H
LBL I
LBL J
Compute
model.
Compute
model.
for logarithmic
for exponential
for power model.
ˆ
x
ˆ
x
10
Compute
ˆ
x
Example: Loop Control With (i).
An index value in i is used by the program "Solutions of Simultaneous
Equations—Matrix Inversion Method" in chapter 15. This program uses the
looping instructions
and
in conjunction with the
ꢊ ꢆ L
ꢍ ꢈ L
indirect instructions
and
to fill and manipulate a matrix .
ꢁꢃꢂ1L2
!ꢑ1L2
The first part of this program is routine A, which stores the initial loop–control
number in i.
Program lines:
Description:
The starting point for data input.
Loop–control number: loop from 1 to 12 in intervals of
1.
ꢀꢕꢔ ꢂꢌꢂ ꢀ
ꢀꢕꢏ ꢔ)ꢕꢔꢏ
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Stores loop–control number in i.
ꢀꢕꢖ !ꢑ L
The next routine is L, a loop to collect all 12 known values for a 3x3
coefficient matrix (variables A – I) and the three constants (J – L) for the
equations.
Program Lines:
Description:
This routine collects all known values in three
equations.
ꢂꢕꢔ ꢂꢌꢂ ꢂ
Prompts for and stores a number into the variable
ꢂꢕꢏ ꢊꢄꢅ"!1L2
ꢂꢕꢖ ꢊ ꢆ L
addressed by i.
Adds 1 to i and repeats the loop until i reaches
13.012.
ꢂꢕꢒ ꢆ!ꢑ ꢂ
ꢂꢕꢗ ꢆ!ꢑ ꢀ
When i exceeds the final counter value, execution
branches back to A.
Label J is a loop that completes the inversion of the 3 3 matrix.
×
Program Lines:
Description:
This routine completes inverse by dividing by
determinant.
ꢛꢕꢔ ꢂꢌꢂ ꢛ
Divides element.
Decrements index value so it points closer to A
Loops for next value.
ꢛꢕꢏ !ꢑª1L2
ꢛꢕꢖ ꢍ ꢈ L
ꢛꢕꢒ ꢆ!ꢑ ꢛ
ꢛꢕꢗ ꢁ!ꢄ
Returns to the calling program or to
.
ꢅꢁꢆꢇ !ꢑꢅ
Equations with (i)
You can use (i) in an equation to specify a variable indirectly. Notice that
means the variable specified by the number in variable i (an indirect
1L2
reference), but that i or
means variable i.
1L2
The following program uses an equation to find the sum of the squares of
variables A through Z.
Program Lines:
Description:
Begins the program.
Sets equations for execution.
ꢈꢕꢔ ꢂꢌꢂ ꢈ
ꢈꢕꢏ ꢃꢋ ꢔꢕ
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Disables equation prompting.
Sets counter for 1 to 26.
Stores counter.
ꢈꢕꢖ ꢃꢋ ꢔꢔ
ꢈꢕꢒ ꢔ)ꢕꢏ
ꢈꢕꢗ !ꢑ L
ꢈꢕ ꢕ
Initializes sum.
Checksum and length: EA5F 017.0
Program Lines:
Description:
Starts summation loop.
ꢋꢕꢔ ꢂꢌꢂ ꢋ
Equation to evaluate the ith square.
(Press { G to start the equation.)
ꢋꢕꢏ 1L2:ꢏ
Ckecksum and length of equation: 48AD 006.0
Adds ith square to sum.
ꢋꢕꢖ -
Tests for end of loop.
ꢋꢕꢒ ꢊ ꢆ L
Branches for next variable.
ꢋꢕꢗ ꢆ!ꢑ ꢋ
Ends program.
ꢋꢕ ꢁ!ꢄ
Checksum and length of program: 19A8 013.5
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14
Solving and Integrating
Programs
Solving a Program
In chapter 7 you saw how you can enter an equation — it's added to the
equation list — and then solve it for any variable. You can also, enter a
program that calculates a function, and then solve it for any variable. This is
especially useful if the equation you're solving changes for certain conditions
or if it requires repeated calculations.
To solve a programmed function:
1. Enter a program that defines the function. (See "To write a program for
SOLVE" below.)
2. Select the program to solve: press
label. (You can skip this step
{ V
if you're re–solving the same program.)
3. Solve for the unknown variable: press
variable.
{ œ
Notice that FN= is required if you're solving a programmed function, but not
if you're solving an equation from the equation list.
To halt a calculation, press
is in the unknown variable; use
or
{ ‰
. The current best estimate of the root
to view it without disturbing the
†
f
stack. To resume the calculation, press
.
f
To write a program for SOLVE:
The program can use equations and RPN operations — in whatever
combination is most convenient.
1. Begin the program with a label. This label identifies the function shat you
want SOLVE to evaluate (
ꢋꢄ/
label).
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2. Include an INPUT instruction for each variable, including the unknown.
INPUT instructions enable you to solve for any variable in a multi–variable
function. INPUT for the unknown is ignored by the calculator, so you need
to write only one program that contains a separate INPUT instruction for
every variable (including the unknown).
If you include no INPUT instructions, the program uses the values stored in
the variables or entered at equation prompts.
3. Enter the instructions to evaluate the function.
ꢀ
A function programmed as a multi–line RPN sequence must be in the
form of an expression that goes to zero at the solution. If your equation
is f(x) = g(x), your program should calculate f(x) – g(x). "=0" is
implied.
ꢀ
A function programmed as an equation can be any type of
equation—equality, assignment, or expression. The equation is
evaluated by the program, and its value goes to zero at the solution. If
you want the equation to prompt for variable values instead of
including INPUT instructions, make sure flag 11 is set.
4. End the program with a RTN. Program execution should end with the value
of the function in the X–register.
SOLVE works only with real numbers. However, if you have a complex–valued
function that can be written to isolate its real and imaginary parts, SOLVE can
solve for the parts separately.
Example: Program Using RPN.
Write a program using RPN operations that solves for any unknown in the
equation for the "Ideal Gas Law." The equation is:
P x V= N x R x T
where
2
P = Pressure (atmospheres or N/m ).
V = Volume (liters).
N = Number of moles of gas.
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R = The universal gas constant
(0.0821 liter–atm/mole–K or 8.314 J/mole–K).
T = Temperature (kelvins; K = °C + 273.1).
To begin, put the calculator in Program mode; if necessary, position the
program pointer to the top of program memory.
Keys:
Display:
Description:
z d z
U Œ Œ
Sets Program mode.
ꢎ
ꢅꢁꢆꢇ !ꢑꢅꢎ
Type in the program:
Program Lines:
ꢆꢕꢔ ꢂꢌꢂ ꢆ
Description:
Identifies the programmed function.
Stores P.
Stores V.
Stores N.
Stores R.
Stores T.
Pressure.
ꢆꢕꢏ ꢊꢄꢅ"! ꢅ
ꢆꢕꢖ ꢊꢄꢅ"! #
ꢆꢕꢒ ꢊꢄꢅ"! ꢄ
ꢆꢕꢗ ꢊꢄꢅ"! ꢁ
ꢆꢕ ꢊꢄꢅ"! !
ꢆꢕꢘ ꢁꢃꢂ ꢅ
Pressure × volume.
Number of moles of gas.
Moles × gas constant.
Moles × gas constant × temp.
ꢆꢕꢙ ꢁꢃꢂº #
ꢆꢕꢓ ꢁꢃꢂ ꢄ
ꢆꢔꢕ ꢁꢃꢂº ꢁ
ꢆꢔꢔ ꢁꢃꢂº !
_
ꢆꢔꢖ ꢁ!ꢄ
(P V) – (N R T).
Ends the program.
×
×
×
ꢆꢔꢏ
Checksum and length: 053B 019.5
Press to cancel Program–entry mode.
†
Use program "G" to solve for the pressure of 0.005 moles of carbon dioxide
in a 2–liter bottle at 24 °C.
Keys:
Display:
Description:
G
Selects "G"—the program. SOLVE
evaluates to find the value of the
{ Vꢁ
ꢎ
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unknown variable.
{ œ P
value
Selects P; prompts for V.
Stores 2 in V; prompts for N.
Stores .005 in N; prompts for R.
Stores .0821 in R; prompts for T.
Calculates T.
#@
ꢎ
2
f
ꢄ@valueꢎ
ꢁ@valueꢎ
!@valueꢎ
ꢎ
.005
f
.0821
f
24 273.1
šꢁ
™
!@ꢏꢓꢘ)ꢔꢕꢕꢕꢎ
ꢑꢂ#ꢊꢄꢆꢎ
ꢅ/ꢕ)ꢕ ꢔꢕꢎ
Stores 297.1 in T; solves for P.
f
Pressure is 0.0610 atm.
Example: Program Using Equation.
Write a program that uses an equation to solve the "Ideal Gas Law."
Keys:
Display:
Description:
Selects Program–entry mode.
Moves program pointer to top of
the list of programs.
z d z
U Œ Œ
ꢎ
ꢅꢁꢆꢇ !ꢑꢅꢎ
H
Labels the program.
z “
ꢐꢕꢔ ꢂꢌꢂ ꢐꢎ
{ }
ꢋ ꢐꢕꢏ ꢋ ꢔꢔꢎ
Enables equation prompting.
{ x
1
Œ
Evaluates the equation, clearing
flag 11. (Checksum and length:
13E3 015.0).
{ G
K P y
V
ꢎ
ꢎ
K {
K N y
ꢎ
ꢎ
ꢎ
R
K y
K T š
{ ”
†
ꢐꢕꢖ ꢅº#/ꢄºꢁºꢎ
Ends the program.
Cancels Program–entry mode.
ꢐꢕꢒ ꢁ!ꢄꢎ
ꢕ)ꢕ ꢔꢕꢎ
Checksum and length of program: 8AD6 19.5
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Now calculate the change in pressure of the carbon dioxide if its temperature
drops by 10 °C from the previous example.
Keys:
Display:
Description:
L
Stores previous pressure.
H
ꢕ)ꢕ ꢔꢕꢎ
ꢕ)ꢕ ꢔꢕꢎ
H
Enters the limits of integration
(lower limit first).
{ V
P
Selects variable P; prompts for V.
Retains 2 in V; prompts for N.
Retains .005 in N; prompts for R.
Retains .0821 in R; prompts for T.
Calculates new T.
{ œ
f
#@ꢏ)ꢕꢕꢕꢕꢎ
ꢄ@ꢕ)ꢕꢕꢗꢕꢎ
ꢁ@ꢕ)ꢕꢙꢏꢔꢎ
!@ꢏꢓꢘ)ꢔꢕꢕꢕꢎ
!@ꢏꢙꢘ)ꢔꢕꢕꢕꢎ
ꢑꢂ#ꢊꢄꢆꢎ
f
f
10
š
f
„
Stores 287.1 in T; solves for new P.
ꢅ/ꢕ)ꢕꢗꢙꢓꢎ
.ꢕ)ꢕꢕꢏꢔꢎ
L
K „
Calculates pressure change of the
gas when temperature drops from
297.1 K to 287.1 K (negative
result indicates drop in pressure).
Using SOLVE in Program
You can use the SOLVE operation as part of a program.
If appropriate, include or prompt for initial guesses (into the unknown
variable and into the X–register) before executing the SOLVE variable
instruction. The two instructions for solving an equation for an unknown
variable appear in programs as:
label
ꢋꢐ/
variable
ꢑꢂ#ꢈ
The programmed SOLVE instruction does not produce a labeled display
(variable = value) since this might not be the significant output for your
program (that is, you might wart to do further calculations with this number
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before displaying it). If you do want this result displayed, add a VIEW
variable instruction after the SOLVE instruction.
If no solution is found for the unknown variable, then the next program line is
skipped (in accordance with the "Do if True" rule, explained in chapter 13).
The program should then handle the case of not finding a root, such as by
choosing new initial estimates or changing an input value.
Example: SOLVE in a Program.
The following excerpt is from a program that allows you to solve for x or y by
pressing
X or Y.
W
Program Lines:
Description:
Setup for X.
Index for X.
%ꢕꢔ ꢂꢌꢂ %
%ꢕꢏ ꢏꢒ
Branches to main routine. Checksum and
length: CCEC 004.5
%ꢕꢖ ꢆ!ꢑ ꢂ
Setup for Y.
&ꢕꢔ ꢂꢌꢂ &
Index for Y.
&ꢕꢏ ꢏꢗ
Branches to main routine.
&ꢕꢖ ꢆ!ꢑ ꢂ
Checksum and length. 2E48 004.5
Main routine.
Stores index in i.
Defines program to solve.
Solves for appropriate variable.
Displays solution.
ꢂꢕꢔ ꢂꢌꢂ ꢂ
ꢂꢕꢏ !ꢑ L
ꢂꢕꢖ ꢋꢄ/ ꢋ
ꢂꢕꢒ ꢑꢂ#ꢈ1L2
ꢂꢕꢗ #ꢊꢈ$1L2
ꢂꢕ ꢁ!ꢄ
Ends program. Checksum and length:
E159 009.0
Calculates f (x,y). Include INPUT or
equation prompting as required.
ꢋꢕꢔ ꢂꢌꢂ ꢋꢎ
)ꢎ
)ꢎ
)
RTN
ꢋꢔꢕ
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Integrating a Program
In chapter 8 you saw how you can enter an equation (or expression) — it's
added to the list of equations — and then integrate it with respect to any
variable. You can also enter a program that calculates a function, and then
integrate it with respect to any variable. This is especially useful if the function
you're integrating changes for certain conditions or if it requires repeated
calculations.
To integrate a programmed function:
1. Enter a program that defines the integrand's function. (See "To write a
program for ∫ FN" below.)
2. Select the program that defines the function to integrate: press { V
label. (You can skip this step if you're reintegrating the same program.)
3. Enter the limits of integration: key in the lower limit and press
key in the upper limit.
then
š
4. Select the variable of integration and start the calculation: press
œ variable.
{
Notice that FN= is required if you're integrating a programmed function, but
list.
riot if you're integrating an equation from the equation
You can halt a running integration calculation by pressing
or
.
†
f
However, no information about the integration is available until the
calculation finishes normally. To resume the calculation, press again.
f
Pressing
while an integration calculation is running cancels the ∫ FN
W
operation. In this case, you should start FN again from the beginning.
∫
To write a program for FN;
∫
The program can use equations and RPN operations — in whatever
combination is most convenient.
1. Begin the program with a label. This label identifies the function that you
want to integrate (
ꢋꢄ/
label).
2. Include an INPUT instruction for each variable, including the variable of
integration. INPUT instructions enable you to integrate with respect to any
variable in a multi–variable function. INPUT for the variable of integration
Solving and Integrating Programs 14–7
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is ignored by the calculator, so you need to write only one program that
contains a separate INPUT instruction for every variable (including the
variable of integration).
If you include no INPUT instructions, the program uses the values stored in
the variables or entered at equation prompts.
3. Enter the instructions to evaluate the function.
ꢀ
A function programmed as a multi–line RPN sequence must calculate
the function values you want to integrate.
ꢀ
A function programmed as an equation is usually included as an
expression specifying the integrand — though it can be any type of
equation. If you want the equation to prompt for variable values
instead of including INPUT instructions, make sure flag 11 is set.
4. End the program with a RTN. Program execution should end with the value
of the function in the X–register.
Example: Program Using Equation.
The sine integral function in the example in chapter 8 is
t
sin x
Si (t) = (
)dx
∫
0
x
This function can be evaluated by integrating a program that defines the
integrand:
Defines the function.
ꢕꢔ ꢂꢌꢂ
The function as an expression. (Checksum and length:
ꢕꢏ ꢊꢄ1%2ª%
4914 009.0).
Ends the subroutine
ꢕꢖ ꢁ!ꢄ
Checksum and length of program: C62A 012.0
Enter this program and integrate the sine integral function with respect to x
from 0 to 2 (t = 2).
Keys:
Display:
Description:
14–8 Solving and Integrating Programs
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{
{
}
ꢁꢍ ꢎ
Selects Radians mode.
z Ÿ
S
Selects label S as the integrand.
Enters lower and upper limits of
integration.
{ V
ꢎ
0
2
š
ꢏ_ꢎ
X
Integrates function from 0 to 2;
{ )
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
displays result.
Restores Degrees mode.
ꢔ) ꢕꢗꢒꢎ
}
ꢍꢆ ꢔ) ꢕꢗꢒꢎ
z Ÿ
Using Integration in a Program
Integration can be executed from a program. Remember to include or prompt
for the limits of integration before executing the integration, and remember
that accuracy and execution time are controlled by the display format at the
time the program runs. The two integration instructions appear in the program
as:
label
ꢋꢄ/
∫
variable
ꢋꢄ G
The programmed ∫ FN instruction does not produce a labeled display ( ∫ =
value) since this might riot be the significant output for your program (that is,
you might want to do further calculations with this number before displaying
it). If you do want this result displayed, add a PSE (
) or STOP
{ꢁe
(f) instruction to display the result in the X–register after the ∫ FN
instruction.
Example: FN in a Program.
∫
The "Normal and Inverse–Normal Distributions" program in chapter 16
includes an integration of the equation of the normal density function
D−M
S
2
/
MD e−(
)
2dD.
1
S 2π
∫
Solving and Integrating Programs 14–9
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2
The ((D−M)÷S) ÷2 function is calculated by the routine labeled F. Other
e
routines prompt for the known values and do the other calculations to find
Q(D), the upper–tail area of a normal curve. The integration itself is set up
and executed from routine Q:
ꢉꢕꢔ ꢂꢌꢂ ꢉ
Recalls lower limit of integration.
ꢉꢕꢏ ꢁꢃꢂ ꢇ
Recalls upper limit of integration. (X = D.)
ꢉꢕꢖ ꢁꢃꢂ %
Specifies the function.
ꢉꢕꢒ ꢋꢄ/ ꢋ
Integrates the normal function using the dummy variable
∫
ꢉꢕꢗ ꢋꢄ G ꢍ
D.
Restrictions o Solving and Integrating
he SOLVE variable and ∫ FN d variable instructions cannot call a routine that
contains another SOLVE or ∫ FN instruction. That is, neither of these
T
instructions can be used recursively. For example, attempting to calculate a
multiple integral will result in an ∫ ∫
1 ꢋꢄ2
error. Also, SOLVE and ∫ FN cannot
call a routine that contains an
label instruction; if attempted, a
ꢋꢄ/
ꢑꢂ#ꢈ
or ∫
error will be returned. SOLVE cannot call a routine
ꢀꢃ!ꢊ#ꢈ
ꢋꢄ ꢀꢃ!ꢊ#ꢈ
that contains an FN instruction (produces a
error), just as FN
∫
∫
∫
ꢑꢂ#ꢈ1 ꢋꢄ2
cannot call a routine that contains a SOLVE instruction (produces an
error).
∫
1 ꢑꢂ#ꢈ2
The SOLVE variable and FN d variable instructions in a program use one of
∫
the seven pending subroutine returns in the calculator. (Refer to "Nested
Subroutines" in chapter 13.)
The SOLVE and ∫FN operations automatically set Decimal display format.
14–10 Solving and Integrating Programs
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15
Mathematics Programs
Vector Operations
This program performs the basic vector operations of addition, subtraction,
cross product, and dot (or scalar) product. The program uses
three–dimensional vectors and provides input and output in rectangular or
polar form. Angles between vectors can also be found.
Z
P
R
Y
T
X
Mathematics Programs 15–1
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This program uses the following equations. Coordinate conversion:
X2 + Y2 + Z2
X = R sin(P) cos(T)
Y = R sin(P) sin(T)
R =
T = arctan (Y/X)
Z
Z = R cos(P)
P = arctan
X2 + Y2
Vector addition and subtraction:
v + v = (X + U)i + (Y + V)j + (Z + W)k
1
2
v – v = (U – X)i + (V – Y)j + (W – Z)k
2
1
Cross product:
v × v = (YW – ZV )i + (ZU – XW)j + (XV – YU)k
1
2
Dot Product:
D = XU + YV + ZW
Angle between vectors (γ):
D
R1× R2
G = arccos
where
v = X i + Y j + Z k
1
and
v =U i + V j + W k
2
The vector displayed by the input routines (LBL P and LBL R) is V .
1
Program Listing:
15–2 Mathematics Programs
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Program Lines:
Description
Defines the beginning of the rectangular input/display
routine.
ꢁꢕꢔ ꢂꢌꢂ ꢁꢎ
Displays or accepts input of X.
ꢁꢕꢏ ꢊꢄꢅ"! %ꢎ
Displays or accepts input of Y.
ꢁꢕꢖ ꢊꢄꢅ"! &ꢎ
Displays or accepts input of Z.
ꢁꢕꢒ ꢊꢄꢅ"! 'ꢎ
Checksum and length: F8AB 006.0
Defines beginning of rectangular–to–polar conversion
process.
ꢉꢕꢔ ꢂꢌꢂ ꢉꢎ
ꢉꢕꢏ ꢁꢃꢂ &ꢎ
ꢉꢕꢖ ꢁꢃꢂ %ꢎ
(X2 + Y2)
and arctan(Y/X).
´
θ
ꢉꢕꢒ ¸8º
8TꢎCalculates
ꢉꢕꢗ º65¸ꢎ
ꢉꢕ !ꢑ !ꢎ
Saves T = arctan(Y/X).
(X2 + Y2)
¶
ꢉꢕꢘ ꢁ ꢎ
Gets
back.
ꢉꢕꢙ ꢁꢃꢂ 'ꢎ
(X2 + Y2 + Z2)
and P.
´
θ
Calculates
Saves R.
ꢉꢕꢓ ¸8º
8Tꢎ
ꢉꢔꢕ !ꢑ ꢁꢎ
ꢉꢔꢔ º65¸ꢎ
ꢉꢔꢏ !ꢑ ꢅꢎ
Saves P
Checksum and length: 3D28 018.0
Defines the beginning of the polar input/display
routine.
ꢅꢕꢔ ꢂꢌꢂ ꢅꢎ
Displays or accepts input of R.
ꢅꢕꢏ ꢊꢄꢅ"! ꢁꢎ
Displays or accepts input of T.
ꢅꢕꢖ ꢊꢄꢅ"! !ꢎ
Displays or accepts input of P.
ꢅꢕꢒ ꢊꢄꢅ"! ꢅꢎ
ꢅꢕꢗ ꢁꢃꢂ !ꢎ
ꢅꢕ ꢁꢃꢂ ꢅꢎ
ꢅꢕꢘ ꢁꢃꢂ ꢁꢎ
´
ꢅꢕꢙ 8T ¸8ºꢎ
Calculates R cos(P) and R sin(P).
Stores Z = R cos(P).
θ
ꢅꢕꢓ !ꢑ 'ꢎ
¶
ꢅꢔꢕ ꢁ ꢎ
´
ꢅꢔꢔ 8T ¸8ºꢎ
θ
ꢅꢔꢏ !ꢑ 'ꢎ
Calculates R sin(P) cos(T) and R sin(P) sin(T).
Saves X = R sin(P) cos(T).
Mathematics Programs 15–3
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Program Listing:
Program Lines:
Description
ꢅꢔꢖ º65¸ꢎ
ꢅꢔꢒ !ꢑ &ꢎ
ꢅꢔꢗ ꢆ!ꢑ ꢅꢎ
Saves Y = R sin(P) sin(T).
Loops back for another display of polar form.
Checksum and length: D518 022.5
Defines the beginning of the vector–enter routine.
Copies values in X , Y and Z to U, V and W
respectively.
ꢈꢕꢔ ꢂꢌꢂ ꢈꢎ
ꢈꢕꢏ ꢁꢃꢂ %ꢎ
ꢈꢕꢔ !ꢑ "ꢎ
ꢈꢕꢒ ꢁꢃꢂ &ꢎ
ꢈꢕꢗ !ꢑ #ꢎ
ꢈꢕ ꢁꢃꢂ 'ꢎ
ꢈꢕꢘ !ꢑ $ꢎ
ꢈꢕꢙ ꢆ!ꢑ ꢉꢎ
Loops back for polar conversion and display/input.
Checksum and length: 1032 012.0
Defines beginning of vector–exchange routine.
Exchanges X, Y and Z with U, V and W respectively.
%ꢕꢔ ꢂꢌꢂ %ꢎ
%ꢕꢏ ꢁꢃꢂ %ꢎ
%ꢕꢖ %65 "ꢎ
%ꢕꢒ !ꢑ %ꢎ
%ꢕꢗ ꢁꢃꢂ &ꢎ
%ꢕ %65 #ꢎ
%ꢕꢘ !ꢑ &ꢎ
%ꢕꢙ ꢁꢃꢂ 'ꢎ
%ꢕꢓ %65 $ꢎ
%ꢔꢕ !ꢑ 'ꢎ
%ꢔꢔ ꢆ!ꢑ ꢉꢎ
Loops back for polar conversion and display/input.
Checksum and length: DACE 016.5
Defines beginning of vector–addition routine.
ꢀꢕꢔ ꢂꢌꢂ ꢀꢎ
ꢀꢕꢏ ꢁꢃꢂ %ꢎ
ꢀꢕꢖ ꢁꢃꢂ- "ꢎ
15–4 Mathematics Programs
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Program Listing:
Program Lines:
Description
Saves X + U in X.
Saves V + Y in Y.
Saves Z + W in Z.
ꢀꢕꢒ !ꢑ %ꢎ
ꢀꢕꢗ ꢁꢃꢂ #ꢎ
ꢀꢕ ꢁꢃꢂ- &ꢎ
ꢀꢕꢘ !ꢑ &ꢎ
ꢀꢕꢙ ꢁꢃꢂ 'ꢎ
ꢀꢕꢓ ꢁꢃꢂ- $ꢎ
ꢀꢔꢕ !ꢑ 'ꢎ
ꢀꢔꢔ ꢆ!ꢑ ꢉꢎ
Loops back for polar conversion and display/input.
Checksum and length: 641B 016.5
Defines the beginning of the vector–subtraction
routine.
ꢕꢔ ꢂꢌꢂ ꢎ
Multiplies X, Y and Z by (–1) to change the sign.
ꢕꢏ .ꢔꢎ
ꢕꢖ !ꢑº %ꢎ
ꢕꢒ !ꢑº &ꢎ
ꢕꢗ !ꢑº 'ꢎ
ꢕ ꢆ!ꢑ ꢀꢎ
Goes to the vector–addition routine.
Checksum and length: D051 017.0
Defines the beginning of the cross–product routine.
ꢃꢕꢔ ꢂꢌꢂ ꢃꢎ
ꢃꢕꢏ ꢁꢃꢂ &ꢎ
ꢃꢕꢖ ꢁꢃꢂº $ꢎ
ꢃꢕꢒ ꢁꢃꢂ 'ꢎ
ꢃꢕꢗ ꢁꢃꢂº #ꢎ
ꢃꢕ .ꢎ
Calculates (YW – ZV), which is the X component.
Calculates (ZU – WX), which is the Y component.
ꢃꢕꢘ ꢁꢃꢂ 'ꢎ
ꢃꢕꢙ ꢁꢃꢂº #ꢎ
ꢃꢕꢓ ꢁꢃꢂ %ꢎ
ꢃꢔꢕ ꢁꢃꢂº $ꢎ
ꢃꢔꢔ .ꢎ
ꢃꢔꢏ ꢁꢃꢂ %ꢎ
ꢃꢔꢖ ꢁꢃꢂº "ꢎ
ꢃꢔꢒ ꢁꢃꢂ &ꢎ
Mathematics Programs 15–5
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Program Listing:
Program Lines:
Description
ꢃꢔꢗ ꢁꢃꢂº #ꢎ
ꢃꢔ .ꢎ
Stores (XV – YU), which is the Z component.
Stores Y component.
ꢃꢔꢘ !ꢑ 'ꢎ
¶
ꢃꢔꢙ ꢁ ꢎ
ꢃꢔꢓ !ꢑ &ꢎ
¶
ꢃꢏꢕ ꢁ ꢎ
Stores X component.
Loops back for polar conversion and display/input.
ꢃꢏꢔ !ꢑ %ꢎ
ꢃꢏꢏ ꢆ!ꢑ ꢉꢎ
Checksum and length: FEB2 033.0
Defines beginning of dot–product and vector–angle
routine.
ꢍꢕꢔ ꢂꢌꢂ ꢍꢎ
ꢍꢕꢏ ꢁꢃꢂ %ꢎ
ꢍꢕꢖ ꢁꢃꢂº "ꢎ
ꢍꢕꢒ ꢁꢃꢂ &ꢎ
ꢍꢕꢗ ꢁꢃꢂº #ꢎ
ꢍꢕ -ꢎ
ꢍꢕꢘ ꢁꢃꢂ 'ꢎ
ꢍꢕꢙ ꢁꢃꢂº $ꢎ
ꢍꢕꢓ -ꢎ
Stores the dot product of XU + YV + ZW.
Displays the dot product.
ꢍꢔꢕ !ꢑ ꢍꢎ
ꢍꢔꢔ #ꢊꢈ$ ꢍꢎ
ꢍꢔꢏ ꢁꢃꢂ ꢍꢎ
ꢍꢔꢖ ꢁꢃꢂª ꢁꢎ
Divides the dot product by the magnitude of the X–,
Y–, Z–vector.
ꢍꢔꢒ ꢁꢃꢂ $ꢎ
ꢍꢔꢗ ꢁꢃꢂ #ꢎ
ꢍꢔ ꢁꢃꢂ "ꢎ
´
θ
ꢍꢔꢙ º65¸ꢎ
ꢍꢔꢘ ¸8º
8Tꢎ
¶
ꢍꢔꢓ ꢁ ꢎ
´
Calculates the magnitude of the U, V, W vector.
θ
ꢍꢏꢔ º65¸ꢎ
ꢍꢏꢕ ¸8º
8Tꢎ
15–6 Mathematics Programs
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Program Listing:
Program Lines:
Description
¶
ꢍꢏꢏ ꢁ ꢎ
Divides previous result by the magnitude.
Calculates angle.
ꢍꢏꢖ ªꢎ
ꢍꢏꢒ ꢀꢃꢑ ꢎ
ꢍꢏꢗ !ꢑ ꢆꢎ
ꢍꢏ #ꢊꢈ$ ꢆꢎ
ꢍꢏꢘ ꢆ!ꢑ ꢅꢎ
Displays angle.
Loops back for polar display/input.
Checksum and length: 1DFC 040.5
Flags Used:
None.
Memory Required:
270 bytes: 182 for program, 88 for variables.
Remarks:
The length of routine S can be shortened by 6.5 bytes. The value –1 as shown
uses 9.5 bytes. If it appears as 1 followed by +/– , it will require only 3 bytes.
To do this, you can press 1
.
{ _
The terms "polar" and "rectangular," which refer to two–dimensional systems,
are used instead of the proper three–dimensional terms of "spherical" and
"Cartesian." This stretch of terminology allows the labels to be associated
with their function without confusing conflicts. For instance, if LBL C had been
associated with Cartesian coordinate input, it would not have been available
for cross product.
Program Instructions:
1. Key in the program routines; press
2. If your vector is in rectangular form, press W R and go to step 4. If your
when done.
†
vector is in polar form, press
P and continue with step 3.
W
Mathematics Programs 15–7
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3. Key in R and press
, key in T and press
, key in Y and press
, then key in P and press
, and key in Z and press
f
f
Continue at step 5.
f
4. Key in X and press
f
f
.
f
5. To key in a second vector, press W E (for enter), then go to step 2.
6. Perform desired vector operation:
a. Add vectors by pressing
A;
W
b. Subtract vector one from vector two by pressing
S;
W
c. Compute the cross product by pressing
d. Compute the dot product by pressing W D and the angle between
C;
W
vectors by pressing
.
f
7. Optional: to review v in polar form, press W P, then press f
1
repeatedly to see the individual elements.
8. Optional: to review v in rectangular form, press
R, then press
W
f
1
repeatedly to see the individual elements.
9. If you added, subtracted, or computed the cross product, v has been
1
replaced by the result, v is not altered. To continue calculations based on
2
the result, remember to press
E before keying in a new vector.
W
10.Go to step 2 to continue vector calculations.
Variables Used:
X, Y, Z
U, V, W
R, T, P
The rectangular components of v .
1
The rectangular components of v .
2
The radius, the angle in the x–y plane (θ), and the angle
from the Z axis of v (U).
1
D
The dot product
G
The angle between vector (γ)
Example 1
A microwave antenna is to be pointed at a transmitter which is 15.7
kilometers North, 7.3 kilometers East and 0.76 kilometers below. Use the
15–8 Mathematics Programs
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rectangular to polar conversion capability to find the total distance and the
direction to the transmitter.
N
(y)
7.3
Transmitter
15.7
Antenna
E
(x)
W
S
Keys:
Display:
Description:
z Ÿ {
}
Sets Degrees mode.
ꢎ
ꢍꢆ
R
W
%@valueꢎ
&@valueꢎ
7.3
Starts rectangular input/display
routine.
fꢁ
15.7
Sets X equal to 7.3. Sets Y equal
to 15.7.
fꢁ
'@valueꢎ
.76
Sets Z equal to –0.76 and
calculates R, the radius.
Calculates T, the angle in the x/y
plane.
_ fꢁ
ꢁ@ꢔꢘ)ꢖꢖꢕꢙꢎ
!@ ꢗ)ꢕ ꢖꢔꢎ
ꢅ@ꢓꢏ)ꢗꢔꢖꢒꢎ
fꢁ
fꢁ
Calculates P, the angle from the
z-axis.
Mathematics Programs 15–9
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Example 2:
What is the moment at the origin of the lever shown below What is the
?
?
component of force along the lever What is the angle between the resultant
?
of the force vectors and the lever
F = 17
1
o
T = 215
o
P = 17
Z
F = 23
1.07m
2
o
T = 80
P = 74
o
63
o
Y
o
125
X
First, add the force vectors.
Keys:
Display:
Description:
W P
Starts polar input routine.
Sets radius equal to 17.
Sets T equal to 215.
Sets P equal to 17.
ꢁ@valueꢎ
!@valueꢎ
ꢅ@value
17
f
215 f
17
f
ꢁ@ꢔꢘ)ꢕꢕꢕ
W E
Enters vector by copying it into v .
ꢁ@ꢔꢘ)ꢕꢕꢕ
2
23
Sets radius of v , equal to 23.
f
!@.ꢔꢒꢗ)ꢕꢕꢕꢕꢎ
ꢅ@ꢔꢘ)ꢕꢕꢕꢕꢎ
1
80
Sets T equal to 80.
f
15–10 Mathematics Programs
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74
Sets P equal to 74.
f
ꢁ@ꢏꢖ)ꢕꢕꢕꢕꢎ
ꢁ@ꢏꢓ)ꢒꢘꢒꢔꢎ
A
Adds the vectors and displays the
resultant R.
W
Displays T of resultant vector.
Displays P of resultant vector.
Enters resultant vector.
f
f
W
!@ꢓꢕ)ꢘꢕꢖꢏꢎ
ꢅ@ꢖꢓ)ꢓꢒꢒꢗꢎ
ꢁ@ꢏꢓ)ꢒꢘꢒꢔꢎ
E
Since the moment equals the cross product of the radius vector and the force
vector (r × F), key in the vector representing the lever and take the cross
product.
Keys:
Display:
Description:
1.07
125
63
Sets R equal to 1.07.
Sets T equal to 125.
f
!@ꢓꢕ)ꢘꢕꢖꢏꢎ
ꢅ@ꢖꢓ)ꢓꢒꢒꢗꢎ
ꢁ@ꢔ)ꢕꢘꢕꢕꢎ
ꢁ@ꢔꢙ)ꢕꢏꢕꢓꢎ
f
Sets P equal to 63.
f
C
Calculates cross product and
displays R of result.
W
Displays T of cross product.
Displays P of cross product.
Displays rectangular form of cross
product.
f
f
W
!@ꢗꢗ)ꢖꢘꢔꢓꢎ
ꢅ@ꢔꢏꢒ)ꢖꢒꢔꢏꢎ
%@ꢙ)ꢒꢗꢗꢒꢎ
R
f
f
&@ꢔꢏ)ꢏꢒꢖꢓꢎ
'@.ꢔꢕ)ꢔ ꢕꢎ
The dot product can be used to resolve the force (still in v ) along the axis of
2
the lever.
Keys:
Display:
Description:
W P
Starts polar input routine.
Defines the radius as one unit
vector.
ꢁ@ꢔꢙ)ꢕꢏꢕꢓꢎ
!@ꢗꢗ)ꢖꢘꢔꢓꢎ
1
f
125
63
Sets T equal to 125.
Sets P equal to 63.
f
ꢅ@ꢔꢏꢒ)ꢖꢒꢔꢏꢎ
ꢁ@ꢔ)ꢕꢕꢕꢕꢎ
f
Mathematics Programs 15–11
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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D
Calculates dot product.
W
f
ꢍ/ꢏꢒ)ꢔꢙꢙꢏꢎ
ꢆ/ꢖꢒ)ꢙꢒꢓꢕꢎ
Calculates angle between
resultant force vector and lever.
Gets back to input routine.
f
ꢁ@ꢔ)ꢕꢕꢕꢕꢎ
Solutions of Simultaneous Equations
This program solves simultaneous linear equations in two or three unknowns.
It does this through matrix inversion and matrix multiplication.
A system of three linear equations
AX + DY + GZ = J
BX + EY + HZ = K
CX + FY + IZ = L
can be represented by the matrix equation below.
A D G X
J
B E H Y = K
C F I Z
L
The matrix equation may be solved for X, Y, and Z by multiplying the result
matrix by the inverse of the coefficient matrix.
′
′
′
′
′
′
A D G J
X
′
B E H K = Y
′
′
C F I L
Z
Specifics regarding the inversion process are given in the comments for the
inversion routine, I.
15–12 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Listing:
Program Lines:
Description
Starting point for input of coefficients.
ꢀꢕꢔ ꢂꢌꢂ ꢀꢎ
Loop–control value: loops from I to 12, one at a
time.
ꢀꢕꢏ ꢔ)ꢕꢔꢏꢎ
ꢀꢕꢖ !ꢑ Lꢎ
Stores control value in index variable.
Checksum and length: 9F76 012.5
Starts the input loop.
ꢂꢕꢔ ꢂꢌꢂ ꢂꢎ
Prompts for and stores the variable addressed by
i.
ꢂꢕꢏ ꢊꢄꢅ"!1L2ꢎ
Adds one to i.
If i is less than 13, goes back to LBL L and gets the
next value.
ꢂꢕꢖ ꢊ ꢆ Lꢎ
ꢂꢕꢒ ꢆ!ꢑ ꢂꢎ
Returns to LBL A to review values.
ꢂꢕꢗ ꢆ!ꢑ ꢀꢎ
Checksum and length: 8356 007.5
This routine inverts a 3 3 matrix.
Calculates determinant and saves value for the
division loop, J.
×
ꢊꢕꢔ ꢂꢌꢂ ꢊꢎ
ꢊꢕꢏ %ꢈꢉ ꢍꢎ
ꢊꢕꢖ !ꢑ $ꢎ
ꢊꢕꢒ ꢁꢃꢂ ꢀꢎ
ꢊꢕꢗ ꢁꢃꢂº ꢊꢎ
ꢊꢕ ꢁꢃꢂ ꢃꢎ
ꢊꢕꢘ ꢁꢃꢂº ꢆꢎ
ꢊꢕꢙ .ꢎ
Calculates E' × determinant = AI – CG.
Calculates F' × determinant = CD – AF.
ꢊꢕꢓ !ꢑ %ꢎ
ꢊꢔꢕ ꢁꢃꢂ ꢃꢎ
ꢊꢔꢔ ꢁꢃꢂº ꢍꢎ
ꢊꢔꢏ ꢁꢃꢂ ꢀꢎ
ꢊꢔꢖ ꢁꢃꢂº ꢋꢎ
ꢊꢔꢒ .ꢎ
ꢊꢔꢗ !ꢑ &ꢎ
ꢊꢔ ꢁꢃꢂ ꢌꢎ
ꢊꢔꢘ ꢁꢃꢂº ꢆꢎ
Mathematics Programs 15–13
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
ꢊꢔꢙ ꢁꢃꢂ ꢀꢎ
ꢊꢔꢓ ꢁꢃꢂº ꢐꢎ
ꢊꢏꢕ .ꢎ
Calculates H' × determinant = BG – AH.
ꢊꢏꢔ !ꢑ 'ꢎ
ꢊꢏꢏ ꢁꢃꢂ ꢀꢎ
ꢊꢏꢖ ꢁꢃꢂº ꢈꢎ
ꢊꢏꢒ ꢁꢃꢂ ꢌꢎ
ꢊꢏꢗ ꢁꢃꢂº ꢍꢎ
ꢊꢏ .ꢎ
Calculates I' determinant = AE – BD.
×
ꢊꢏꢘ !ꢑ Lꢎ
ꢊꢏꢙ ꢁꢃꢂ ꢈꢎ
ꢊꢏꢓ ꢁꢃꢂº ꢊꢎ
ꢊꢖꢕ ꢁꢃꢂ ꢋꢎ
ꢊꢖꢔ ꢁꢃꢂº ꢐꢎ
ꢊꢖꢏ .ꢎ
Calculates A' x determinant = EI – FH,
Calculates B' × determinant = CH – BI.
ꢊꢖꢖ !ꢑ ꢀꢎ
ꢊꢖꢒ ꢁꢃꢂ ꢃꢎ
ꢊꢖꢗ ꢁꢃꢂº ꢐꢎ
ꢊꢖ ꢁꢃꢂ ꢌꢎ
ꢊꢖꢘ ꢁꢃꢂº ꢊꢎ
ꢊꢖꢙ .ꢎ
ꢊꢖꢓ ꢁꢃꢂ ꢌꢎ
ꢊꢒꢕ ꢁꢃꢂº ꢋꢎ
ꢊꢒꢔ ꢁꢃꢂ ꢃꢎ
ꢊꢒꢏ ꢁꢃꢂº ꢈꢎ
ꢊꢒꢖ .ꢎ
Calculates C' × determinant = BF – CE.
Stores B'.
ꢊꢒꢒ !ꢑ ꢃꢎ
¶
ꢊꢒꢗ ꢁ ꢎ
ꢊꢒ !ꢑ ꢌꢎ
ꢊꢒꢘ ꢁꢃꢂ ꢋꢎ
ꢊꢒꢙ ꢁꢃꢂº ꢆꢎ
ꢊꢒꢓ ꢁꢃꢂ ꢍꢎ
ꢊꢗꢕ ꢁꢃꢂº ꢊꢎ
ꢊꢗꢔ .ꢎ
Calculates D' × determinant = FG – DI.
15–14 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
ꢊꢗꢏ ꢁꢃꢂ ꢍꢎ
ꢊꢗꢖ ꢁꢃꢂº ꢐꢎ
ꢊꢗꢒ ꢁꢃꢂ ꢈꢎ
ꢊꢗꢗ ꢁꢃꢂº ꢆꢎ
ꢊꢗ .ꢎ
Calculates G', × determinant = DH – EG.
Stores D'.
ꢊꢗꢘ !ꢑ ꢆꢎ
¶
ꢊꢗꢙ ꢁ ꢎ
ꢊꢗꢓ !ꢑ ꢍꢎ
ꢊ ꢕ ꢁꢃꢂ Lꢎ
Stores I'.
Stores E'.
Stores F'.
Stores H'.
ꢊ ꢔ !ꢑ ꢊꢎ
ꢊ ꢏ ꢁꢃꢂ %ꢎ
ꢊ ꢖ !ꢑ ꢈꢎ
ꢊ ꢒ ꢁꢃꢂ %ꢎ
ꢊ ꢗ !ꢑ ꢋꢎ
ꢊ
ꢁꢃꢂ 'ꢎ
ꢊ ꢘ !ꢑ ꢐꢎ
ꢊ ꢙ ꢓꢎ
Sets index value to point to last element of matrix.
Recalls value of determinant.
ꢊ ꢓ !ꢑ Lꢎ
ꢊꢘꢕ ꢁꢃꢂ $ꢎ
Checksum and length: 4C14 105.0
This routine completes inverse by dividing by
determinant.
ꢛꢕꢔ ꢂꢌꢂ ꢛꢎ
Divides element.
Decrements index value so it points closer to A.
Loops for next value.
ꢛꢕꢏ !ꢑª1L2ꢎ
ꢛꢕꢖ ꢍ ꢈ Lꢎ
ꢛꢕꢒ ꢆ!ꢑ ꢛꢎ
ꢛꢕꢗ ꢁ!ꢄꢎ
Returns to the calling program or to
.
ꢅꢁꢆꢇ !ꢑꢅ
Checksum and length: 9737 007.5
This routine multiplies a column matrix and a 3
×
ꢇꢕꢔ ꢂꢌꢂ ꢇꢎ
ꢇꢕꢏ ꢘꢎ
3 matrix.
Sets index value to point, to last clement in first
row.
Mathematics Programs 15–15
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
ꢇꢕꢖ %ꢈꢉ ꢄꢎ
Sets index value to point to last element in second
row.
ꢇꢕꢒ ꢙꢎ
ꢇꢕꢗ %ꢈꢉ ꢄꢎ
ꢇꢕ ꢓꢎ
Sets index value to point to last element in third
row.
Checksum and length: C1D3 009.0
This routine calculates product of column vector
ꢄꢕꢔ ꢂꢌꢂ ꢄꢎ
and row pointed to by index value.
Saves index value in i.
ꢄꢕꢏ !ꢑ Lꢎ
ꢄꢕꢖ ꢁꢃꢂ ꢛꢎ
ꢄꢕꢒ ꢁꢃꢂ ꢚꢎ
ꢄꢕꢗ ꢁꢃꢂ ꢂꢎ
ꢄꢕ ꢁꢃꢂº1L2ꢎ
ꢄꢕꢘ %ꢈꢉ ꢅꢎ
ꢄꢕꢙ %ꢈꢉ ꢅꢎ
ꢄꢕꢓ ꢏꢖꢎ
Recalls J from column vector.
Recalls K from column vector.
Recalls L from column vector.
Multiplies by last element in row.
Multiplies by second element in row and adds.
Multiplies by first element in row and adds.
Sets index value to display X, Y, or Z based on
input row.
ꢄꢔꢕ !ꢑ Lꢎ
¶
ꢄꢔꢔ ꢁ ꢎ
Gets result back.
Stores result.
Displays result.
ꢄꢔꢏ !ꢑ1L2ꢎ
ꢄꢔꢖ #ꢊꢈ$1L2ꢎ
ꢄꢔꢒ ꢁ!ꢄꢎ
Returns to the calling program or to
.
ꢅꢁꢆꢇ !ꢑꢅ
Checksum and length: 4E9D 021.0
This routine multiples and adds values within a
row.
ꢅꢕꢔ ꢂꢌꢂ ꢅꢎ
Gets next column value.
Sets index value to point to next row value.
ꢅꢕꢏ º65¸ꢎ
ꢅꢕꢖ ꢍ ꢈ Lꢎ
ꢅꢕꢒ ꢍ ꢈ Lꢎ
ꢅꢕꢗ ꢍ ꢈ Lꢎ
ꢅꢕ ꢁꢃꢂº1L2ꢎ
ꢅꢕꢘ -ꢎ
Multiples column value by row value.
Adds product to previous sum.
Returns to the calling program.
ꢅꢕꢙ ꢁ!ꢄꢎ
15–16 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
Checksum and length: 4E79 012.0
This routine calculates the determinant.
ꢍꢕꢔ ꢂꢌꢂ ꢍꢎ
ꢍꢕꢏ ꢁꢃꢂ ꢀꢎ
ꢍꢕꢖ ꢁꢃꢂº ꢈꢎ
ꢍꢕꢒ ꢁꢃꢂº ꢊꢎ
ꢍꢕꢗ ꢁꢃꢂ ꢍꢎ
ꢍꢕ ꢁꢃꢂº ꢐꢎ
ꢍꢕꢘ ꢁꢃꢂº ꢃꢎ
ꢍꢕꢙ -ꢎ
Calculates A E I.
×
×
Calculates (A × E × I) + (D × H × C).
ꢍꢕꢓ ꢁꢃꢂ ꢆꢎ
ꢍꢔꢕ ꢁꢃꢂº ꢋꢎ
ꢍꢔꢔ ꢁꢃꢂº ꢌꢎ
ꢍꢔꢏ -ꢎ
Calculates (A E I) + (D H C) + (G F B).
×
×
×
×
×
×
ꢍꢔꢖ ꢁꢃꢂ ꢆꢎ
ꢍꢔꢒ ꢁꢃꢂº ꢈꢎ
ꢍꢔꢗ ꢁꢃꢂº ꢃꢎ
ꢍꢔ .ꢎ
(A × E × I) + (D × H × C) + (G × F × B) – (G × E ×
C).
ꢍꢔꢘ ꢁꢃꢂ ꢀꢎ
ꢍꢔꢙ ꢁꢃꢂº ꢋꢎ
ꢍꢔꢓ ꢁꢃꢂº ꢐꢎ
ꢍꢏꢕ .ꢎ
(A × E × I) + (D × H × C) + (G × F × B) –(G × E ×
C) – (A F H).
×
×
ꢍꢏꢔ ꢁꢃꢂ ꢍꢎ
ꢍꢏꢏ ꢁꢃꢂº ꢌꢎ
ꢍꢏꢖ ꢁꢃꢂº ꢊꢎ
ꢍꢏꢒ .ꢎ
(A × E × I) + (D × H × C) + (G × F × B) – (G × E ×
B) – (A F H) – (D B I).
×
×
×
×
Returns to the calling program or to
.
ꢍꢏꢗ ꢁ!ꢄꢎ
ꢅꢁꢆꢇ !ꢑꢅ
Checksum and length: 44B2 037.5
Mathematics Programs 15–17
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Flags Used:
None.
Memory Required:
348 bytes: 212 for program, 136 for variables.
Program Instructions:
1. Key in the program routines; press
2. Press W A to input coefficients of matrix and column vector.
when done.
†
3. Key in coefficient or vector value (A through L) at each prompt and press
.
f
4. Optional: press
D to compute determinant of 3 3 system.
W
×
5. Press
6. Optional: press W A and repeatedly press f to review the values of
I to compute inverse of 3 × 3 matrix.
W
the inverted matrix.
7. Press W M to multiply the inverted matrix by the column vector and to
see the value of X . Press
to see the value of Z.
to see the value of Y, then press
again
f
f
8. For a new case, go back to step 2.
Variables Used:
A through I
J through L
W
X through Z
i
Coefficients of matrix.
Column vector values.
Scratch variable used to store the determinant.
Output vector values; also used for scratch.
Loop–control value (index variable); also used for
scratch.
Remarks:
For 2 × 2 solutions use zero for coefficients C, F, H, G and for L. Use 1 for
coefficient I.
Not all systems of equations have solutions.
15–18 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Example:
For the system below, compute the inverse and the system solution. Review the
inverted matrix. Invert the matrix again and review the result to make sure that
the original matrix is returned.
23X + 15Y + 17Z = 31
8X + 11Y – 6Z = 17
4X + 15Y + 12Z = 14
Keys:
Display:
Description:
A
Starts input routine.
W
ꢀ@valueꢎ
ꢌ@valueꢎ
23
Sets first coefficient, A, equal to
23.
f
8
4
Sets B equal to 8.
f
ꢃ@valueꢎ
ꢍ@valueꢎ
ꢈ@valueꢎ
.
.
.
Sets C equal to 4.
f
15
.
.
.
Sets D equal to 15.
Continues entry for E through L.
f
14
Returns to first coefficient entered.
Calculates the inverse and displays
the determinant.
f
ꢀ@ꢏꢖ)ꢕꢕꢕꢕꢎ
I
W
ꢒ8ꢗꢓꢙ)ꢕꢕꢕꢕꢎ
M
A
Multiplies by column vector to
compute X.
W
%/ꢕ)ꢓꢖꢕ ꢎ
Calculates and displays Y.
Calculates and displays Z.
Begins review of the inverted
matrix.
f
f
W
¸/ꢕ)ꢘꢓꢒꢖꢎ
'/.ꢕ)ꢔꢖ ꢒꢎ
ꢀ@ꢕ)ꢕꢒꢙꢖꢎ
f
f
f
f
f
Displays next value.
ꢌ@.ꢕ)ꢕꢏ ꢔꢎ
ꢃ@ꢕ)ꢕꢔ ꢗꢎ
ꢍ@ꢕ)ꢕꢔ ꢖꢎ
ꢈ@ꢕ)ꢕꢒꢗꢏꢎ
ꢋ@.ꢕ)ꢕ ꢏꢕꢎ
Displays next value.
Displays next value.
Displays next value.
Displays next value.
Mathematics Programs 15–19
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Displays next value.
Displays next value.
Displays next value.
Inverts inverse to produce original
matrix.
f
ꢆ@.ꢕ)ꢕ ꢕꢏꢎ
ꢐ@ꢕ)ꢕꢗꢓ ꢎ
ꢊ@ꢕ)ꢕꢏꢙꢓꢎ
ꢕ)ꢕꢕꢕꢏꢎ
f
f
W I
A
Begins review of inverted matrix.
Displays next value, ...... and so
on.
W
f
ꢀ@ꢏꢖ)ꢕꢕꢕꢕꢎ
ꢌ@ꢙ)ꢕꢕꢕꢕꢎ
.
.
.
.
.
.
Polynomial Root Finder
This program finds the roots of a polynomial of order 2 through 5 with real
coefficients. It calculates both real and complex roots.
For this program, a general polynomial has the form
n
n–1
x + a x
+ ... + a x + a = 0
n–1
1
0
where n = 2, 3, 4, or 5. The coefficient of the highest–order term (a ) is
n
assumed to be 1. If the leading coefficient is not 1, you should make it I by
dividing all the coefficients in the equation by the leading coefficient. (See
example 2.)
The routines for third– and fifth–order polynomials use SOLVE to find one real
root of the equation, since every odd–order polynomial must have at least one
real root. After one root is found, synthetic division is performed to reduce the
original polynomial to a second– or fourth–order polynomial.
To solve a fourth–order polynomial, it is first necessary to solve the resolvant
cubic polynomial:
3
2
y + b y + b y + b = 0
2
1
0
where b = – a
2
2
b = a a – 4a
0
1
3 1
15–20 Mathematics Programs
File name 32sii-Manual-E-0424
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Size : 17.7 x 25.2 cm
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2
2
b = a (4a – a ) – a .
0
0
2
3
1
Let y be the largest real root of the above cubic. Then the fourth–order
0
polynomial is reduced to two quadratic polynomials:
2
x + (J + L)× + (K + M) = 0
2
x + (J – L)x + (K – M) = 0
where J = a /2
3
K = y /2
0
J2 − a2 + y0
L =
× (the sign of JK – a /2)
1
K2 − a2
M =
Roots of the fourth degree polynomial are found by solving these two
quadratic polynomials.
2
A quadratic equation x + a x + a = 0 is solved by the formula
1
0
a1
2
a1
2
x1,2 = −
( )2 − a0
2
If the discriminant d = (a /2) – a ≥ 0, the roots are real; if d < 0, the roots
1
o
u iv = −(a 2) i − d
are complex, being
.
1
Program Listing:
Program Lines:
Description
Defines the beginning of the polynomial root finder
routine.
ꢅꢕꢔ ꢂꢌꢂ ꢅꢎ
Prompts for and stores the order of the polynomial.
ꢅꢕꢏ ꢊꢄꢅ"! ꢋꢎ
Uses order as loop counter.
ꢅꢕꢖ !ꢑ Lꢎ
Checksum and length: 699F 004.5
Starts prompting routine.
ꢊꢕꢔ ꢂꢌꢂ ꢊꢎ
Prompts for a coefficient.
ꢊꢕꢏ ꢊꢄꢅ"!1L2ꢎ
Counts down the input loop.
Repeats until done.
ꢊꢕꢖ ꢍ ꢈ Lꢎ
ꢊꢕꢒ ꢆ!ꢑ ꢊꢎ
ꢊꢕꢗ ꢁꢃꢂ ꢋꢎ
ꢊꢕ !ꢑ Lꢎ
Uses order to select root finding routine.
Mathematics Programs 15–21
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
Download from Www.Somanuals.com. All Manuals Search And Download.
Program Lines:
Description
Starts root finding routine.
ꢊꢕꢘ ꢆ!ꢑ1L2ꢎ
Checksum and length: CE86 010.5
Evaluates polynomials using Horner's method, and
synthetically reduces the order of the polynomial using
the root.
ꢐꢕꢔ ꢂꢌꢂ ꢐꢎ
ꢐꢕꢏ ꢁꢃꢂ ꢐꢎ
ꢐꢕꢖ !ꢑ Lꢎ
ꢐꢕꢒ ꢔꢎ
Uses pointer to polynomial as index.
Starting value for Horner's method.
Checksum and length: B85F 006.0
Starts the Horner's method loop.
ꢛꢕꢔ ꢂꢌꢂ ꢛꢎ
ꢛꢕꢏ ꢈꢄ!ꢈꢁꢎ
ꢛꢕꢖ ꢁꢃꢂº %ꢎ
Saves synthetic division coefficient.
Multiplies current sum by next power of x.
Adds new coefficient.
ꢛꢕꢒ ꢁꢃꢂ-1L2ꢎ
Counts down the loop.
Repeats until done.
ꢛꢕꢗ ꢍ ꢈ Lꢎ
ꢛꢕ ꢆ!ꢑ ꢛꢎ
ꢛꢕꢘ ꢁ!ꢄꢎ
Checksum and length: 139C 010.5
Starts solver setup routine.
Stores location of coefficients to use.
ꢕꢔ ꢂꢌꢂ ꢎ
ꢕꢏ !ꢑ ꢐꢎ
ꢕꢖ ꢏꢗꢕꢎ
ꢕꢒ !ꢑ %ꢎ
ꢕꢗ -+.ꢎ
ꢕ ꢋꢄ/ ꢐꢎ
First initial guess.
Second initial guess.
Specifies routine to solve.
Solves for a real root.
ꢕꢘ ꢑꢂ#ꢈ %ꢎ
Gets synthetic division coefficients for next lower order
polynomial.
ꢕꢙ ꢆ!ꢑ ꢐꢎ
ꢕꢓ ꢕꢎ
ꢔꢕ ªꢎ
Generates DIVIDE BY 0 error if no real root found.
Checksum and length: 27C3 015.0
Starts quadratic solution routine.
ꢉꢕꢔ ꢂꢌꢂ ꢉꢎ
ꢉꢕꢏ º65¸ꢎ
Exchanges a and a .
0
1
15–22 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
ꢉꢕꢖ ꢏꢎ
a /2.
1
ꢉꢕꢒ ªꢎ
–a /2.
ꢉꢕꢗ -+.ꢎ
ꢉꢕ ꢈꢄ!ꢈꢁꢎ
ꢉꢕꢘ ꢈꢄ!ꢈꢁꢎ
ꢉꢕꢙ !ꢑ ꢋꢎ
ꢉꢕꢓ ºꢏꢎ
1
Saves – a /2.
1
Stores real part if complex root.
2
(a /2) .
1
µ
ꢉꢔꢕ ꢁ ꢎ
a .
0
2
(a /2) – a .
ꢉꢔꢔ .ꢎ
1
o
Initializes flag 0.
Discriminant (d) < 0
Sets flag 0 if d < 0 (complex roots).
d
ꢉꢔꢏ ꢃꢋ ꢕꢎ
ꢉꢔꢖ º6ꢕ@ꢎ
ꢉꢔꢒ ꢋ ꢕꢎ
ꢉꢔꢗ ꢀꢌ ꢎ
d
ꢉꢔ ꢉꢁ!ꢎ
Stores imaginary part if complex root.
ꢉꢔꢘ !ꢑ ꢆꢎ
ꢉꢔꢙ ꢋ @ꢎ
ꢉꢔꢓ ꢁ!ꢄꢎ
?
Complex roots
Returns if complex roots.
d
ꢉꢏꢕ !ꢑ. ꢋꢎCalculates – a /2 –
1
¶
ꢉꢏꢔ ꢁ ꢎ
d
ꢉꢏꢏ !ꢑ- ꢆꢎCalculates – a /2 +
1
ꢉꢏꢖ ꢁ!ꢄꢎ
Checksum and length= E454 034.5
Starts second–order solution routine.
Gets L.
Gets M.
ꢌꢕꢔ ꢂꢌꢂ ꢌꢎ
ꢌꢕꢏ ꢁꢃꢂ ꢌꢎ
ꢌꢕꢖ ꢁꢃꢂ ꢀꢎ
ꢌꢕꢒ ꢆ!ꢑ !ꢎ
Calculates and displays two roots.
Checksum and length: 52B9 006.0
Starts third–order solution routine.
ꢃꢕꢔ ꢂꢌꢂ ꢃꢎ
ꢃꢕꢏ ꢖꢎ
Indicates cubic polynomial to be solved.
Solves for one real root and puts a and a for
second–order polynomial on stack.
ꢃꢕꢖ %ꢈꢉ ꢎ
0
1
¶
ꢃꢕꢒ ꢁ ꢎ
Discards polynomial function value.
Mathematics Programs 15–23
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
Solves remaining second–order polynomial and stores
roots.
ꢃꢕꢗ %ꢈꢉ ꢉꢎ
Displays real root of cubic.
Displays remaining roots.
ꢃꢕ #ꢊꢈ$ %ꢎ
ꢃꢕꢘ ꢆ!ꢑ ꢄꢎ
Checksum and length: CCF5 010.5
Starts fifth–order solution routine.
Indicates fifth–order polynomial to be solved.
ꢈꢕꢔ ꢂꢌꢂ ꢈꢎ
ꢈꢕꢏ ꢗꢎ
Solves for one real root and puts three synthetic
division coefficients for fourth–order polynomial on
stack.
ꢈꢕꢖ %ꢈꢉ ꢎ
¶
ꢈꢕꢒ ꢁ ꢎ
Discards polynomial function value.
Stores coefficient.
ꢈꢕꢗ !ꢑ ꢀꢎ
¶
ꢈꢕ ꢁ ꢎ
Stores coefficient.
Stores coefficient.
ꢈꢕꢘ !ꢑ ꢌꢎ
¶
ꢈꢕꢙ ꢁ ꢎ
ꢈꢕꢓ !ꢑ ꢃꢎ
ꢈꢔꢕ ꢁꢃꢂ ꢈꢎ
ꢈꢔꢔ ꢁꢃꢂ- %ꢎ
ꢈꢔꢏ !ꢑ ꢍꢎ
ꢈꢔꢖ #ꢊꢈ$ %ꢎ
Calculates a .
3
Stores a .
3
Displays real root of fifth–order polynomial.
Checksum and length: 0FE9 019.5
Starts fourth–order solution routine.
ꢍꢕꢔ ꢂꢌꢂ ꢍꢎ
ꢍꢕꢏ ꢒꢎ
2
4a .
3
ꢍꢕꢖ ꢁꢃꢂº ꢃꢎ
ꢍꢕꢒ ꢁꢃꢂ ꢍꢎ
ꢍꢕꢗ ºꢏꢎ
a .
2
a .
3
2
4a – a .
ꢍꢕ .ꢎ
2
3
2
a (4a – a ).
ꢍꢕꢘ ꢁꢃꢂº ꢀꢎ
ꢍꢕꢙ ꢁꢃꢂ ꢌꢎ
ꢍꢕꢓ ºꢏꢎ
o
2
3
a .
1
2
a .
1
2
2
b =a (4a – a ) – a .
ꢍꢔꢕ .ꢎ
o
o
0
3
1
Stores b .
a .
2
ꢍꢔꢔ !ꢑ ꢈꢎ
ꢍꢔꢏ ꢁꢃꢂ ꢃꢎ
0
15–24 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
b = –a .
ꢍꢔꢖ -+.ꢎ
ꢍꢔꢒ !ꢑ ꢆꢎ
ꢍꢔꢗ ꢁꢃꢂ ꢍꢎ
ꢍꢔ ꢁꢃꢂº ꢌꢎ
ꢍꢔꢘ ꢒꢎ
2
2
Stores b .
2
a .
3
a a .
3 1
4a .
b = a a – 4a .
1
ꢍꢔꢙ ꢁꢃꢂº ꢀꢎ
ꢍꢔꢓ .ꢎ
0
3 1
0
Stores b .
ꢍꢏꢕ !ꢑ ꢋꢎ
ꢍꢏꢔ ꢒꢎ
1
To enter lines D21 and D22
Press 4 3.
{
ꢍꢏꢏ ꢖꢎ
ꢍꢏꢖ ꢔꢕ%ꢎ
ꢍꢏꢒ ªꢎ
ꢍꢏꢗ ꢘꢎ
Creates 7.004 as a pointer to the cubic coefficients.
ꢍꢏ -ꢎ
Solves for real root and puts a and a for
ꢍꢏꢘ %ꢈꢉ ꢎ
0
1
second–order polynomial on stack.
¶
ꢍꢏꢙ ꢁ ꢎ
Discards polynomial function value.
Solves for remaining roots of cubic and stores roots.
Gets real root of cubic.
Stores real root.
ꢍꢏꢓ %ꢈꢉ ꢉꢎ
ꢍꢖꢕ ꢁꢃꢂ %ꢎ
ꢍꢖꢔ !ꢑ ꢈꢎ
ꢍꢖꢏ ꢋ @ ꢕꢎ
ꢍꢖꢖ ꢆ!ꢑ ꢋꢎ
?
Complex roots
Calculate four roots of remaining fourth–order
polynomial.
If not complex roots, determine largest real root (y )
ꢍꢖꢒ ꢁꢃꢂ ꢋꢎ
ꢍꢖꢗ º6¸@ꢎ
ꢍꢖ º65¸ꢎ
ꢍꢖꢘ ꢁꢃꢂ ꢆꢎ
ꢍꢖꢙ º6¸@ꢎ
ꢍꢖꢓ º65¸ꢎ
ꢍꢒꢕ !ꢑ ꢈꢎ
0
Stores largest real root of cubic.
Checksum and length: C333 060.0
Starts fourth–order solution routine.
ꢋꢕꢔ ꢂꢌꢂ ꢋꢎ
ꢋꢕꢏ ꢏꢎ
J = a /2
ꢋꢕꢖ !ꢑª ꢍꢎ
3
Mathematics Programs 15–25
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
K = y /2
ꢋꢕꢒ !ꢑª ꢈꢎ
0
ꢋꢕꢗ ꢓꢎ
ꢋꢕ ꢔꢕ%ꢎ
ꢋꢕꢘ ꢔ+ºꢎ
ꢋꢕꢙ ꢁꢃꢂ ꢈꢎ
ꢋꢕꢓ ºꢏꢎ
–9
2
Creates 10 as a lower bound for M
K
K .
2
2
2
M = K –a .
ꢋꢔꢕ ꢁꢃꢂ. ꢀꢎ
ꢋꢔꢔ º6¸@ꢎ
ꢋꢔꢏ ꢃꢂºꢎ
ꢋꢔꢖ ꢉꢁ!ꢎ M = K2 − a0
0
2
–9
2
If M < 10 , use 0 for M .
Stores M.
ꢋꢔꢒ !ꢑ ꢀꢎ
J.
ꢋꢔꢗ ꢁꢃꢂ ꢍꢎ
JK.
ꢋꢔ ꢁꢃꢂº ꢈꢎ
a .
1
ꢋꢔꢘ ꢁꢃꢂ ꢌꢎ
ꢋꢔꢙ ꢏꢎ
a /2
1
ꢋꢔꢓ ªꢎ
JK – a /2.
ꢋꢏꢕ .ꢎ
1
ꢋꢏꢔ º/ꢕ@ꢎ
ꢋꢏꢏ ꢔꢎ
Use 1 if JK – a /2 = 0
1
Stores 1 or JK – a /2.
ꢋꢏꢖ !ꢑ ꢌꢎ
ꢋꢏꢒ ꢀꢌ ꢎ
1
Calculates sign of C.
J.
ꢋꢏꢗ !ꢑª ꢌꢎ
ꢋꢏ ꢁꢃꢂ ꢍꢎ
ꢋꢏꢘ ºꢏꢎ
2
J
2
J -– a .
ꢋꢏꢙ ꢁꢃꢂ. ꢃꢎ
ꢋꢏꢓ ꢁꢃꢂ- ꢈꢎ
ꢋꢖꢕ ꢁꢃꢂ- ꢈꢎ
ꢋꢖꢔ ꢉꢁ!ꢎ C =
2
2
J -– a +y .
2
0
J2 − a2 + y0
.
Stores C with proper sign.
J.
J + L.
K.
K + M.
ꢋꢖꢏ !ꢑº ꢌꢎ
ꢋꢖꢖ ꢁꢃꢂ ꢍꢎ
ꢋꢖꢒ ꢁꢃꢂ- ꢌꢎ
ꢋꢖꢗ ꢁꢃꢂ ꢈꢎ
ꢋꢖ ꢁꢃꢂ- ꢀꢎ
ꢋꢖꢘ %ꢈꢉ !ꢎ
Calculate and display two roots of the fourth–order
15–26 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
polynomial.
J.
J – L.
K.
ꢋꢖꢙ ꢁꢃꢂ ꢍꢎ
ꢋꢖꢓ ꢁꢃꢂ. ꢌꢎ
ꢋꢒꢕ ꢁꢃꢂ ꢈꢎ
ꢋꢒꢔ ꢁꢃꢂ. ꢀꢎ
K – M.
Checksum and length: 9133 061.5
Starts routine to calculate and display two roots.
Uses quadratic routine to calculate two roots.
!ꢕꢔ ꢂꢌꢂ !ꢎ
!ꢕꢏ %ꢈꢉ ꢉꢎ
Checksum and length: 0019 003.0
Starts routine to display two real roots or two roots.
Gets the first real root.
Stores the first real root.
Displays real root or real part of complex root.
Gets the second real root or imaginary part of
complex root.
ꢄꢕꢔ ꢂꢌꢂ ꢄꢎ
ꢄꢕꢏ ꢁꢃꢂ ꢋꢎ
ꢄꢕꢖ !ꢑ %ꢎ
ꢄꢕꢒ #ꢊꢈ$ %ꢎ
ꢄꢕꢗ ꢁꢃꢂ ꢆꢎ
?
Were there any complex roots
Displays complex roots if any.
Stores second real root.
Displays second real root.
Returns to calling routine.
ꢄꢕ ꢋ @ ꢕꢎ
ꢄꢕꢘ ꢆ!ꢑ "ꢎ
ꢄꢕꢙ !ꢑ %ꢎ
ꢄꢕꢓ #ꢊꢈ$ %ꢎ
ꢄꢔꢕ ꢁ!ꢄꢎ
Checksum and length: BE87 015.0
Starts routine to display complex roots.
"ꢕꢔ ꢂꢌꢂ "ꢎ
"ꢕꢏ !ꢑ Lꢎ
"ꢕꢖ #ꢊꢈ$ Lꢎ
"ꢕꢒ #ꢊꢈ$ %ꢎ
"ꢕꢗ ꢁꢃꢂ Lꢎ
"ꢕ -+.ꢎ
Stores the imaginary part of the first complex root.
Displays the imaginary part of the first complex root.
Displays the real part of the second complex root.
Gets the imaginary part of the complex roots.
Generates the imaginary part of the second complex
root.
Stores the imaginary part of the second complex root.
Displays the imaginary part of the second complex
root.
"ꢕꢘ !ꢑ Lꢎ
"ꢕꢙ #ꢊꢈ$ Lꢎ
Checksum and length: OEE4 012.0
Mathematics Programs 15–27
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Flags Used:
Flag 0 is used to remember if the root is real or complex (that is, to remember
the sign of d). If d is negative, then flag 0 is set. Flag 0 is tested later in the
program to assure that both the real and imaginary parts are displayed if
necessary.
Memory Required:
382.0 bytes: 268.5 for programs, 33.5 for SOLVE, 80 for variables.
Remarks:
The program accommodates polynomials of order 2, 3, 4, and 5. It does not
check if the order you enter is valid.
The program requires that the constant term a is nonzero for these
0
polynomials. (If a is 0, then 0 is a real root. Reduce the polynomial by one
0
order by factoring out x.)
The order and the coefficients are not preserved by the program.
Because of round–off error in numerical computations, the program may
produce values that are not true roots of the polynomial. The only way to
confirm the roots is to evaluate the polynomial manually to see if it is zero at
the roots.
For a third– or higher–order polynomial, if SOLVE cannot find a real root, the
error
is displayed.
ꢍꢊ#ꢊꢍꢈ ꢌ& ꢕ
You can save time and memory by omitting routines you don't need. If you're
not solving fifth–order polynomials, you can omit routine E. If you're not
solving fourth– or fifth–order polynomials, yoga can omit routines D, E, and F.
If you're not solving third–, fourth–, or fifth–order polynomials, you can omit
routines C, D, E, and F.
Program Instructions:
1. Press
{
} to clear all programs and variables. This
z b
ꢀꢂꢂ
program requires all but 2 bytes of memory while running.
15–28 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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2. Key in the program routines; press
when done.
†
3. Press
P to start the polynomial root finder.
W
4. Key in F, the order of the polynomial, and press
f
5. At each prompt, key in the coefficient and press
. You're not
f
prompted for the highest–order coefficient — it's assumed to be 1. You
must enter 0 for coefficients that are 0. Coefficient A must not be 0.
Terms mid Coefficients
5
4
3
2
Order
x
1
x
x
x
x
B
B
B
B
Constant
5
4
3
2
E
1
D
D
1
C
A
A
A
A
C
C
1
6. After you enter the coefficients, the first root is calculated. A real root is
displayed as real value. A complex root is displayed as real part,
%/
%/
(Complex roots always occur in pairs of the for u i v, and are labeled in
the output as
next step.)
real part and i =imaginary part, which you'll see in the
%/
7. Press
repeatedly to see the other roots, or to see i = imaginary part,
f
the imaginary part of a complex root. The order of the polynomial is same
as the number of roots you get.
8. For a new polynomial, go to step 3.
A through E Coefficients of tints of polynomial; scratch.
F
Order of polynomial; scratch.
Scratch.
Pointer to polynomial coefficients.
The value f a real root, or the real part of complex root
The imaginary part of a complex root; also used as are
index variable.
G
H
X
i
Mathematics Programs 15–29
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Example 1:
Find the roots of x – x – 101x +101x + 100x – 100 = 0.
5
4
3
2
Keys:
Display:
Description:
P
Starts the polynomial root finder;
prompts for order.
W
ꢋ@valueꢎ
5
Stores 5 its F; prompts for E.
Stores –1 in E; prompts for D.
Store –101 in D. prompts for C.
Stores 101 in C; prompts for B.
Stores 100 in B; prompts for A.
Stores –100 in A; calculates the
first root.
f
ꢈ@valueꢎ
ꢍ@value
1
_ f
101 f
ꢃ@value
101
f
ꢌ@value
100
f
ꢀ@valueꢎ
%/ꢔ)ꢕꢕꢕꢕꢎ
100
_ f
Calculates the second root.
Displays the third root.
f
f
f
f
%/ꢔ)ꢕꢕꢕꢕꢎ
%/ꢔꢕ)ꢕꢕꢕꢕꢎ
%/.ꢔꢕ)ꢕꢕꢕꢕꢎ
L/.ꢔ)ꢕꢕꢕꢕꢎ
Displays the fourth root.
Displays the fifth root.
Example 2:
4
3
2
Find the roots of 4x – 8x – 13x – 10x + 22 = 0. Because the coefficient of
the highest–order term must be 1, divide that coefficient into each of the other
coefficients.
Keys:
Display:
Description:
P
Starts the polynomial root finder;
prompts for order.
W
ꢋ@valueꢎ
4
Stores 4 its F; prompts for D.
Stores –8/4 in D; prompts for C.
f
ꢍ@valueꢎ
ꢎ
8
p f
4
_ š
ꢃ@value
ꢎ
13
Store –13/4 in C. prompts for B.
_ š
4
p f
ꢌ@value
15–30 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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22
4
Stores –10/4 in B; prompts for A.
Stores 22/4 in A; calculates the
first root.
š
ꢀ@value
p f
%/ꢕ)ꢙꢙꢏꢕꢎ
Calculates the second root.
Displays the real part of the third
root.
f
%/ꢖ)ꢔꢔꢙꢕꢎ
%/.ꢔ)ꢕꢕꢕꢕꢎ
f
f
f
f
Displays the imaginary part of the
third root.
%/ꢔ)ꢕꢕꢕꢕꢎ
%/.ꢔ)ꢕꢕꢕꢕꢎ
L/.ꢔ)ꢕꢕꢕꢕꢎ
Displays the real part of the fourth
root.
Displays the imaginary part of the
fourth root.
The third and fourth roots are –1.00 1.00 i.
Example 3:
Find the roots of the following quadratic polynomial:
2
x + x – 6 = 0
Keys:
Display:
Description:
P
Starts the polynomial root finder;
prompts for order.
W
ꢋ@valueꢎ
2
Stores 2 its F; prompts for B.
Stores 4 its B; prompts for A.
Stores –6 its A; calculates the first
root.
f
ꢋ@valueꢎ
1
6
f
ꢋ@valueꢎ
_f
%/.ꢖ)ꢕꢕꢕꢕꢎ
Calculates the second root.
f
%/ꢏ)ꢕꢕꢕꢕꢎ
Coordinate Transformations
This program provides two–dimensional coordinate translation and rotation.
Mathematics Programs 15–31
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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The following formulas are used to convert a point P from the Cartesian
coordinate pair (x, y) in the old system to the pair (u, v) in the new, translated,
rotated system.
u = (x – m) cosθ + (y – n) sinθ
v = (y – n) cos – (y – n) sin
θ
θ
The inverse transformation is accomplished with the formulas below.
x = u cos – v sin + m
θ
θ
y = u sinθ + v cosθ + n
The HP 32SII complex and polar–to–rectangular functions make these
computations straightforward.
15–32 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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y
y'
x
u
Old coordinate
system
P
x'
v
y
[0, 0]
x
θ
[m, n]
New coordinate
system
Program Listing:
Program Lines:
Description
This routine defines the new coordinate system.
ꢍꢕꢔ ꢂꢌꢂ ꢍꢎ
Prompts for and stores M, the new origin's
x–coordinate.
ꢍꢕꢏ ꢊꢄꢅ"! ꢇꢎ
Prompts for and stores N, the new origin's
y–coordinate.
ꢍꢕꢖ ꢊꢄꢅ"! ꢄꢎ
Prompts for and stores T, the angle θ.
ꢍꢕꢒ ꢊꢄꢅ"! !ꢎ
Loops for review of inputs.
ꢍꢕꢗ ꢆ!ꢑ ꢍꢎ
Checksum and length: 2ED3 007.5
This routine converts from the old system to the new
system.
ꢄꢕꢔ ꢂꢌꢂ ꢄꢎ
Mathematics Programs 15–33
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
Prompts for and stores X, the old x–coordinate.
Prompts for and stores Y, the old y–coordinate.
Pushes Y up and recalls X to the X–register.
Pushes X and Y up and recalls N to the X–register.
Pushes N, X, and Y up and recalls M.
ꢄꢕꢏ ꢊꢄꢅ"! %ꢎ
ꢄꢕꢖ ꢊꢄꢅ"! &ꢎ
ꢄꢕꢒ ꢁꢃꢂ %ꢎ
ꢄꢕꢗ ꢁꢃꢂ ꢄꢎ
ꢄꢕ ꢁꢃꢂ ꢇꢎ
ꢄꢕꢘ ꢃꢇꢅꢂ%.ꢎ
ꢄꢕꢙ ꢁꢃꢂ !ꢎ
ꢄꢕꢓ -+.ꢎ
Calculates (X – M) and (Y – N).
Pushes (X – M) and (Y – N) up and recalls T.
Charges the sign of T because sin(–T) equals –sin(T).
Sets radius to 1 for computation of cos(T) and –sin(T).
Calculates cost (T) and –sin(T) in X– and Y–registers.
ꢄꢔꢕ ꢔꢎ
´
θ
ꢄꢔꢔ 8T ¸8ºꢎ
Calculates (X – M) cos (T) + (Y–N) sin (T) and (Y – N)
ꢄꢔꢏ ꢃꢇꢅꢂ%ºꢎ
cos (T) – (X – M) sin(T).
Stores x–coordinate in variable U.
Swaps positions of the coordinates.
Stores y–coordinate in variable V.
Swaps positions of coordinates back.
Halts program to display U.
ꢄꢔꢖ !ꢑ "ꢎ
ꢄꢔꢒ º65¸ꢎ
ꢄꢔꢗ !ꢑ "ꢎ
ꢄꢔ º65¸ꢎ
ꢄꢔꢘ #ꢊꢈ$ "ꢎ
ꢄꢔꢙ #ꢊꢈ$ #ꢎ
ꢄꢔꢓ ꢆ!ꢑ ꢄꢎ
Halts program to display V.
Goes back for another calculation.
Checksum and length: 3A46 028.5
This routine converts from the new system to the old
system.
ꢑꢕꢔ ꢂꢌꢂ ꢑꢎ
Prompts for and stores U.
Prompts for and stores V.
Pushes V up and recalls U.
Pushes U and V up and recalls T.
Sets radius to 1 for the computation of sin(T) and
cos(T).
ꢑꢕꢏ ꢊꢄꢅ"! "ꢎ
ꢑꢕꢖ ꢊꢄꢅ"! #ꢎ
ꢑꢕꢒ ꢁꢃꢂ "ꢎ
ꢑꢕꢗ ꢁꢃꢂ !ꢎ
ꢑꢕ ꢔꢎ
´
ꢑꢕꢘ 8T ¸8ºꢎ
θ
Calculates cos(T) and sin(T).
Calculates U cos(T) V sin(T) and U sin(T) + V cos(T).
Pushes up previous results and recalls N.
Pushes up results and recalls M.
Completes calculation by adding M and N to
previous results.
ꢑꢕꢙ ꢃꢇꢅꢂ%ºꢎ
ꢑꢕꢓ ꢁꢃꢂ ꢄꢎ
ꢑꢔꢕ ꢁꢃꢂ ꢇꢎ
ꢑꢔꢔ ꢃꢇꢅꢂ%-ꢎ
15–34 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
Stores the x–coordinate in variable X.
Swaps the positions of the coordinates.
Stores the y–coordinate in variable Y.
Swaps the positions of the coordinates back.
Halts the program to display X.
ꢑꢔꢏ !ꢑ %ꢎ
ꢑꢔꢖ º65¸ꢎ
ꢑꢔꢒ !ꢑ &ꢎ
ꢑꢔꢗ º65¸ꢎ
ꢑꢔ #ꢊꢈ$ %ꢎ
ꢑꢔꢘ #ꢊꢈ$ &ꢎ
ꢑꢔꢙ ꢆ!ꢑ ꢑꢎ
Halts the program to display Y.
Goes back for another calculation.
Checksum and length: 7C14 027.0
Flags Used:
None.
Memory Required:
119 bytes: 63 for program, 56 for variables.
Program Instructions:
1. Key in the program routines; press
2. Press W D to start the prompt sequence which defines the coordinate
when done.
†
transformation.
3. Key in the x–coordinate of the origin of the new system M and press f.
4. Key in the y–coordinate of the origin of the new system N and press
f
5. Key in the rotation angle T and press
.
f
6. To translate from the old system to the new system, continue with step 7. To
translate from the new system to the old system, skip to step 12.
7. Press
N to start the old–to–new transformation routine.
W
8. Key in X and press
9. Key in Y, press
.
f
, and see the x–coordinate, U, in the new system.
and see the y–coordinate, V, in the new system.
f
10.Press
f
11. For another old–to–new transformation, press
and go to step 8. For a
f
new–to–old transformation, continue with step 12.
12.Press O to start the new–to–old transformation routine.
W
Mathematics Programs 15–35
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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13. Key in U (the x–coordinate in the new system) and press
14.Key in V (the y–coordinate in the new system) and press
.
f
to see X.
f
15. Press
to see Y.
f
16. For another new–to–old transformation, press
and go to step 13. For
f
an old–to–new transformation, go to step 7.
Variables Used:
M
N
T
X
Y
The x–coordinate of the origin of the new system.
The y–coordinate of the origin of the new system.
The rotation angle, θ, between the old and new systems.
The x–coordinate f a point in the old system.
The y–coordinate of a point in the old system.
The x–coordinate of a point in the new system.
The y–coordinate of a point in the new system.
U
V
Remark:
For translation only, key in zero for T. For rotation only, key in zero for M and
N.
Example:
For the coordinate stems shorn below, convert points P , P and P ,which are
1
2
3
currently in the (X, Y) system, to points in the (X', Y') system. Convert point P' ,
4
which is lid the (X',Y') system, to the (X,Y) system.
15–36 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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y
y'
P
(6, 8)
3
_
P
( 9, 7)
1
x
_ _
( 5, 4)
2
T
P
(M, N)
_
P' (2.7, 3.6)
4
_
(M, N) = (7, 4)
T = 27 o
Keys:
Display:
Description:
{ }
ꢍꢆ ꢎ
Sets Degrees mode since T is given
in degrees.
z Ÿ
D
Starts the routine that defines the
transformation.
W
ꢇ@valueꢎ
7
Store 7 in M.
f
ꢄ@value
4
Store –4 in N.
_ f
!@value
27
Stores 27 in T.
f
ꢇ@ꢘ)ꢕꢕꢕꢕꢎ
N
Starts the old–to–new routine.
W
%@valueꢎ
Mathematics Programs 15–37
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
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9
7
Stores –9 in X.
_ f
f
&@valueꢎ
Stores 7 in Y and calculates U.
Calculates V.
"/.ꢓ)ꢏ ꢏꢏꢎ
#/ꢔꢘ)ꢕ ꢒꢓꢎ
%@.ꢓ)ꢕꢕꢕꢕꢎ
f
f
Resumes the old–to–new routine
for next problem.
5
Stores –5 in X.
_ f
&@ꢘ)ꢕꢕꢕꢕꢎ
"/.ꢔꢕ) ꢓꢏꢔꢎ
#/ꢗ)ꢒꢒꢘꢓꢎ
%@.ꢗ)ꢕꢕꢕꢎ
4
Stores –4 in Y.
_ f
Calculates V.
f
f
Resumes the old–to–new routine
for next problem.
6
Stores 6 in X .
f
&@.ꢒ)ꢕꢕꢕꢎ
"/ꢒ)ꢗꢗ ꢓꢎ
#/ꢔꢔ)ꢔꢒ ꢔꢎ
"@ꢒ)ꢗꢗ ꢓꢎ
#@ꢔꢔ)ꢔꢒ ꢔꢎ
%/ꢔꢔ)ꢕꢒꢕꢔꢎ
&/.ꢗ)ꢓꢙꢔꢙꢎ
8
Stores 8 in Y and calculates U.
Calculates V.
f
f
O
Starts the new–to–old routine.
Stores 2.7 in U.
W
2.7 f
3.6
Stores –3.6 in V and calculates X.
Calculates Y.
_ f
f
15–38 Mathematics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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16
Statistics Programs
Curve Fitting
This program can be used to fit one of four models of equations to your data.
These models are the straight line, the logarithmic curve, the exponential
curve and the power curve. The program accepts two or more (x, y) data
pairs and then calculates the correlation coefficient, r, and the two regression
coefficients, m and b. The program includes a routine to calculate the
ˆ
y
estimates
Regression" in chapter 11.)
and
. (For definitions of these values, see "Linear
ˆ
x
Samples of the curves and the relevant equations are shown below. The
internal regression functions of the HP 32SII are used to compute the
regression coefficients.
Statistics Programs 16–1
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Exponential Curve Fit
Straight Line Fit
E
S
y
y
_
Mx
y = B Mx
y = Be
x
x
Logarithmic Curve Fit
Power Curve Fit
L
P
y
y
M
y = B + MIn x
y = Bx
x
x
To fit logarithmic curves, values of x must be positive. To fit exponential curves,
values of y must be positive. To fit power curves, both x and y must be positive.
A
error will occur if a negative number is entered for these
ꢂꢑꢆ1ꢄꢈꢆ2
cases.
Data values of large magnitude but relatively small differences can incur
problems of precision, as can data values of greatly different magnitudes.
Refer to "Limitations in Precision of Data" in chapter 11.
16–2 Statistics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Listing:
Program Lines:
Description
This routine set, the status for the straight–line model.
ꢕꢔ ꢂꢌꢂ ꢎ
Enters index value for later storage in i (for indirect
addressing).
ꢕꢏ ꢔꢎ
Clears flag 0, the indicator for ln X.
Clears flag 1, the indicator for In Y.
Branches to common entry point Z.
ꢕꢖ ꢃꢋ ꢕꢎ
ꢕꢒ ꢃꢋ ꢔꢎ
ꢕꢗ ꢆ!ꢑ 'ꢎ
Checksum and length: EBD2 007.5
This routine sets the status fog the logarithmic model.
Enters index value for later storage in i (for indirect
addressing).
ꢂꢕꢔ ꢂꢌꢂ ꢂꢎ
ꢂꢕꢏ ꢏꢎ
Sets flag 0, the indicator for ln X.
Clears flag 1, the indicator ln Y
Branches to common entry point Z.
ꢂꢕꢖ ꢋ ꢕꢎ
ꢂꢕꢒ ꢃꢋ ꢔꢎ
ꢂꢕꢗ ꢆ!ꢑ 'ꢎ
Checksum and length: 7462 007.5
This routine sets the status for the exponential model.
Enters index value for later storage in i (for indirect
addressing).
ꢈꢕꢔ ꢂꢌꢂ ꢈꢎ
ꢈꢕꢏ ꢖꢎ
Clears flag 0, the indicator for ln X.
Sets flag 1, the indicator for ln Y
Branches to common entry point Z.
ꢈꢕꢖ ꢃꢋ ꢕꢎ
ꢈꢕꢒ ꢋ ꢔꢎ
ꢈꢕꢗ ꢆ!ꢑ 'ꢎ
Checksum and length: DCEA 007.5
This routine sets the status for the power model.
Enters index value for later storage in i (for indirect
addressing.)
ꢅꢕꢔ ꢂꢌꢂ ꢅꢎ
ꢅꢕꢏ ꢒꢎ
Sets flag 0, the indicator for ln X.
Sets flag 1 the indicator for ln Y.
ꢅꢕꢖ ꢋ ꢕꢎ
ꢅꢕꢒ ꢋ ꢔꢎ
Checksum and length: F399 006.0
Defines common entry point for all models.
Clears the statistics registers.
'ꢕꢔ ꢂꢌꢂ 'ꢎ
'ꢕꢏ ꢃꢂ´ꢎ
Statistics Programs 16–3
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
Stores the index value in i for indirect addressing.
'ꢕꢖ !ꢑ Lꢎ
Sets the loop counter to zero for the first input.
'ꢕꢒ ꢕꢎ
Checksum and length: 8C2F 006.0
Defines the beginning of the input loop.
Adjusts the loop counter by one to prompt for input.
$ꢕꢔ ꢂꢌꢂ $ꢎ
$ꢕꢏ ꢔꢎ
$ꢕꢖ -ꢎ
Stores loop counter in X so that it will appear with the
prompt for X.
$ꢕꢒ !ꢑ %ꢎ
Displays counter with prompt and stores X input.
$ꢕꢗ ꢊꢄꢅ"! %ꢎ
If flag 0 is set . . .
$ꢕ ꢋ @ ꢕꢎ
$ꢕꢘ ꢂꢄꢎ
. . . takes the natural log of the X–input.
Stores that value for the correction routine.
Prompts for and stores Y.
$ꢕꢙ !ꢑ ꢌꢎ
$ꢕꢓ ꢊꢄꢅ"!ꢎ
$ꢔꢕ ꢋ @ ꢔꢎ
$ꢔꢔ ꢂꢄꢎ
If flag 1 is set . . .
. . . takes the natural log of the Y–input.
$ꢔꢏ !ꢑ ꢁꢎ
$ꢔꢖ ꢁꢃꢂ ꢌꢎ
$ꢔꢒ ´-ꢎ
Accumulates B and R as x,y–data pair in statistics
registers.
Loops for another X, Y pair.
$ꢔꢗ ꢆ!ꢑ $ꢎ
Checksum and length: AAD5 022.5
Defines the beginning of the "undo" routine.
Recalls the most recent data pair.
"ꢕꢔ ꢂꢌꢂ "ꢎ
"ꢕꢏ ꢁꢃꢂ ꢁꢎ
"ꢕꢖ ꢁꢃꢂ ꢌꢎ
"ꢕꢒ ´.ꢎ
Deletes this pair from the statistical accumulation.
Loops for another X, Y pair.
"ꢕꢗ ꢆ!ꢑ $ꢎ
Checksum and length: AFAA 007.5
Defines the start f the output routine
Calculates the correlation coefficient.
Stores it in R.
ꢁꢕꢔ ꢂꢌꢂ ꢁꢎ
ꢁꢕꢏ Tꢎ
ꢁꢕꢖ !ꢑ ꢁꢎ
Displays the correlation coefficient.
ꢁꢕꢒ #ꢊꢈ$ ꢁꢎ
Calculates the coefficient b.
ꢁꢕꢗ Eꢎ
16–4 Statistics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
If flag 1 is seta takes the natural antilog of b.
ꢁꢕ ꢋ @ ꢔꢎ
ꢁꢕꢘ H%ꢎ
Stores b in B.
ꢁꢕꢙ !ꢑ ꢌꢎ
Displays value,
ꢁꢕꢓ #ꢊꢈ$ ꢌꢎ
Calculates coefficient m.
ꢁꢔꢕ Pꢎ
Stores m in M.
ꢁꢔꢔ !ꢑ ꢇꢎ
Displays value.
ꢁꢔꢏ #ꢊꢈ$ ꢇꢎ
Checksum aril length: EBF3 018.0
Defines the beginning of the estimation (projection)
loop.
&ꢕꢔ ꢂꢌꢂ &ꢎ
Displays, prompts for, and, if changed, stores x–value
&ꢕꢏ ꢊꢄꢅ"! %ꢎ
in X.
ˆ
y
Calls subroutine to compute
&ꢕꢖ %ꢈꢉ1L2ꢎ
.
ˆ
y
Stores –value in Y.
&ꢕꢒ !ꢑ &ꢎ
Displays, prompts for, and, if changed, stores y–value
&ꢕꢗ ꢊꢄꢅ"! &ꢎ
in Y.
&ꢕ
ꢎ
Adjusts index value to address the appropriate
&ꢕꢘ !ꢑ- Lꢎ
subroutine.
Calls subroutine to compute
.
ˆ
x
&ꢕꢙ %ꢈꢉ1L2ꢎ
Stores
in X for next loop.
ˆ
x
&ꢕꢓ !ꢑ %ꢎ
&ꢔꢕ ꢆ!ꢑ &ꢎ
Loops for another estimate.
Checksum and length: BA07 015.
ˆ
y
This subroutine calculates
model.
for the straight–line
ꢀꢕꢔ ꢂꢌꢂ ꢀꢎ
ꢀꢕꢏ ꢁꢃꢂ ꢇꢎ
ꢀꢕꢖ ꢁꢃꢂº %ꢎ
ˆ
y
Calculates
= MX + B.
ꢀꢕꢒ ꢁꢃꢂ- ꢌꢎ
Returns to the calling routine.
Checksum and length: 2FDA 007.5
ꢀꢕꢗ ꢁ!ꢄꢎ
This subroutine calculates
model.
for the straight–line
ˆ
x
ꢆꢕꢔ ꢂꢌꢂ ꢆꢎ
Statistics Programs 16–5
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
Restores index value to its original value.
ꢆꢕꢏ !ꢑ. Lꢎ
ꢆꢕꢖ ꢁꢃꢂ &ꢎ
ꢆꢕꢒ ꢁꢃꢂ. ꢌꢎ
Calculates
=(Y – B) ÷ M.
ˆ
x
ꢆꢕꢗ ꢁꢃꢂª ꢇꢎ
Returns to the calling routine.
Checksum and length: 0D3F 009.0
ꢆꢕ ꢁ!ꢄꢎ
ˆ
y
This subroutine calculates
model.
for the logarithmic
ꢌꢕꢔ ꢂꢌꢂ ꢌꢎ
ꢌꢕꢏ ꢁꢃꢂ %ꢎ
ꢌꢕꢖ ꢂꢄꢎ
ꢌꢕꢒ ꢁꢃꢂº ꢇꢎ
Calculates
ˆ
y
= M In X + B.
ꢌꢕꢗ ꢁꢃꢂ- ꢌꢎ
Returns to the calling routine.
Checksum and length: 7AB7 009.0
ꢌꢕ ꢁꢃꢂꢎ
This subroutine calculates
for the logarithmic
ˆ
x
ꢐꢕꢔ ꢂꢌꢂ ꢐꢎ
model.
Restores index value to its original value.
ꢐꢕꢏ !ꢑ. Lꢎ
ꢐꢕꢖ ꢁꢃꢂ &ꢎ
ꢐꢕꢒ ꢁꢃꢂ. ꢌꢎ
ꢐꢕꢗ ꢁꢃꢂª ꢇꢎ
(Y – B) ÷ M
ꢐꢕ H%ꢎ
Calculates
= e
ˆ
x
Returns to the calling routine.
ꢐꢕꢘ ꢁ!ꢄꢎ
Checksum and length: B00D 010.5
ˆ
y
This subroutine calculates
model.
for the exponential
ꢃꢕꢔ ꢂꢌꢂ ꢃꢎ
ꢃꢕꢏ ꢁꢃꢂ ꢇꢎ
ꢃꢕꢖ ꢁꢃꢂº %ꢎ
ꢃꢕꢒ H%ꢎ
MX
ˆ
y
Calculates = Be .
ꢃꢕꢗ ꢁꢃꢂº ꢌꢎ
ꢃꢕ ꢁ!ꢄꢎ
Returns to the calling routine.
Checksum and length: AA19 009.0
16–6 Statistics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
This subroutine calculates
for the exponential
ˆ
x
ꢊꢕꢔ ꢂꢌꢂ ꢊꢎ
model.
Restores index value to its original value.
ꢊꢕꢏ !ꢑ. Lꢎ
ꢊꢕꢖ ꢁꢃꢂ &ꢎ
ꢊꢕꢒ ꢁꢃꢂª ꢌꢎ
ꢊꢕꢗ ꢂꢄꢎ
Calculates
Returns to the calling routine.
= (ln (Y ÷ B)) ÷ M.
ˆ
x
ꢊꢕ ꢁꢃꢂª ꢇꢎ
ꢊꢕꢘ ꢁ!ꢄꢎ
Checksum and length: 7D3B 010.5
ˆ
y
This subroutine calculates
for the power model.
ꢍꢕꢔ ꢂꢌꢂ ꢍꢎ
ꢍꢕꢏ ꢁꢃꢂ %ꢎ
ꢍꢕꢖ ꢁꢃꢂ ꢇꢎ
ꢍꢕꢒ ¸%ꢎ
M
Calculates Y= B(X ).
ꢍꢕꢗ ꢁꢃꢂº ꢌꢎ
Returns to the calling routine.
ꢍꢕ ꢁ!ꢄꢎ
Checksum and length: 30CD 009.0
This subroutine calculates
for the power model.
ˆ
x
ꢛꢕꢔ ꢂꢌꢂ ꢛꢎ
Restores index value to its original value.
ꢛꢕꢏ !ꢑ. Lꢎ
ꢛꢕꢖ ꢁꢃꢂ &ꢎ
ꢛꢕꢒ ꢁꢃꢂª ꢌꢎ
ꢛꢕꢗ ꢁꢃꢂ ꢇꢎ
ꢛꢕ ꢔ+ºꢎ
1/M
ꢛꢕꢘ ¸%ꢎ
Calculates
= (Y/B)
ˆ
x
Returns to the calling routine.
ꢛꢕꢙ ꢁ!ꢄꢎ
Checksums and length: 7139 012.0
Flags Used:
Flag 0 is set if a natural log is required of the X input. Flag 1 is set if a natural
log is required of the Y input.
Statistics Programs 16–7
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Memory Required:
270 bytes: 174 for program, 96 for data (statistic. registers 48).
Program instructions:
1. Key in the program routines; press
when done.
†
2. Press
and select the type of curve you wish to fit by pressing:
W
ꢀ
ꢀ
ꢀ
ꢀ
S for a straight line;
L for a logarithmic curvy.;
E for an exponential curve; or
P for a power curve.
3. Key in x–value and press
4. Key in y–value and press
.
.
f
f
5. Repeat steps 3 and 4 for each data pair. If you discover that you have
value
made an error after you have pressed f in step 3 (with the
&@
prompt still visible), press
again (displaying the
value prompt)
f
%@
and press
U to undo (remove) the last data pair. If you discover that
W
you made an error after step 4, press
step 3.
6. After all data are keyed in, press W R to see the correlation coefficient,
U. In either case continue at
W
R.
7. Press
8. Press
9. Press
10.ff you wish to estimate
then press
11. If you wish to estimate
to see the regression coefficient B.
to see the regression coefficient M.
f
f
f
ˆ
y
to see the
value prompt for the
,
–estimation routine.
ˆ
x
%@ˆ
y
based on x, key in x at the value prompt,
%@
ˆ
y
to see
(
).
f
&@
based on y, press
until you see the
).
f
ˆ
x
value prompt, key in y, then press
to see
(
f
ˆ
x %@
&@
12. For more estimations, go to step 10 or 11.
13. For a new case, go to step 2.
Variables Used:
B
Regression coefficient (y–intercept of a straight line);
16–8 Statistics Programs
File name 32sii-Manual-E-0424
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also used for scratch.
M
R
X
Regression coefficient (slope of a straight line).
Correlation coefficient; also used for scratch.
The x–value of a data pair when entering data; the
ˆ
y
hypothetical x when projecting ; or
ˆ
x
(x–estimate) when given a hypothetical y.
Y
i
The y–value of a data pair when entering data; the
ˆ
y
hypothetical y when projecting
; or
ˆ
x
(y–estimate) when given a hypothetical x.
Index variable used to indirectly address the correct
ˆ
y
–, –projection equation.
ˆ
x
Statistics registers Statistical accumulation and computation.
Example 1:
Fit a straight line to the data below. Make an intentional error when keying in
the third data pair and correct it with the undo routine. Also, estimate y for an
x value of 37. Estimate x for a y value of 101.
X
Y
40.5
38.6
102
37.9
1.00
36.2
97.5
35.1
95.5
34.6
94
104.5
Keys:
Display:
Description:
S
Starts straight–line routine.
Enters x–value of data pair.
Enters y–value of data pair.
Enters x–value of data pair.
Enters y–value of data pair.
W
%@ꢔ)ꢕꢕꢕꢕꢎ
40.5
f
&@valueꢎ
104.5
38.6
f
%@ꢏ)ꢕꢕꢕꢎ
&@ꢔꢕꢒ)ꢗꢕꢕꢕꢎ
%@ꢖ)ꢕꢕꢕꢎ
f
102 f
Now intentionally enter 379 instead of 37.9 so that you can see how to
correct incorrect entries.
Keys:
Display:
Description:
379
Enters wrong x–value of data pair.
f
&@ꢔꢕꢏ)ꢕꢕꢕꢕꢎ
Statistics Programs 16–9
File name 32sii-Manual-E-0424
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Retrieves
prompt.
f
%@ꢒ)ꢕꢕꢕꢕꢎ
%@ꢖ)ꢕꢕꢕꢕꢎ
%@
U
Deletes the last pair. Now proceed
with the correct data entry.
Enters correct x–value of data pair.
Enters y–value of data pair.
Enters x–value of data pair.
Enters y–value of data pair.
Enters x–value of data pair.
Enters y–value of data pair.
Enters x–valise of data pair.
Enters y–value of data pair.
Calculates the correlation
coefficient.
W
37.9
100
f
&@ꢔꢕꢏ)ꢕꢕꢕꢕꢎ
%@ꢒ)ꢕꢕꢕꢕꢎ
&@ꢔꢕꢕ)ꢕꢕꢕꢕꢎ
%@ꢗ)ꢕꢕꢕꢕꢎ
&@ꢓꢘ)ꢗꢕꢕꢕꢎ
%@ )ꢕꢕꢕꢕꢎ
&@ꢓꢗ)ꢗꢕꢕꢕꢎ
%@ꢘ)ꢕꢕꢕꢕꢎ
ꢁ/ꢕ)ꢓꢓꢗꢗꢎ
f
f
f
f
f
f
36.2
97.5
35.1
95.5
34.6
94
f
R
W
Calculates regression coefficient B.
Calculates regression coefficient
M.
f
f
ꢌ/ꢖꢖ)ꢗꢏꢘꢔꢎ
ꢇ/ꢔ)ꢘ ꢕꢔꢎ
Prompts for hypothetical x–value.
f
%@ꢘ)ꢕꢕꢕꢕꢎ
&@ꢓꢙ) ꢗꢏ ꢎ
%@ꢖꢙ)ꢖꢖꢖ ꢎ
ˆ
y
37
Stores 37 in X and calculates
.
f
101 f
Stores 101 in Y and calculates
.
ˆ
x
Example 2:
Repeat example 1 (using the same data) for logarithmic, exponential, and
power curve fits. The table below gives you the starting execution label and
the results (the correlation and regression coefficients and the x– and y–
estimates) for each type of curve. You will need to reenter the data values
each time you run the program for a different curve fit.
16–10 Statistics Programs
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Logarithmic
W L
Exponential
W E
Power
W P
0.9959
8.9730
0.6640
98.6845
38.3151
To start:
R
M
B
0.9965
0.9945
–139.0088
65.8446
98.7508
38.2857
51.1312
0.0177
ˆ
y
Y ( when X=37)
98.5870
38.3628
X ( when Y=101)
ˆ
x
Normal and Inverse–Normal Distributions
Normal distribution is frequently used to model the behavior of random
variation about a mean. This model assumes that the sample distribution is
symmetric about the mean, M, with a standard deviation, S, and
approximates the shape of the bell–shaped curve shown below. Given a
value x, this program calculates the probability that a random selection from
the sample data will have a higher value. This is known as the upper tail area,
Q(x). This program also provides the inverse: given a value Q(x), the
program calculates the corresponding value x.
y
"Upper tail"
area
Q [x]
x
x
Statistics Programs 16–11
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
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x e−((x−x)÷σ ) ÷2dx
2
1
σ 2π
Q(x) = 0.5−
∫
x
This program uses the built–in integration feature of the HP 32SIl to integrate
the equation of the normal frequency curve. The inverse is obtained using
Newton's method to iteratively search for a value of x which yields the given
probability Q(x).
Program Lines:
Description
This routine initializes the standard–deviation program.
Stores default value for mean.
ꢕꢔ ꢂꢌꢂ ꢎ
ꢕꢏ ꢕꢎ
ꢕꢖ !ꢑ ꢇꢎ
Prompts for and stores mean, M.
ꢕꢒ ꢊꢄꢅ"! ꢇꢎ
Stores default value for standard deviation.
ꢕꢗ ꢔꢎ
ꢕ !ꢑ ꢎ
Prompts for and stores standard deviation, S.
ꢕꢘ ꢊꢄꢅ"! ꢎ
Stops displaying value of standard deviation.
ꢕꢙ ꢁ!ꢄꢎ
Checksum and length: E5FA 012.0
This routine calculates Q(X) given X.
ꢍꢕꢔ ꢂꢌꢂ ꢍꢎ
Prompts for and stores X.
ꢍꢕꢏ ꢊꢄꢅ"! %ꢎ
Calculates upper tail area.
ꢍꢕꢖ %ꢈꢉ ꢉꢎ
Stores value in Q so VIEW function can display it.
Displays Q(X).
ꢍꢕꢒ !ꢑ ꢉꢎ
ꢍꢕꢗ #ꢊꢈ$ ꢉꢎ
Loops to calculate another Q(X).
ꢍꢕ ꢆ!ꢑ ꢍꢎ
Checksum and length: 2D6A 009.0
This routine calculates X given Q(X).
ꢊꢕꢔ ꢂꢌꢂ ꢊꢎ
Prompts for and stores Q(X).
ꢊꢕꢏ ꢊꢄꢅ"! ꢉꢎ
Recalls the mean.
ꢊꢕꢖ ꢁꢃꢂ ꢇꢎ
Stores the mean as the guess for X, called X
.
ꢊꢕꢒ !ꢑ %ꢎ
guess
Checksum and length: 35BF 006.0
This label defines the start of the iterative loop.
!ꢕꢔ ꢂꢌ! !ꢎ
!ꢕꢏ %ꢈꢉ ꢉꢎ
Calculates (Q( X
– Q(X)).
guess
16–12 Statistics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
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Program Lines:
Description
!ꢕꢖ ꢁꢃꢂ. ꢉꢎ
!ꢕꢒ ꢁꢃꢂ %ꢎ
!ꢕꢗ !ꢑ ꢍꢎ
¶
!ꢕ ꢁ ꢎ
Calculates the derivative at X
.
!ꢕꢘ %ꢈꢉ ꢋꢎ
!ꢕꢙ ꢁꢃꢂª !ꢎ
guess
Calculates the correction for X
!ꢕꢓ ªꢎ
guess
Adds the correction to yield a new X
.
!ꢔꢕ !ꢑ- %ꢎ
guess
!ꢔꢔ ꢀꢌ ꢎ
!ꢔꢏ ꢕ)ꢕꢕꢕꢔꢎ
Tests to see if the correction is significant.
Goes back to start of loop if correction is significant.
Continues if correction is not significant.
!ꢔꢖ º6¸@ꢎ
!ꢔꢒ ꢆ!ꢑ !ꢎ
!ꢔꢗ ꢁꢃꢂ %ꢎ
Displays the calculated value of X.
!ꢔ #ꢊꢈ$ %ꢎ
Loops to calculate another X.
!ꢔꢘ ꢆ!ꢑ ꢊꢎ
Checksum and length: C2AD 033.5
This subroutine calculates the upper–tail area Q(x).
Recalls the lower limit of integration.
Recalls the upper limit of integration.
ꢉꢕꢔ ꢂꢌꢂ ꢉꢎ
ꢉꢕꢏ ꢁꢃꢂ ꢇꢎ
ꢉꢕꢖ ꢁꢃꢂ %ꢎ
ꢉꢕꢒ ꢋꢄ/ ꢋꢎ
∫
Selects the function defined by LBL F for integration.
Integrates the normal function using the dummy
ꢉꢕꢗ ꢋꢄ G ꢍꢎ
variable D.
ꢉꢕ ꢏꢎ
π
ꢉꢕꢘ ꢎ
ꢉꢕꢙ ºꢎ
ꢉꢕꢓ ꢉꢁ!ꢎ
Calculates S ×
ꢉꢔꢕ ꢁꢃꢂº ꢎ
.
2π
Stores result temporarily for inverse routine.
ꢉꢔꢔ !ꢑ !ꢎ
ꢉꢔꢏ ªꢎ
ꢉꢔꢖ -+.ꢎ
ꢉꢔꢒ ꢕ)ꢗꢎ
ꢉꢔꢗ -ꢎ
Adds half the area under the curve since we integrated
using the mean as the lower limit.
Statistics Programs 16–13
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
Returns to the calling routine.
ꢉꢔ ꢁ!ꢄꢎ
Checksum and length: F79E 032.0
This subroutine calculates the integrand for the normal
ꢋꢕꢔ ꢂꢌꢂ ꢋꢎ
−((X−M)÷S)2 ÷2
function
e
ꢋꢕꢏ ꢁꢃꢂ ꢍꢎ
ꢋꢕꢖ ꢁꢃꢂ. ꢇꢎ
ꢋꢕꢒ ꢁꢃꢂª ꢎ
ꢋꢕꢗ ºꢏꢎ
ꢋꢕ ꢏꢎ
ꢋꢕꢘ ªꢎ
ꢋꢕꢙ -+.ꢎ
ꢋꢕꢓ H%ꢎ
Returns to the calling routine.
ꢋꢔꢕ ꢁ!ꢄꢎ
Checksum and length: 3DC2 015.0
Flags Used:
None.
Memory Required:
155.5 bytes: 107.5 for program, 48 for variables.
Remarks:
The accuracy of this program is dependent on the display setting. For inputs
in the rare between 3 standard deviations a display of four or more
significant figures is adequate for most application.
At full precision, the input limit becomes 5 standard deviations.
Computation time is significantly less with a lower number of displayed digits.
In routine N, the constant 0.5 may be replaced by 2 and
6.5 byte at the expense of clarity.
. This will save
3
16–14 Statistics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
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Yom do riot need to key in the inverse routine (in routines I and T) if you are
not interested in the inverse capability.
Program Instructions:
1. Key in the program routines; press † when done.
2. Press
S.
W
3. After the prompt for M, key in the population mean and press
. (If the
f
mean is zero, just press
.)
f
4. After the prompt for S, key in the population standard deviation and press
. (If the standard deviation is 1, just press
)
f
f
5. To calculate X given Q(X), skip to step 9 of these instructions.
6. To calculate Q(X) given X, W D.
7. After the prompt, key in the value of X and press
. The result, Q(X), is
f
displayed.
8. To calculate Q(X) for a new X with the same mean and standard deviation,
press and go to step 7.
f
9. To calculate X given Q(X), press
I.
W
10.After the prompt, key in the value of Q(X) and press
. The result, X, is
f
displayed.
11. To calculate X for a new Q(X) with the same mean and standard deviation,
press and go to step 10.
f
Variables Used:
D
M
Q
S
Dummy variable of integration.
Population mean, default value zero.
Probability corresponding to the upper–tail area.
Population standard deviation, default value of 1.
T
Variable used temporarily to pass the value S ×
inverse program.
to the
2π
X
Input value that defines the left side of the upper–tail area.
Statistics Programs 16–15
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
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Example 1:
Your good friend informs you that your blind date has "3σ" intelligence. You
interpret this to mean that this person is more intelligent than the local
population except for people more than three standard deviations above the
mean.
Suppose that you intuit that the local population contains 10,000 possible
?
blind dates. How many people fall into the "3 " band Since this problem is
σ
stated in terms of standard deviations, use the default value of zero for M and
1 for S.
Keys:
Display:
Description:
W S
Starts the initialization routine.
Accepts the default value of zero
for M.
ꢇ@ꢕ)ꢕꢕꢕꢕꢎ
@ꢔ)ꢕꢕꢕꢕꢎ
f
f
W
3
Accepts the default value of 1 for
S.
ꢔ)ꢕꢕꢕꢕꢎ
%@valueꢎ
ꢉ/ꢕ)ꢕꢕꢔꢒꢎ
D
Starts the distribution program and
prompts for X.
Enters 3 for X and starts
f
computation of Q(X). Displays the
ratio of the population smarter than
everyone within three standard
deviations of the mean.
10000
Multiplies by the population.
Displays the approximate number
of blind dates in the local
population that meet the criteria.
y
ꢔꢖ)ꢗꢕꢒꢓꢎ
Since your friend has been known to exaggerate from time to tame, you
decide to see how rare a "2σ" date might be. Note that the program may be
rerun simply by pressing
.
f
Keys:
Display:
Description:
16–16 Statistics Programs
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Resumes program.
f
%@ꢖ)ꢕꢕꢕꢕꢎ
ꢉ/ꢕ)ꢕꢏꢏꢘꢎ
2
Enters X–value of 2 and calculates
Q(X).
f
10000
Multiplies by the population for the
revised estimate.
y
ꢏꢏꢘ)ꢒꢓꢖꢘꢎ
Example 2:
The mean of a set of test scores is 55. The standard deviation is 15.3.
Assuming that the standard normal curve adequately models the distribution,
?
what is the probability that a randomly selected student scored 90 What is
the score that only 10 percent of the students would be expected to have
?
surpassed What would he the score that only 20 percent of the students
?
would have failed to achieve
Keys:
Display:
Description:
W S
Starts the initialization routine.
Stores 55 for the mean.
Stores 15.3 for the standard
deviation.
ꢇ@ꢕ)ꢕꢕꢕꢕꢎ
@ꢔ)ꢕꢕꢕꢕꢎ
ꢔꢗ)ꢖꢕꢕꢕꢎ
55
f
15.3
W
90
f
D
Starts the distribution program and
prompts for X.
%@valueꢎ
Enters 90 for X and calculates
Q(X).
f
ꢉ/ꢕ)ꢕꢔꢔꢔꢎ
Thus, we would expect that only about 1 percent of the students would do
better than score 90.
Keys:
Display:
Description:
I
Starts the inverse routine.
Stores 0.1 (10 percent) in Q(X)
and calculates X.
W
ꢉ@ꢕ)ꢕꢔꢔꢔꢎ
%/ꢘꢒ) ꢕꢘꢙꢎ
0.01
f
Resumes the inverse routine.
f
ꢉ@ꢕ)ꢔꢕꢕꢕꢎ
Statistics Programs 16–17
File name 32sii-Manual-E-0424
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0.8
Stores 0.8 (100 percent minus 20
f
%/ꢒꢏ)ꢔꢏꢖꢏꢎ
percent) in Q(X) and calculates X.
Grouped Standard Deviation
The standard deviation of grouped data, S , is the standard deviation of
xy
data points x , x , ... , x , occurring at positive integer frequencies f1, f2, ... ,
1
2
n
fn.
( xf )2
x2 −
∑
i i
∑
i
f
∑
i
Sxg =
( f )−1
∑
i
This program allows you to input data, correct entries, and calculate the
standard deviation and weighted mean of the grouped data.
Program Lines:
Description
Start grouped standard deviation program.
Clears statistics registers (28 through 33).
ꢕꢔ ꢂꢌꢂ ꢎ
ꢕꢏ ꢃꢂ;ꢎ
ꢕꢖ ꢕꢎ
Clears the count N.
ꢕꢒ !ꢑ ꢄꢎ
Checksum and length: 104F 006.0
Input statistical data points.
ꢊꢕꢔ ꢂꢌꢂ ꢊꢎ
Stores data point in X.
ꢊꢕꢏ ꢊꢄꢅ"! %ꢎ
Stores data–point frequency in F.
ꢊꢕꢖ ꢊꢄꢅ"! ꢋꢎ
Enters increment for N.
Recalls data–point frequency f .
ꢊꢕꢒ ꢔꢎ
ꢊꢕꢗ ꢁꢃꢂ ꢋꢎ
i
Checksum and length: 4060 007.5
Accumulate summations.
ꢋꢕꢔ ꢂꢌꢂ ꢋꢎ
ꢋꢕꢏ ꢏꢙꢎ
Stores index for register 28.
ꢋꢕꢖ !ꢑ Lꢎ
¶
ꢋꢕꢒ ꢁ ꢎ
16–18 Statistics Programs
File name 32sii-Manual-E-0424
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Program Lines:
Description
f
Updates
in register 28.
∑
i
ꢋꢕꢗ !ꢑ-1L2ꢎ
ꢋꢕ ꢁꢃꢂº %ꢎxifi
ꢋꢕꢘ ꢏꢓꢎ
Stores index for register 29.
ꢋꢕꢙ !ꢑ Lꢎ
¶
ꢋꢕꢓ ꢁ ꢎ
xf
Updates
ꢋꢔꢔ ꢁꢃꢂº %ꢎ i
in register 29.
ꢋꢔꢕ !ꢑ-1L2ꢎ
x 2f
ꢋꢔꢏ ꢖꢔꢎ
∑
i i
Stores index for register 31.
ꢋꢔꢖ !ꢑ Lꢎ
¶
ꢋꢔꢒ ꢁ ꢎ
x 2f
Updates
ꢋꢔꢗ !ꢑ-1L2ꢎ
ꢋꢔ º65¸ꢎ
in register 31.
∑
i
i
Gets 1 (or –1).
Increments (or decrements) N.
ꢋꢔꢘ !ꢑ- ꢄꢎ
ꢋꢔꢙ ꢁꢃꢂ ꢄꢎ
Displays current number of data pairs.
ꢋꢔꢓ #ꢊꢈ$ ꢄꢎ
Goes to label I for next data input.
ꢋꢏꢕ ꢆ!ꢑ ꢊꢎ
Checksum and length: 214E 030.0
Calculates statistics for grouped data.
Grouped standard deviation.
ꢆꢕꢔ ꢂꢌꢂ ꢆꢎ
ꢆꢕꢏ Uºꢎ
ꢆꢕꢖ !ꢑ ꢎ
Display grouped standard deviation.
ꢆꢕꢒ #ꢊꢈ$ ꢎ
Weighted mean.
ꢆꢕꢗ
ꢎ
º
ꢆꢕ !ꢑ ꢇꢎ
Displays weighted mean.
ꢆꢕꢘ #ꢊꢈ$ ꢇꢎ
Goes back for more points
ꢆꢕꢙ ꢆ!ꢑ ꢊꢎ
Checksum and length: 4A4A 012.0
Undo data–entry error.
Enters decrement for N.
"ꢕꢔ ꢂꢌꢂ "ꢎ
"ꢕꢏ .ꢔꢎ
Recalls last data frequency input.
"ꢕꢖ ꢁꢃꢂ ꢋꢎ
"ꢕꢒ -+.ꢎ
"ꢕꢗ ꢆ!ꢑ ꢋꢎ
Changes sign of f .
i
Adjusts court and summations.
Checksum and length: 615A 015.5
Statistics Programs 16–19
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Flags Used:
None.
Memory Required:
143 bytes: 71 for programs, 72 for data.
Program Instructions:
1. Key in the program routines; press
when done.
†
2. Press
S to start entering new data.
W
3. Key in x –value (data point) and press
.
f
i
4. Key in f –value (frequency) and press
.
f
i
f
5. Press
after VIEWing the number of points entered.
6. Repeat steps 3 through 5 for each data point.
If you discover that you have made a data-entry error ( x or f ) after you
i
i
f
have pressed
in step 4, press
U and then press
again.
f
W
Then go back to step 3 to enter the correct data.
7. When the last data pair has been input, press
G to calculate and
W
display the grouped standard deviation.
8. Press to display the weighted mean of the grouped data.
f
9. To add additional data points, press f and continue at step 3.
To start a new problem, start at step 2.
Variables Used:
X
Data point.
F
N
S
Data–point frequency.
Data–pair counter.
Grouped standard deviation.
Weighted mean.
M
16–20 Statistics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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i
Index variable used to indirectly address the correct
statistics register.
Register 28
Register 29
Register 31
Summation f .
Σ
i
Summation Σx f .
i i
2
Summation x f .
Σ
i i
Example:
Enter the following data and calculate the grouped standard deviation.
Group
1
2
3
4
5
6
x
5
17
8
26
13
37
15
43
22
73
37
115
i
f
i
Keys:
Display:
Description:
S
Prompts for the first x .
W
%@valueꢎ
ꢋ@valueꢎ
ꢄ/ꢔ)ꢕꢕꢕꢕꢎ
i
5 f
Stores 5 in X; prompts for first f .
i
17
Stores 17 in F; displays the
f
counter.
Prompts for the second x .
f
%@ꢗ)ꢕꢕꢕꢕꢎ
ꢋ@ꢔꢘ)ꢕꢕꢕꢕꢎ
ꢄ/ꢏ)ꢕꢕꢕꢕꢎ
%@ꢙ)ꢕꢕꢕꢕꢎ
ꢋ@ꢏ )ꢕꢕꢕꢕꢎ
ꢄ/ꢖ)ꢕꢕꢕꢕꢎ
i
8
Prompts for second f .
f
i
26
Displays the counter.
f
Prompts for the third x .
f
i
14
Prompts for the third f .
f
i
37
Displays the counter.
f
You erred by entering 14 instead of 13 for x . Undo your error by executing
3
routine U:
W U
Removes the erroneous data;
displays the revised counter.
ꢄ/ꢏ)ꢕꢕꢕꢕꢎ
f
Prompts for new third x .
%@ꢔꢒ)ꢕꢕꢕꢕꢎ
ꢋ@ꢖꢘ)ꢕꢕꢕꢕꢎ
i
13
Prompts for the new third f .
f
i
Statistics Programs 16–21
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Keys:
Display:
Description:
f
f
Displays the counter.
ꢄ/ꢖ)ꢕꢕꢕꢕꢎ
%@ꢔꢖ)ꢕꢕꢕꢕꢎ
ꢋ@ꢖꢘ)ꢕꢕꢕꢕꢎ
ꢄ/ꢒ)ꢕꢕꢕꢕꢎ
%@ꢔꢗ)ꢕꢕꢕꢕꢎ
ꢋ@ꢒꢖ)ꢕꢕꢕꢕꢎ
ꢄ/ꢗ)ꢕꢕꢕꢕꢎ
%@ꢏꢏ)ꢕꢕꢕꢕꢎ
ꢋ@ꢘꢖ)ꢕꢕꢕꢕꢎ
ꢄ/ )ꢕꢕꢕꢕꢎ
/ꢔꢔ)ꢒꢔꢔꢙꢎ
Prompts for the fourth x .
i
15 f
Prompts for the fourth f .
i
43
Displays the counter.
f
f
Prompts for the fifth x .
1
22
Prompts for the fifth f .
f
i
73
Displays the counter.
f
Prompts for the sixth x .
f
i
37
Prompts for the sixth f .
f
i
115
Displays the counter.
f
G
Calculates and displays the
grouped standard deviation
(sx) of the six data points.
Calculates and displays
W
f
†
ꢇ/ꢏꢖ)ꢒꢕꢙꢒꢎ
ꢏꢖ)ꢒꢕꢙꢒꢎ
weighted mean ( ).
x
Clears VIEW.
16–22 Statistics Programs
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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17
Miscellaneous Programs and
Equations
Time Value of Money
Given any four of the five values in the "Time–Value–of–Money equation"
(TVM), you can solve for the fifth value. This equation is useful in a wide
variety of financial applications such as consumer and home loans and
savings accounts.
The TVM equation is:
−N
1− (1+ I 100
P
+ F(1+ (I 100))−N + B = 0
I 100
Balance, B
Payments, P
3
N
_
N 1
1
2
Future Value, F
The signs of the cash values (balance, B; payment, P; and future balance, F)
correspond to the direction of the cash flow. Money that you receive has a
positive sign while money that you pay has a negative sign. Note that any
Miscellaneous Programs and Equations 17–1
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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problem can he viewed from two perspectives. The lender and the borrower
view the same problem with reversed signs.
Equation Entry:
Key in this equation:
ꢅºꢔꢕꢕº1ꢔ.1ꢔ-ꢊªꢔꢕꢕ2:.ꢄ2ªꢊ-ꢋº1ꢔ-ꢊªꢔꢕꢕ2:.ꢄ-ꢌꢎ
Keys:
Display:
Description:
Selects Equation mode.
{ G
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅ
or current equationꢎ
P
K y
100
Starts entering equation.
ꢅº ꢔꢕꢕ_ꢎ
y { \ 1 „
ꢅºꢔꢕꢕº1ꢔ.¾ꢎ
ºꢔꢕꢕº1ꢔ.1ꢔ-¾ꢎ
ꢔ.1ꢔ-ꢊªꢔꢕꢕ_ꢎ
.1ꢔ-ꢊªꢔꢕꢕ2:¾ꢎ
-ꢊªꢔꢕꢕ2:.ꢄ2¾ꢎ
ꢕ2:.ꢄ2ªꢊ-ꢋº¾ꢎ
ꢄ2ªꢊ-ꢋº1ꢔ-ꢊ¾ꢎ
ꢋº1ꢔ-ꢊªꢔꢕꢕ2¾ꢎ
ꢔ-ꢊªꢔꢕꢕ2:.ꢄ¾ꢎ
ꢊªꢔꢕꢕ2:.ꢄ-ꢌ¾ꢎ
1
y { \ ™
I
K p
100
{ ] 0
„ K N { ]
I
p K ™ K y
F
1
{ \ ™ K
I
100
p
{ ]
0 „ K N
B
™ K
Terminates the equation.
ꢅºꢔꢕꢕº1ꢔ.1ꢔ-ꢎ
š
(hold)
Checksum and length.
ꢃꢚ/ꢒꢗꢃꢒ ꢕꢗꢒ)ꢕꢎ
{
Memory Required:
94 bytes: 54 bytes for the equation, 40 bytes for variables.
17–2 Miscellaneous Programs and Equations
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Remarks:
The TVM equation requires that I must be non–zero to avoid a
ꢍꢊ#ꢊꢍꢈ ꢌ& ꢕ
error. If you're solving for I and aren't sure of its current value, press 1
H
I before you begin the SOLVE calculation (
I ).
{ œ
The order in which you're prompted for values depends upon the variable
you're solving for.
SOLVE instructions:
1. If your first TVM calculation is to solve for interest rate, I, press 1 H I.
2. Press
. If necessary, press
{ G
or
to scroll
z — z ˜
through the equation list until you come to the TVM equation.
3. Do one of the following five operations:
a. Press
periods.
b. Press
N to calculate the number of compounding
{ œ
{ œ
I to calculate periodic interest.
For monthly payments, the result returned for I is the monthly interest
rate, i; press 12 to see the annual interest rate.
y
c. Press
B to calculate initial balance of a loan or savings
{ œ
account.
d. Press
e. Press
P to calculate periodic payment.
F to calculate future value or balance of a
{ œ
loan.
{ œ
4. Key in the values of the four known variables as they are prompted for;
press f after each value.
5. When you press the last
, the value of the unknown variable is
f
calculated and displayed.
6. To calculate a new variable, or recalculate the carne variable using
different data, go back to step 2.
SOLVE works effectively in this application without initial guesses.
Miscellaneous Programs and Equations 17–3
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Variables Used:
N
I
The number of compounding periods.
The periodic interest rate as a percentage. (For example, if
the annual interest rate is 15% and there are 12 payments
per year, the periodic interest rate, i, is 15 12=1.25%.)
÷
B
P
F
The initial balance of loan or savings account.
The periodic payment.
The future value of a savings account or balance of a loan.
Example:
Part 1. You are financing the purchase of a car with a 3–year (36–montld)
loan at 10.5% annual interest compounded monthly. The purchase price of
the car is $7,250. Your down payment is $1,500.
_
B = 7,250 1,500
I = 10.5% per year
N = 36 months
F = 0
P =
?
Keys:
Display:
Description:
{
ꢋ%
}
Selects FIX 2 display format.
z ž
ꢎ
2
(
Displays the leftmost part of the
TVM equation.
{ G z
Rºꢔꢕꢕº1ꢔ.1ꢔ-ꢎ
as needed )
˜
P
Selects P; prompts for I.
Converts your annual interest rate
input to the equivalent monthly
rate.
{ œ
ꢊ@valueꢎ
10.5
p
12
š
ꢊ@ꢕ)ꢙꢙꢎ
f
Stores 0.88 in I; prompts for N.
ꢄ@valueꢎ
17–4 Miscellaneous Programs and Equations
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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36
0
Stores 36 in N; prompts for F.
Stores 0 in F; prompts for D.
Calculates B, the beginning loan
balance.
f
ꢋ@valueꢎ
f
ꢌ@valueꢎ
7250
š
ꢌ@ꢗ8ꢘꢗꢕ)ꢕꢕꢎ
1500 „
Stores 5750 in B; calculates
monthly payment, P.
f
ꢑꢂ#ꢊꢄꢆꢎ
ꢅ/.ꢔꢙ )ꢙꢓꢎ
The answer is negative since the loan has been viewed from the borrower's
perspective. Money received by the borrower (the beginning balance) is
positive, while money paid out is negative.
?
Part 2. What interest rate would reduce the monthly payment by $10
Keys:
Display:
Description:
Displays the leftmost hart of the
Rºꢔꢕꢕº1ꢔ.1ꢔ-ꢎ
{ G
TVM equation.
I
Selects I; prompts for P.
Rounds the payment to two
decimal places.
{ œ
z I
R@.ꢔꢙ )ꢙꢓꢎ
R@.ꢔꢙ )ꢙꢓꢎ
10
Calculates new payment.
Stores –176,89 in P; prompts for
N.
™
R@.ꢔꢘ )ꢙꢓꢎ
ꢄ@ꢖ )ꢕꢕꢎ
f
Retains 36 in N; prompts for F.
Retains 0 in F; prompts for B.
Retains 5750 in B; calculates
monthly interest rate.
f
f
f
ꢋ@ꢕ)ꢕꢕꢎ
ꢌ@ꢗ8ꢘꢗꢕ)ꢕꢕꢎ
ꢑꢂ#ꢊꢄꢆꢎ
ꢊ/ꢕ)ꢗ ꢎ
)ꢘꢗꢎ
12
Calculates annual interest, rate.
y
Part 3. Using the calculated interest rate (6.75%), assume that you sell the
?
car after 2 years. What balance will you still owe In other words, what is the
?
future balance in 2 years
Miscellaneous Programs and Equations 17–5
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Note that the interest rate, I, from part 2 is not zero, so you won't get a
error when you calculate the new I.
ꢍꢊ#ꢊꢍꢈ ꢌ& ꢕ
Keys:
Display:
Description:
Displays leftmost part of the TVM
Rºꢔꢕꢕº1ꢔ.1ꢔ-ꢎ
{ G
equation.
F
Selects F; prompts for P.
{ œ
f
R@.ꢔꢘ )ꢙꢓꢎ
ꢊ@ꢕ)ꢗ ꢎ
Retains P; prompts for I.
Retains 0.56 in I; prompts for N.
Stores 24 in N; prompts for B.
Retains 5750 in B; calculates F, the
future balance. Again, the sign is
f
ꢄ@ꢖ )ꢕꢕꢎ
ꢌ@ꢗ8ꢘꢗꢕ)ꢕꢕꢎ
ꢑꢂ#ꢊꢄꢆꢎ
24
f
f
ꢋ/.ꢏ8ꢕꢒꢘ)ꢕꢗꢎ
negative, indicating that you must,
pay out this money.
{
ꢋ%
} 4
Sets FIX 4 display format.
z ž
ꢎ
Prime Number Generator
This program accepts any positive integer greater than 3. If the number is a
prime number (not evenly divisible by integers other than itself and 1), then
the program returns the input value. If the input is not a prime number, then
the program returns the first prime number larger than the input.
The program identifies non–prime numbers by exhaustively trying all possible
factors. If a number is riot prime, the program adds 2 (assuring that the value
is still odd) and tests to see if it, has found a prime. This process continues until
a prime number is found.
17–6 Miscellaneous Programs and Equations
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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LBL Y
VIEW Prime
LBL Z
Note: x is the
value in the
X-register.
P + 2
→
x
LBL P
Start
x
3
→
→
P
D
LBL X
FP [
P
/
D
]
→
x
yes
x = 0
?
no
yes
D >
√
P
?
no
D
+ 2
→
D
Miscellaneous Programs and Equations 17–7
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Listing:
Program Lines:
Description
This routine displays prime number P.
&ꢕꢔ ꢂꢌꢂ &ꢎ
&ꢕꢏ #ꢊꢈ$ ꢅꢎ
Checksum and length: 5D0B 003.0
This routine adds 2 to P.
'ꢕꢔ ꢂꢌꢂ 'ꢎ
'ꢕꢏ ꢏꢎ
'ꢕꢖ ꢁꢃꢂ- ꢅꢎ
Checksum and length: 0C68 004.5
This routine stores the input value for P.
ꢅꢕꢔ ꢂꢌꢂ ꢅꢎ
ꢅꢕꢏ !ꢑ ꢅꢎ
ꢅꢕꢖ ꢏꢎ
ꢅꢕꢒ ªꢎ
ꢅꢕꢗ ꢋꢅꢎ
ꢅꢕ ꢕꢎ
Tests for even input.
ꢅꢕꢘ º/¸@ꢎ
ꢅꢕꢙ ꢔꢎ
ꢅꢕꢓ !ꢑ- ꢅꢎ
Increments P if input an even number.
Stores 3 in test divisor, D.
ꢅꢔꢕ ꢖꢎ
ꢅꢔꢔ !ꢑ ꢍꢎ
Checksum and length: 40BA 016.5
This routine tests P to see if it is prime.
%ꢕꢔ ꢂꢌꢂ %ꢎ
%ꢕꢏ ꢁꢃꢂ ꢅꢎ
%ꢕꢖ ꢁꢃꢂª ꢍꢎ
Finds the fractional part of P ÷ D.
%ꢕꢒ ꢋꢅꢎ
Tests for a remainder of zero (not prime).
If the number is not prime, tries next possibility.
%ꢕꢗ º/ꢕ@ꢎ
%ꢕ ꢆ!ꢑ 'ꢎ
%ꢕꢘ ꢁꢃꢂ ꢅꢎ
%ꢕꢙ ꢉꢁ!ꢎ
%ꢕꢓ ꢁꢃꢂ ꢍꢎ
Tests to see whether all possible factors have been
tried.
>
%ꢔꢕ º ¸@ꢎ
17–8 Miscellaneous Programs and Equations
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Program Lines:
Description
If all factors have been tried, branches to the display
routine.
%ꢔꢔ ꢆ!ꢑ &ꢎ
Calculates the next possible factor, D + 2.
%ꢔꢏ ꢏꢎ
%ꢔꢖ !ꢑ- ꢍꢎ
Branches to test potential prime with new factor.
%ꢔꢒ ꢆ!ꢑ %ꢎ
Checksum and length: 061F 021.0
Flags Used:
None.
Memory Required:
61 bytes: 45 for program, 16 for variables.
Program Instructions:
1. Key in the program routines; press
when done.
†
2. Key in a positive integer greater than 3.
3. Press P to run program. Prime number, P will b e displayed.
W
4. To see the next prime number, press
.
f
Variables Used:
P
Prime value and potential prime values.
D
Divisor used to test the current value of P.
Remarks:
No test is made to ensure that the input is greater than 3.
Example.
?
?
What is the first prime number after 789 What is the next prime number
Keys:
Display:
Description:
Miscellaneous Programs and Equations 17–9
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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789
P
Calculates next prime number after
W
ꢅ/ꢘꢓꢘ)ꢕꢕꢕꢕꢎ
ꢅ/ꢙꢕꢓ)ꢕꢕꢕꢕꢎ
789.
Calculates next prime number after
797.
f
17–10 Miscellaneous Programs and Equations
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Part 3
Appendixes and Reference
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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A
Support, Batteries,
and Service
Calculator Support
You can obtain answers to questions about using your calculator from our
Calculator Support Department. Our experience shows that many customers
have similar questions about our products, so we have provided the following
section, "Answers to Common Questions." If you don't find an answer to your
question, contact us at the address or phone number listed on the inside back
cover.
Answers to Common Questions
?
Q: How can I determine if the calculator is operating properly
A: Refer to page A–5, which describes the diagnostic self–test.
Q. My numbers contain commas instead of periods as decimal points. How
?
do I restore the periods
A: Use the
{ } function (page 1–14).
)
z Ÿ
?
Q: How do l change the number of decimal places in the display
A: Use the menu (page 1–15).
z ž
?
Q; How do 1 clear all or portions of memory
A: displays the CLEAR menu, which allows you to clear all
z b
variables, all programs (in program entry only), the statistics registers, or all
of user memory (not during program entry).
?
Q: What does an "E" in a number (for example,
) mean
ꢏ)ꢗꢔꢈ.ꢔꢖ
Support, Batteries, and Service
A–1
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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–13
A: Exponent of ten; that is, 2.51 10
.
×
Q: The calculator has displayed the message
. What should I
ꢇꢈꢇꢑꢁ& ꢋ"ꢂꢂ
?
do
A: You must clear a portion of memory before proceeding. (See appendix B.)
Q: Why does calculating the sine (or tangent) of π radians display a very
?
small number instead of 0
A: cannot be represented exactly with the 12–digit precision of the
π
calculator.
?
Q: Why do I get incorrect answers when I use the trigonometric functions
A: You must make sure the calculator is using the correct angular mode (
z
{ }, { }, or { } ).
ꢍꢆ ꢁꢍ ꢆꢁ
Ÿ
?
Q. What does the symbol in the display mean
A: This is an annuncidor, and it indicates something about the status of the
calculator. See "Annunciators" in chapter 1.
?
Q: Numbers show up as fractions. How do I get decimal numbers
A: Press
.
z Š
Environmental Limits
To maintain product reliability, observe the following temperature and
humidity limits:
ꢀ
ꢀ
ꢀ
Operating temperature: 0 to 45 °C (32 to 113 °F).
°
Storage temperature: –20 to 65 C (–4 to 149 °F).
°
°
Operating and storage humidity: 90% relative humidity at 40 C (104 F)
maximum.
A–2
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Changing the Batteries
Replace the batteries as soon as possible when the low battery annunciator
(
) appears. If the battery annunciator is on, and the display dims, you
¤ꢁ
may lose data. If data is lost, the
message is displayed.
ꢇꢈꢇꢑꢁ& ꢃꢂꢈꢀꢁ
Once you've removed the batteries, replace them within 2 minutes to avoid
losing stored information. (Have the new batteries readily at hand before you
open the battery compartment.) Use any brand of fresh I.E.C LR44 (or
manufacturer's equivalent) button–cell batteries.
Equivalent 1.5–volt, button–cell batteries you might find from various
manufacturers are LR44, A76, V13GA, KA76, 357, SP357, V357, and
SR44W.
1. Have three fresh button–cell batteries at hand. Avoid touching the battery
terminals — handle batteries only by their edges.
2. Make sure the calculator is OFF. Do not press ON (
) again
†
until the entire battery–changing procedure is completed.
If the calculator is ON when the batteries are removed,
the contents of Continuous Memory will be erased.
3. Remove the battery–compartment door by pressing down and outward on
it until the door slides off (left illustration).
A-3 picture
4. Turn the calculator over and shake the batteries out.
Do not mutilate, puncture, or dispose of
batteries in fire. The batteries can burst or
explode, releasing hazardous chemicals.
Warning
5. Insert the new batteries (right illustration). Stack them according to the
diagram inside the battery compartment.
6. Replace the battery–compartment door (slide the tab on the door back
into the slot in the calculator case).
Support, Batteries, and Service
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Testing Calculator Operation
Use the following guidelines to determine if the calculator is working properly.
Test the calculator after every step to see if its operation has been restored. If
your calculator requires service, refer to page A–7.
ꢀ
The calculator won't turn on (steps 1–4) or doesn't
respond when you press the keys (steps 1–3):
1. Reset the calculator. Hold down the
key and press
. It may be
-
†
necessary to repeat these reset keystrokes several times.
2. Erase memory. Press and hold down , then press and hold down
†
both < and 6, Memory is cleared and the
ꢇꢈꢇꢑꢁ& ꢃꢂꢈꢀꢁ
message is displayed when you release all three keys.
3. Remove the batteries (see "Changing the Batteries") and lightly press
a coin against both battery contacts in the calculator. Replace the
batteries and turn on the calculator. It should display
ꢇꢈꢇꢑꢁ&
.
ꢃꢂꢈꢀꢁ
4. Install new batteries (see "Changing the Batteries").
If these steps fail to restore calculator operation, it requires service.
ꢀ
If the calculator responds to keystrokes but you suspect
that it is malfunctioning:
1. Do the self–test described in the next section. If the calculator fails the
self test, it requires service.
2. If the calculator passes the self–test, you may have made a mistake
operating the calculator. Reread portions of the manual and check
"Answers to Common Questions" (page A–1).
3. Contact the Calculator Support Department. The address and phone
number are listed on the inside back cover.
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The Self–Test
If the display can be turned on, but the calculator does not seem to be
operating properly, do the following diagnostic self–test.
1. Hold down the
key, then press
, at the same time.
0
†
2. Press any key eight times and watch the various patterns displayed. After
you've pressed the key eight times, the calculator displays the copyright
message
and then the message
.
ꢃꢑꢅꢁ)ꢐꢅꢙꢘ8ꢓꢕ
ꢚꢌꢍ ꢕꢔ
3. Starting at the upper left corner (
) and moving from left to right, press
<
each key in the top row. Then, moving left to right, press each key in the
second row, the third row, and so on, until you've pressed every key.
ꢀ
If you press the keys in the proper order and they are functioning
properly, the calculator displays followed by two–digit numbers.
ꢚꢌꢍ
(The calculator is counting the keys using hexadecimal base.)
If you press a key out of order, or if a key isn't functioning properly, the
next keystroke displays a fail message (see step 4).
ꢀ
4. The self–test produces one of these two results:
ꢀ
The calculator displays
5.
if it passed the self–test. Go to step
ꢖꢏꢊꢊ.ꢑꢚ
ꢀ
The calculator displays
followed by a one–digit
ꢖꢏ ꢊꢊ.ꢋꢀꢊꢂ
number, if it failed the self–test. If you received the message because
you pressed a key out of order, reset the calculator (hold down
,
†
press - ) and do the self test again. If you pressed the keys in order,
but got this message, repeat the self–test to verify the results. If the
calculator fails again, it requires service (see page A–7). Include a
copy of the fail message with the calculator when you ship it for
service.
5. To exit the self–test, reset the calculator (hold down † and press -).
Pressing and starts a continuous self–test that is used at the factory.
†
3
You can halt this factory test by pressing any key.
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Limited One–Year Warranty
What Is Covered
The calculator (except for the batteries, or damage caused by the batteries) is
warranted by Hewlett–Packard against defects in materials and workmanship
for one year from the dale of original purchase. If you sell your unit or give it
as a gift, the warranty is automatically transferred to the new owner and
remains in effect for the original one–year period. During the warranty period,
we will repair or, at our option, replace at no charge a product that proves to
be defective, provided you return the product, shipping prepaid, to a
Hewlett–Packard service center. (Replacement may be with a newer model of
equivalent or better functionality.
This warranty gives you specific legal rights, and you may also have other
rights that vary from state to state, province to province, or country to country.
What Is Not Covered
Batteries, and damage caused by the batteries, are not covered by the
Hewlett–Packard warranty. Check with the battery manufacturer about
battery and battery leakage warranties.
This warranty does not apply if the product has been damaged by accident or
misuse or as the result of service or modification by other than an authorized
Hewlett–Packard service center.
No other express warranty is given. The repair or replacement of a product is
your exclusive remedy. ANY OTHER IMPLIED WARRANTY OF
MERCHANTABILITY OR FITNESS IS LIMITED TO THE ONE–YEAR
DURATION OF THIS WRITTEN WARRANTY. Some states, provinces,
or countries do not allow limitations on how long an implied warranty lasts,
so the above limitation may not apply to you. IN NO EVENT SHALL
HEWLETT–PACKARD
COMPANY
BE
LIABLE
FOR
CONSEQUENTIAL DAMAGES. Some states, provinces, or countries do
not allow the exclusion or limitation of incidental or consequential damages,
so the above limitation or exclusion may not apply to you.
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Products are sold on the basis of specifications applicable at the time of
manufacture. Hewlett–Packard shall have no obligation to modify or update
products once sold.
Consumer Transaction in the United Kingdom
This warranty shall not apply to consumer transactions and shall not affect the
statutory rights of a consumer. In relation to such transactions, the rights and
obligations of Seller and Buyer shall be determined by statute.
If the Calculator Requires Service
Hewlett–Packard maintains service centers in many countries. These centers
will repair a calculator or replace it (with an equivalent or newer model),
whether it is under warranty or not. There is a charge for service after the
warranty period. Calculators normally are serviced and reshipped within 5
working days.
ꢀ
In the United States: Send the calculator to the Calculator Service
Center listed on the inside of the back cover.
ꢀ
In Europe: Contact your HP sales office or dealer, or HP's European
headquarters for the location of the nearest service center. Do not ship
the calculator for service without first contacting a Hewlett–Packard office.
Hewlett–Packard S.A.
150, Route du Nant–d'Avril
P.O. Box CH 1217 Meyrin 2
Geneva, Switzerland
Telephone: 022 780.81.11
ꢀ
In other countries: Contact your HP sales office or dealer or write to
the U.S. Calculator Service Center (listed on the inside of the back cover)
for the location of other service centers. If local service is unavailable,
you can ship the calculator to the U.S. Calculator Service Center for
repair.
Support, Batteries, and Service
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All shipping, reimportation arrangements, and customs costs are your
responsibility.
Service Charge
There is a standard repair charge for out–of–warranty service. The Calculator
Service Center (listed on the inside of the back cover) can tell you how much
this charge is. The full charge is subject to the customer's local sales or
value–added tax wherever applicable.
Calculator products damaged by accident or misuse are not covered by the
fixed service charges. In these cases, charges are individually determined
based on time and material.
Shipping Instructions
If your calculator requires service, ship it to the nearest authorized service
center or collection point. Be sure to:
ꢀ
ꢀ
ꢀ
Include your return address and description of the problem.
Include proof of purchase date if the warranty has not expired.
Include a purchase order, check, or credit card number plus expiration
date (Visa or MasterCard) to cover the standard repair charge. In the
United States and some other countries, the serviced calculator can be
returned C.O.D. if you do not pay in advance.
ꢀ
ꢀ
Ship the calculator in adequate protective packaging to prevent damage.
Such damage is not covered by the warranty, so we recommend that you
insure the shipment.
Pay the shipping charges for delivery to the Hewlett–Packard service
center, whether or not the calculator is under warranty.
Warranty on Service
Service is warranted against defects in materials and workmanship for 90
days from the date of service.
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Service Agreements
In the U.S., a support agreement is available for repair and service. Refer to
the form that was packaged with the manual. For additional information,
contact the Calculator Service Center (see the inside of the back cover).
Regulatory Information
U.S.A. The HP 32SII generates and uses radio frequency energy and may
interfere with radio and television reception. The calculator complies with the
limits for a Class B computing device as specified in Subpart J of Part 15 of
FCC Rules, which provide reasonable protection against such interference in
a residential installation. In the unlikely event that there is interference to radio
or television reception (which can be determined by turning the calculator off
and on or by removing the batteries), try:
ꢀ
ꢀ
Reorienting the receiving antenna.
Relocating the calculator with respect to the receiver.
For more information, consult your dealer, an experienced radio or television
technician, or the following booklet, prepared by the Federal Corrnunications
Commission: How to Identify and Resolve Radio–TV Interference Problems.
This booklet is available from the U.S. Government Printing Office,
Washington, D.C.20402, Stock Number 004=000–00345–4. At the first
printing of this manual, the telephone number was (202) 783–3238.
West Germany. The HP 32SII complies with VFG 1046/84, VDE 0871B,
and similar non–interference standards. If you use equipment that is not
authorized by Hewlett–Packard, that system configuration has to comply with
the requirements of Paragraph 2 of the German Federal Gazette, Order (VFG)
1046/84, dated December 14, 1984.
Noise Declaration. In the operator position under normal operation (per
ISO 7779): LpA<70dB.
Support, Batteries, and Service
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B
User Memory and the Stack
This appendix covers
ꢀ
ꢀ
ꢀ
The allocation and requirements of user memory,
How to reset the calculator without affecting memory,
How to clear (purge) all of user memory and reset the system defaults,
and
ꢀ
Which operations affect stack lift.
Managing Calculator Memory
The HP 32SII has 384 bytes of user memory available to you for any
combination of stored data (variables, equations, or program lines). SOLVE,
∫
FN, and statistical calculations also require user memory. (The ∫ FN operation
is particularly "expensive" to run.)
All of your stored data is preserved until you explicitly clear it. The message
means that there is currently not enough memory available
ꢇꢈꢇꢑꢁ& ꢋ"ꢂꢂ
for the operation you just attempted. You need to clear some (or all) of user
memory. For instance, you can:
ꢀ
ꢀ
ꢀ
Clear the contents of any or all variables (see "Clearing Variables" its
chapter 3).
Clear any or all equations (see "Editing and Clearing Equations" in
chapter 6).
Clear any or all programs (see "Clearing One or More Programs" in
chapter 12).
ꢀ
ꢀ
Clear the statistics registers (press
{Σ} ).
z b
Clear all of user memory (press
{
} ).
z b
ꢀꢂꢂ
User Memory and the Stack
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Memory Requirements
Data or Operation Amount of Memory Used
Variables
8 bytes per non–zero value. (No bytes
for zero values.)
Instructions in program lines
Numbers in program lines
1.5 bytes.
Integers 0 through 254: 1.5 bytes. All
other numbers: 9.5 bytes.
Operations in equations
Numbers in equations
1.5 bytes.
Integers 0 through 254: 1.5 bytes. All
other numbers: 9.5 bytes.
Statistics data
48 bytes maximum (8 bytes for each
non–zero summation register).
SOLVE calculations
∫ FN (integration) calculations
33.5 bytes.
140 bytes.
To see how much memory is available, press
the number of bytes available.
. The display shows
z X
To see the memory requirements of specific equations in the equation list:
1. Press
to activate Equation mode. (
or the left
{ G
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅ
end of the current equation will be displayed.)
2. If necessary, scroll through the equation list (press
˜ ) until you see the desired equation.
or
z —
z
3. Press
to see the checksum (hexadecimal) and length (in
{
bytes) of the equation. For example,
.
ꢃꢚ/ꢘꢋꢒꢓ ꢕꢕꢓ)ꢕ
To see the total memory requirements of specific programs:
1. Press } to display the first label in the program list.
{
z X
ꢅꢆꢇ
2. Scroll through the program list (press
or
until you see
.
z — z ˜
the desired program label and size). For example,
ꢂꢌꢂ ꢋ ꢕꢔꢏ)ꢕ
3. Optional: Press { to see the checksum (hexadecimal) and
length (in bytes) of the program$. For example,
program F.
012.0 for
ꢃꢚ/ꢗꢍꢈꢀ
To see the memory requirements of an equation in a program:
B–2
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1. Display the program line containing the equation.
2. Press to see the checksum and length. For example,
{
.
ꢃꢚ/ꢘꢋꢒꢓ ꢕꢕꢓ)ꢕ
To manually deallocate the memory allocated for a SOLVE or ∫ FN calculation
that has been interrupted, press . This deallocation is done
{ ”
automatically whenever you execute a program or another SOLVE or ∫ FN
calculation.
Resetting the Calculator
If the calculator doesn't respond to keystrokes or if it is otherwise behaving
unusually, attempt to reset it. Resetting the calculator halts the current
calculation and cancels program entry, digit entry, a running program, a
SOLVE calculation, an ∫ FN calculation, a VIEW display, or an INPUT display.
Stored data usually remain intact.
To reset the calculator, hold down the
key and press
. If you are
-
†
unable to reset the calculator, try installing fresh batteries. If the calculator
cannot be reset, or if it still fails to operate properly, you should attempt to
clear memory using the special procedure described in the next section.
The calculator can reset itself if it is dropped or if power is interrupted.
Clearing Memory
The usual way to clear user memory is to press
{
}. However,
z b
ꢀꢂꢂ
there is 1so more powerful clearing procedure that resets additional
information and is useful if e keyboard is not functioning properly.
If the calculator fails to respond to keystrokes, and you are unable to restore
operation by resetting it or changing the batteries, try the following MEMORY
CLEAR procedure. These keystrokes clear all of memory, reset the calculator,
and restore all format and modes to their original, default settings (shown
below):
User Memory and the Stack
B–3
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1. Press and hold down the
2. Press and hold down
key.
†
.
<
3. Press
. (You will be pressing three keys simultaneously). When you
6
release all three keys, the display shows
is successful.
if the operation
ꢇꢈꢇꢑꢁ& ꢃꢂꢈꢀꢁ
Category
CLEAR ALL
MEMORY CLEAR
(Default)
Angular mode
Base mode
Contrast setting
Decimal point
Unchanged
Degrees
Decimal
Medium
Unchanged
Unchanged
Unchanged
" "
)
Denominator (/c value) Unchanged
4095
Display format
Flags
Fraction–display mode Unchanged
Random–number seed Unchanged
Unchanged
Unchanged
FIX 4
Cleared
Off
Zero
Equation pointer
Equation list
FN = label
EQN LIST TOP
Cleared
Null
EQN LIST TOP
Cleared
Null
Program pointer
Program memory
Stack lift
Stack registers
Variables
PRGM TOP
Cleared
Enabled
Cleared to zero
Cleared to zero
PRGM TOP
Cleared
Enabled
Cleared to zero
Cleared to zero
Memory may inadvertently be cleared if the calculator is dropped or if power
is interrupted.
The Status of Stack Lift
The four stack registers are always present, and the stack always has a
stack–lift status. That is to say, the stack lift is always enabled or disabled
regarding its behavior when the next number is placed in the X–register.
(Refer to chapter 2, "The Automatic Memory Stack.")
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All functions except those in the following two lists will enable stack lift.
Disabling Operations
The four operations ENTER, Σ+, Σ–, and CLx disable stack lift. A number
keyed in after one of these disabling operations writes over the number
currently in the X–register. The Y–, Z– and T–registers remain unchanged.
In addition, when
and
act like CLx, they also disable stack lift.
†
@
The INPUT function disables stack lift as it halts a program for prompting (so
any number you then enter writes over the X–register), but it enables stack lift
when the program resumes.
Neutral Operations
The following operations do not affect the status of stack lift:
DEG, RAD,
GRAD
FIX, SCI,
ENG, ALL
DEC, HEX, OCT, CLVARS
BIN
PSE
SHOW
RADIX .
RADIX ,
z —
CLΣ
and STOP
and
* and
*
ꢀ
f
†
a
z ˜
{
}**
{
}**
GTO
d
program entry
label nn
X
EQN
X
FDISP
U Œ Œ
Errors
Œ
#ꢀꢁ
ꢅꢆꢇ
and
Switching
binary windows
Digit entry
ꢀ
Except when used like CLx.
ꢀꢀ
Including all operations performed while the catalog is displayed
except {
#ꢀꢁ
} š and {
ꢅꢆꢇ
} W, which enable stack lift.
User Memory and the Stack
B–5
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The Status of the LAST X Register
The following operations save x in the LAST X register:
2
x
+, –, × , ÷
LN, LOG
SQRT, x
e , 10x
X
x
y
y ,
I/x
ˆ
y
,
SIN, COS, TAN
ASINH, ACOSH,
ATANH
ASIN, ACOS, ATAN
IP, FP, RND, ABS
ˆ
x
SINH, COSH, TANH
%, %CHG
Σ+, Σ–
RCL+, –, ×, ÷
,
,
€
€
€
€
y,x
,r
HR HMS
DEG RAD
θ
€
θ,r y, x
€
Cn,r
x!
CMPLX +/–
Pn,r
x
x
CMPLX +. –, × ,÷
CMPLX e , LN, y ,
1/x
CMPLX SIN, COS,
TAN
€
€
€
€
€
€
kg, lb,
°C, °F
cm, in
€ €
l, gal
Notice that /c does riot affect the LAST X register,
The recall–arithmetic sequence x variable stores a different value in
K ™
the LAST X register than the sequence x K variable ™ does. The former
stores x in LAST X; the latter stores the recalled number in LAST X.
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C
More about Solving
This appendix provides information about the SOLVE operation beyond that
given in chapter 7.
How SOLVE Finds a Root
SOLVE is an iterative operation; that is, it repetitively executes the specified
equation. The value returned by the equation is a function f(x) of the unknown
variable x. (f(x) is mathematical shorthand for a function defined in terms of
the unknown variable x.) SOLVE starts with an estimate for the unknown
variable, x, and refines that estimate with each successive execution of the
function, f(x).
If any two successive estimates of the function f(x) have opposite signs, then
SOLVE presumes that the function f(x) crosses the x–axis in at least one place
between the two estimates. This interval is systematically narrowed until a root
is found.
For SOLVE to find a root, the root has to exist within the range of numbers of
the calculator, and the function must be mathematically defined where the
iterative search occurs. SOLVE always finds a root, provided one exists
(within the overflow bounds), if one or more of these conditions are met:
ꢀ
Two estimates yield f(x) values with opposite signs, and the function's
graph crosses the x–axis in at least one place between those estimates
(figure a, below).
ꢀ
ꢀ
f(x) always increases or always decreases as x increases (figure b,
below).
The graph of f(x) is either concave everywhere or convex everywhere
(figure c, below).
More about Solving
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If f(x) has one or more local minima or minima, each occurs singly
between adjacent roots off f(x) (figure d, below).
ꢀ
f (x)
f (x)
x
x
b
a
f (x)
f (x)
x
x
d
c
Function Whose Roots Can Be Found
In most situations, the calculated root is an accurate estimate of the theoretical,
infinitely precise root of the equation. An "ideal" solution is one for which f(x)
= 0. However, a very small non–zero value for f(x) is often acceptable
because it might result from approximating numbers with limited (12–digit)
precision.
C–2
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Interpreting Results
The SOLVE operation will produce a solution under either of the. following
conditions:
ꢀ
ꢀ
If it finds an estimate for which f(x) equals zero. (See figure a, below.)
If it finds an estimate where f(x) is not equal to zero, but the calculated
root is a 12–digit number adjacent to the place where the function's
graph crosses the x–axis (see figure b, below). This occurs when the two
final estimates are neighbors (that is, they differ by 1 in the 12th digit),
and the function's value is positive for one estimate and negative for the
–499
–499
other. Or they are (0, 10
relatively close to zero.
) or (0, –10
). In most cases, f(x) will be
f (x)
f (x)
x
x
a
b
Cases Where a Root Is Found
To obtain additional information about the result, press
see the previous
9
estimate of the root (x), which was left in the Y–register. Press
again to
9
see the value of f(x), which was left in the Z–register. If f(x) equals zero or is
relatively small, it is very likely that a solution has been found. However, if f(x)
is relatively large, you must use caution in interpreting the results.
Example: An Equation With One Root.
Find the root of the equation:
3
2
–2x + 4x – 6x + 8 = 0
Enter the equation as an expression:
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Keys:
Display:
Description:
{ G
Select Equation mode.
Enters the equation.
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
2
K X 0 3
_ꢁyꢁ
4
™ꢁ yꢁ
X
K 0ꢁ
„ yꢁKꢁ ꢁ
2
6
X
8
™ š
.ꢏº%:ꢖ-ꢒ%:ꢏ.ꢎ
Clecksum and length.
{
†
ꢃꢚ/ꢑꢃ ꢀ ꢕꢖꢗ)ꢕꢎ
Cancels Equation mode.
ꢎ
Now, salve the equation to find the root:
Keys:
Display:
Description:
0
X 10
Initial guesses for the root.
Selects Equation mode;
displays the left end of the
equation.
H
ꢔꢕ_
{ G
.ꢏº%:ꢖ-ꢒº%:ꢏ.ꢎ
X
Solves for X; displays the
result.
{ œ
9
ꢑꢂ#ꢊꢄꢆꢎ
%/ꢔ) ꢗꢕ ꢎ
ꢔ) ꢗꢕ ꢎ
Final two estimates are the
same to four decimal places.
f(x) is very small, so the
approximation is a good
root.
9
.ꢒ)ꢕꢕꢕꢕꢈ.ꢔꢔꢎ
Example: An Equation with Two Roots.
Find the two roots of the parabolic equation:
2
x + x – 6 = 0.
Enter the equation as an expression:
C–4
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Keys:
Display:
Description:
{ G
Selects Equation mode.
Enters the equation.
ꢎ
ꢎ
X
2
K 0 ™
K X „ 6 š
{
†
%:ꢏ-%. ꢎ
ꢃꢚ/ꢕꢖ ꢖ ꢕꢔꢏ)ꢕꢎ
Checksum and length.
Cancels Equation mode.
ꢎ
Now, solve the equation to find its positive and negative roots:
Keys:
Display:
Description:
0
X 10
Your initial guesses for the
positive root.
H
ꢔꢕ_
Selects Equation mode;
displays the equation.
Calculates the positive root
using guesses 0 an 10.
Final two estimates are they
same.
{ G
{ œ
9
%:ꢏ-%. ꢎ
X
ꢑꢂ#ꢊꢄꢆꢎ
%/ꢏ)ꢕꢕꢕꢕꢎ
ꢏ)ꢕꢕꢕꢎ
f(x) = 0.
ꢕ)ꢕꢕꢕꢕꢕꢕꢕꢕꢕꢕꢕꢎ
9 {
0
X 10
Your initial guesses for the
negative root.
H
_
.ꢔꢕ_ꢎ
Redisplays the equation.
Calculates negative root
using guesses 0 and –10.
f(x) = 0.
{ G
%:ꢏ-%. ꢎ
X
{ œ
ꢑꢂ#ꢊꢄꢆꢎ
%/.ꢖ)ꢕꢕꢕꢕꢎ
ꢕ)ꢕꢕꢕꢕꢕꢕꢕꢕꢕꢕꢕꢎ
9 9 {
Certain cases require special consideration:
ꢀ
If the function's graph has a discontinuity that crosses the x–axis, then the
SOLVE operation returns a value adjacent to the discontinuity (see figure
a, below). In this case, f(x) may be: relatively large.
ꢀ
Values of f(x) may be approaching infinity at the location where the
graph changes sign (see figure b, below). This situation is called a pole.
Since the SOLVE operation determines that there is a sign change
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between two neighboring values of x, it returns the possible root.
However, the value for f(x) will be relatively large. If the pole occurs at a
value of x that is exactly represented with 12 digits, then that value
would cause the calculation to halt with an error message.
f (x)
f (x)
x
x
a
b
Special Case: A Discontinuity and a Pole
Example: Discontinuous Function.
Find the root of the equation:
IP(x) = 1.5
Enter the equation:
Keys:
Display:
Description:
Selects Equation mode.
Enter the equation.
z G
ꢎ
[PARTS] {
K X { ] {
}
z
ꢊꢅ
ꢎ
ꢎ
1.5
š
ꢊꢅ1%2/ꢔ)ꢗꢎ
Checksum and length.
ꢃꢚ/ꢙꢀꢗꢗ ꢕꢔꢘ)ꢕꢎ
{
C–6
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Cancels Equation mode.
†
ꢎ
Now, solve to find the root:
Keys:
Display:
Description:
0 H X 5
Your initial guesses for the
root.
ꢗ_ꢎ
{ G
Selects Equation mode;
displays the equation.
Finds a root with guesses 0
and 5.
ꢊꢅ1%2/ꢔ)ꢗꢎ
X
{ œ
{
ꢑꢂ#ꢊꢄꢆꢎ
%/ꢏ)ꢕꢕꢕꢕꢎ
Shows root, to 11 decimal
places.
ꢔ)ꢓꢓꢓꢓꢓꢓꢓꢓꢓꢓꢓꢎ
The previous estimate is
slightly bigger.
9 {
9
ꢏ)ꢕꢕꢕꢕꢕꢕꢕꢕꢕꢕꢕꢎ
f(x) is relatively large.
.ꢗ)ꢕꢕꢕꢎ
Note the difference between the last two estimates, as well as the relatively
large value for f(x). The problem is that there is no value of x for which f(x)
equals zero. However, at x = 1.99999999999, there is a neighboring value
of x that yields ant opposite sign for f(x).
Example: A Pole.
Find the root of the equation
x
−1= 0
x2 − 6
As x approaches
number.
, f(x) becomes a very large positive or negative
6
Enter the equation as an expression.
Keys:
Display:
Description:
Selects Equation mode.
{ G
ꢎ
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X
K pꢁ
{ \ K X
Enters the equation.
ꢎ
ꢎ
2
6
0 „
ꢎ
1
{ ] „
š
ꢎ
%ª1%:ꢏ. 22.ꢔꢎ
Checksum and length.
{
†
ꢃꢚ/ꢃꢋꢘꢃ ꢕꢔꢙ)ꢕꢎ
Cancels Equation mode.
ꢎ
Now, solve to find the root.
Keys:
Display:
Description:
2.3
X 2.7
Your initial guesses for the root.
Selects Equation mode; displays
the equation.
H
ꢏ)ꢘ_ꢎ
{ G
{ œ
9 9
%ª1%:ꢏ. 22.ꢔꢎ
X
Calculates the root using guesses
ꢑꢂ#ꢊꢄꢆꢎ
that bracket
.
%/ꢏ)ꢒꢒꢓꢗꢎ
ꢙꢔ8 ꢒꢓ8 ꢗꢙ8ꢕꢓꢏ)ꢕꢎ
6
f(x) is relatively large.
There is a pole between the final estimates. The initial guesses yielded
opposite signs for f(x), and the interval between successive estimates was
narrowed until two neighbors were found. Unfortunately, these neighbors
made f(x) approach a pole instead of the x–axis. The function does have roots
at –2 and 3, which can be found by entering better guesses.
When SOLVE Cannot Find Root
Sometimes SOLVE fails to find a root. The following conditions cause the
message
:
ꢄꢑ ꢁꢑꢑ! ꢋꢄꢍ
ꢀ
The search terminates near a local minimum or maximum (see figure a,
below). If the ending value of f(x) (stored in the Z–register) is relatively
close to zero, it is possible that a root has been found; the number stored
in the unknown variable might be a 12–digit number very close to a
theoretical root.
C–8
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ꢀ
ꢀ
The search halts because SOLVE is working on a horizontal
asymptote—an area where f(x) is essentially constant for a wide range of
x (see figure b, below). The ending value of f(x) is the value of the
potential asymptote.
The search is concentrated in a local "flat" region of the function (see
figure c, below). The ending value of f(x) is the value of the function in
this region.
f (x)
f (x)
x
x
b
a
f (x)
x
c
Case Where No Root Is Found
The SOLVE operation returns a math error if an estimate produces an
operation that is not allowed — for example, division by zero, a square root
of a negative number, or a logarithm of zero. Keep in mind that SOLVE can
generate estimates over a wide range. You can sometimes avoid math errors
by using good guesses. If a math error occurs, press
unknown variable
K
(or { ‰ variable) to see the value that produced the error.
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Example: A Relative Minimum.
Calculate the root of this parabolic equation:
2
x – 6x + 13 = 0.
It has a minimum at x = 3.
Enter the equation as an expression:
Keys:
Display:
Description:
Selects Equation mode.
Enters the equation.
{ G
ꢎ
X
K 0
„ y K ™
2
ꢎ
6
X
ꢎ
13
š
%:ꢏ. º%-ꢔꢖꢎ
ꢎ
Checksum and length.
{
†
Cancels Equation mode.
ꢃꢚ/ꢗꢋꢃꢃ ꢕꢔꢗ)ꢕꢎ
Now, solve to find the root:
Keys:
Display:
Description:
0
X 10
Your initial guesses for the root.
Selects Equation mode; displays
the equation.
H
ꢔꢕ_ꢎ
{ G
%:ꢏ. º%-ꢔꢖꢎ
ꢄꢑ ꢁꢑꢑ! ꢋꢄꢍꢎ
X
Search fails with guesses 0 and
10
{ œ
Displays the final estimate of x.
@ {
9 {
ꢖ)ꢕꢕꢕꢕꢕꢕꢔꢕꢕꢕꢕꢔꢎ
Previous estimate was not the
ꢖ)ꢕꢕꢕꢕꢕꢒ ꢙꢒꢒꢖꢎ
same.
9
Final value for f(x) is relatively
large.
ꢒ)ꢕꢕꢕꢎ
Example: An Asymptote.
Find the root of the equation
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1
10− = 0
X
Enter the equation as an expression.
Keys:
Display:
Description:
Selects Equation mode.
Enters the equation.
{ G
ꢎ
10 „ 3 K X
{ ] š
{ ꢁ†
ꢎ
ꢔꢕ.ꢊꢄ#1%2ꢎ
Checksum and length.
ꢃꢚ/ ꢃ ꢍ ꢕꢓ)ꢕꢎ
.005
X 5
Cancels Equation mode.
H
ꢗ_ꢎ
{ G
Your positive guesses for the
root.
ꢔꢕ.ꢊꢄ#1%2ꢎ
{ œ X
9
Selects Equation mode; displays
the equation.
%/ꢕ)ꢔꢕꢕꢕꢎ
ꢕ)ꢔꢕꢕꢕꢎ
Solves for x using guesses 0.005
and 5.
Previous estimate is the same.
ꢕ)ꢕꢕꢕꢕꢕꢕꢕꢕꢕꢕꢕꢎ
9 {
Watch what happens when you use negative values for guesses:
Keys:
Display:
Description:
1 _ H X
Your negative guesses for
the root.
.ꢔ)ꢕꢕꢕꢕꢎ
2
Selects Equation mode;
displays the equation.
No root found for f(x).
Displays last estimate of x.
_ { G
ꢔꢕ.ꢊꢄ#1%2ꢎ
X
{ œ
ꢄꢑ ꢁꢑꢑ! ꢋꢄꢍꢎ
@
9
.ꢒ )
8
8 ꢓꢏ)ꢔꢎ
Previous estimate was
much larger.
.ꢗ)ꢘꢘꢗꢕꢈꢔꢗꢎ
f(x) for last estimate is
rather large.
9
ꢔꢕ)ꢕꢕꢕꢕꢎ
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It's apparent from inspecting the equation that if x is a negative number, the
smallest that f(x) can be is 10. f(x) approaches 10 as x becomes a negative
number of large magnitude.
Example: A Math Error.
Find the root of the equation:
[x ÷ (x + 0.3)]− 0.5= 0
Enter the equation as an expression:
Keys:
Display:
Description:
{ G
Selects Equation mode.
Enters the equation.
ꢎ
ꢎ
ꢎ
ꢎ
ꢎ
X
< K p
{ \ K X
3
™ Œ {
] { ] „
5
Œ š
ꢉꢁ!1%ª1%-ꢕ)ꢖꢎ
Checksum and length.
{
†
ꢃꢚ/ꢃꢈꢕꢃ ꢕꢖꢒ)ꢕꢎ
Cancels Equation mode.
ꢎ
First attempt to find a positive root:
Keys:
Display:
Description:
0
X 10
Your positive guesses for the root.
Selects Equation mode; displays
H
ꢔꢕ_ꢎ
{ G
ꢉꢁ!1%ª1%-ꢕ)ꢖꢎ
the left end of the equation.
X
Calculates the root using guesses 0
and 10.
{ œ
%/ꢕ)ꢔꢕꢕꢕꢎ
Now attempt to find a negative root by entering guesses 0 and –10. Notice
that the function is undefined for values of x between 0 and –0.3 since those
values produce a positive denominator but a negative numerator, causing a
negative square root.
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Keys:
Display:
Description:
0 H X 10 _
{ G
.ꢔꢕ_ꢎ
ꢉꢁ!1%ª1%-ꢕ)ꢖꢎ
Selects Equation mode; displays
the left end of the equation.
X
Math error.
{ œ
ꢉꢁ!1ꢄꢈꢆ2ꢎ
ꢎ
† †
Clears error message; cancels
Equation mode.
X
Displays the final estimate of x.
{ ‰
%/.ꢕ)ꢔꢓꢙꢙꢎ
Example : A Local "Flat" Region.
Find the root of the function
f(x) = x + 2 if x< –1,
f(x) = 1 for –1 ≤ x ≤ 1 (a local flat region),
f(x) = –x + 2 if x >1.
Enter the function as the program:
ꢛꢕꢔ ꢂꢌꢂ ꢛꢎ
ꢛꢕꢏ .ꢎ
ꢛꢕꢖ ꢈꢄ!ꢈꢁꢎ
ꢛꢕꢒ ꢏꢎ
ꢛꢕꢗ ꢁꢃꢂ- %ꢎ
ꢛꢕ º6¸ꢎ
ꢛꢕꢘ ꢁ!ꢄꢎ
ꢛꢕꢙ ꢒꢎ
ꢛꢕꢓ .ꢎ
ꢛꢔꢕ -+.ꢎ
ꢛꢔꢔ º5¸@ꢎ
¶
ꢛ ꢔꢏ ꢁ ꢎ
ꢛꢔꢖ ꢁ!ꢄꢎ
Checksum and length: 23C2 019.5
You can subsequently delete line J03 to save memory.
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–8
–8
Solve for X using initial guesses of 10 and –10 .
Keys:
Display:
Description:
8
` _ H
1 _ ` 8 _
X
Enters guesses.
.ꢔꢈ.ꢙ_ꢎ
J
Selects program "J" as the
function.
{ V
.ꢔ)ꢕꢕꢕꢕꢈ.ꢙꢎ
X
No root found using very small
guesses near zero (thereby
restricting the search to the flat
region of the function).
The last two estimates are far
apart, and the final value of f(x) is
large.
{ œ
ꢄꢑ ꢁꢑꢑ! ꢋꢄꢍꢎ
@
9
9
ꢔ)ꢕꢕꢕꢕꢈ.ꢙꢎ
ꢕ)ꢕꢕꢏꢗꢎ
ꢔ)ꢕꢕꢕꢕꢎ
If you use larger guesses, then SOLVE can find the roots, which are outside
the flat region (at x = 2 and x = –2).
Round–Off Error
The limited (12–digit) precision of the calculator can cause errors due to
rounding off, which adversely affect the iterative solutions of SOLVE and
integration. For example,
15 2
[( x +1)+10 ] -1030 = 0
has no roots because f(x) is always greater than zero. However, given initial
guesses of 1 and 2, SOLVE returns the answer 1.0000 due to round–off
error.
Round–off error can also cause SOLVE to fail to find a root. The equation
x2 - 7 = 0
has a root at
. However, no 12–digit number exactly equals
, so
7
7
the calculator can never make the function equal to zero. Furthermore, the
C–14 More about Solving
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function never changes sign SOLVE returns the message
.
ꢄꢑ ꢁꢑꢑ! ꢋꢄꢍ
However, the final estimate of x (press
to see it) is the best possible
@
12–digit approximation of the root when the routine quits.
Underflow
Underflow occurs when the magnitude of a number is smaller than the
calculator can represent, so it substitutes zero. This can affect SOLVE results.
For example, consider the equation
1
x2
whose root is infinite in value. Because of underflow, SOLVE returns a very
large value as a root. (The calculator cannot represent infinity, anyway.)
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D
More about Integration
This appendix provides information about integration beyond that given in
chapter 8.
How the Integral Is Evaluated
The algorithm used by the integration operation, ∫
, calculates the
ꢋꢄ Gº
integral of a function f(x) by computing a weighted average of the function's
values at many values of x (known as sample points) within the interval of
integration. The accuracy of the result of any such sampling process depends
on the number of sample points considered: generally, the more sample
points, the greater the accuracy, if f(x) could be evaluated at an infinite
number of sample points, the algorithm could — neglecting the limitation
imposed by the inaccuracy in the calculated function f(x) — always provide
an exact answer.
Evaluating the function at an infinite number of sample points would take
forever. However, this is not necessary since the maximum accuracy of the
calculated integral is limited by the accuracy of the calculated function values.
Using only a finite number of sample points, the algorithm can calculate an
integral that is as accurate as is justified considering the inherent uncertainty
in f(x).
The integration algorithm at first considers only a few sample points, yielding
relatively inaccurate approximations. If these approximations are not yet as
accurate as the accuracy of f(x) would permit, the algorithm is iterated
(repeated) with a larger number of sample points. These iterations continue,
using about twice as many sample points each time, until the resulting
approximation is as accurate as is justified considering the inherent
uncertainty in f(x).
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As explained in chapter 8, the uncertainty of the final approximation is a
number derived from the display format, which specifies the uncertainty for
the function. At the end of each iteration, the algorithm compares the
approximation calculated during that iteration with the approximations
calculated during two previous iterations. If the difference between any of
these three approximations and the other two is less than the uncertainty
tolerable in the final approximation, the calculations ends, leaving the current
approximation in the X–register and its uncertainty in the Y–register.
It is extremely unlikely that the errors in each of three successive
approximations — that is, the differences between the actual integral and the
approximations — would all be larger than the disparity among the
approximations themselves. Consequently, the error in the final
approximation will be less than its uncertainty (provided that f(x) does not
vary rapidly). Although we can't know the error in the final approximation,
the error is extremely unlikely to exceed the displayed uncertainty of the
approximation. In other words, the uncertainty estimate in the Y–register is an
almost certain "upper bound" on the difference between the approximation
and the actual integral.
Conditions That Could Cause Incorrect Results
Although the integration algorithm in the HP 32SII is one of the best available,
in certain situations it — like all other algorithms for numerical
integration—might give you an incorrect answer. The possibility of this
occurring is extremely remote. The algorithm has been designed to give
accurate results with almost any smooth function. Only for functions that
exhibit extremely erratic behavior is there any substantial risk of obtaining an
inaccurate answer. Such functions rarely occur in problems related to actual
physical situations; when they do, they usually can be recognized and dealt
with ire a straightforward manner.
Unfortunately, since all that the algorithm knows about f(x) are its values at the
sample points, it cannot distinguish between f(x) and any other function that
agrees with f(x) at all the sample points. This situation is depicted below,
D–2
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showing (over a portion of the interval of integration) three functions whose
graphs include the many sample points in common.
f (x)
x
With this number of sample pints, the algorithm will calculate the same
approximation for the integral of any of the functions shown. The actual
integrals of the functions shown with solid blue and black lines are about the
same, so the approximation will be fairly accurate if f(x) is one of these
functions. However, the actual integral of the function shown with a dashed
line is quite different from those of the others, so the current approximation
will be rather inaccurate if f(x) is this function.
The algorithm cores to know the general behavior of the function by sampling
the function at more and more points. If a fluctuation of the function in one
region is not unlike the behavior over the rest of the interval of integration, at
some iteration the algorithm will likely detect the fluctuation. When this
happens, the number of sample points is increased until successive iterations
yield approximations that take into account the presence of the most rapid,
but characteristic, fluctuations.
For example, consider the approximation of
More about Integration
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∞ xe−xdx
∫
0
Since you're evaluating this integral numerically, you might think that you
499
should represent the upper limit of integration as 10 , which is virtually the
largest cumber you ears key into the calculator.
–x
Try it and what happens. Enter the function f(x) = xe .
Keys:
Display:
Description:
{ G
Select equation mode.
Enter the equation.
End of the equation.
ꢎ
X
K y *
%ºꢈ%ꢅ1¾ꢎ
„ K X { ]
š
%ºꢈ%ꢅ1.%2ꢎ
Checksum and length.
{
†
ꢃꢚ/ꢏꢓꢘꢋ ꢕꢔꢕ)ꢗꢎ
Cancels Equation mode.
ꢎ
Set the display format to SCI 3, specify the lower and upper limits of
499
integration as zero and 100 , than start the integration.
Keys:
Display:
Description:
{
} 3
Specifies accuracy level and
limits of integration.
z ž
ꢃꢊ
ꢎ
0 š ` 499
ꢔꢈꢒꢓꢓ_ꢎ
Selects Equation mode; displays
the equation.
{ G
%ºꢈºꢅ1.%2ꢎ
X
Approximation of the integral.
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
{ )
∫ /ꢕ)ꢕꢕꢕꢈꢕꢎ
The answer returned by the calculator is clearly incorrect, since the actual
–x
integral of f(x) = xe from zero to
is exactly 1. But the problem is not that
∞
499
was represented by 10 , since the actual integral of this function from
∞
499
zero to 10
is very close to 1. The reasons or the incorrect answer becomes
apparent from the graph of f(x) over the interval of integration.
D–4
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f (x)
x
The graph is a spike very close to the origin. Because no sample point
happened to discover the spike, the algorithm assumed that f(x) was
identically equal to zero throughout the interval of integration. Even if you
increased the number of sample points by calculating the integral in SCI 11
or ALL format, none of the additional sample points would discover the spike
when this particular function is integrated over this particular interval. (For
better approaches to problems such as this, see the next topic, "Conditions
That Prolong Calculation Time.")
Fortunately, functions exhibiting such aberrations (a fluctuation that is
uncharacteristic of the behavior of the function elsewhere) are unusual
enough that you are unlikely to have to integrate one unknowingly. A function
that could lead to incorrect results can be identified in simple terms by how
rapidly it and its low–order derivatives vary across the interval of integration.
Basically, the more rapid the variation in the function or its derivatives, and
the lower the order of such rapidly varying derivatives, the less quickly will the
calculation finish, and the less reliable will be the resulting approximation.
Note that the rapidity of variation in the function (or its low–order derivatives)
must be determined with respect to the width of the interval of integration.
With a given number of sample points, a function f(x) that has three
More about Integration
D–5
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fluctuations can be better characterized by its samples when these variations
are spread out over most of the interval of integration than if they are
confined to only a small fraction of the interval. (These two situations are
shown in the following two illustrations.) Considering the variations or
fluctuation as a type of oscillation in the function, the criterion of interest is the
ratio of the period of the oscillations to the width of the interval of integration:
the larger this ratio, the more quickly the calculation will finish, and the more
reliable will be, the resulting approximation.
D–6
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f (x)
Calculated integral
of this function
will be accurate.
x
a
b
f (x)
Calculated integral
of this function
may be accurate.
x
a
b
In many cases you will be familiar enough with the function you want to
integrate that you will know whether the function has any quick wiggles
relative to the interval of integration. If you're not familiar with the function,
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and you suspect that it may cause problems, you can quickly plot a few points
by evaluating the function using the equation or program you wrote for that
purpose.
If, for any reason, after obtaining an approximation to an integral, you
suspect its validity, there's a simple procedure to verify it: subdivide the
interval of integration into two or more adjacent subintervals, integrate the
function over each subinterval, then add the resulting approximations. This
causes the function to be sampled at a brand new set of sample points,
thereby more likely revealing any previously hidden spikes. If the initial
approximation was valid, it will equal the sum of the approximations over the
subintervals.
Conditions That Prolong Calculation Time
In the preceding example, the algorithm gave an incorrect answer because it
never detected the spike in the function. This happened because the variation
in the function was too quick relative to the width of the interval of integration.
If the width of the interval were smaller, you would get the correct answer; but
it would take a very long time if the interval were still too wide.
Consider an integral where the interval of integration is wide enough to
require excessive calculation time, but not so wide that it would be calculated
–x
incorrectly. Note that because f(x) = xe approaches zero very quickly as x
approaches , the contribution to the integral of the function at large values
∞
of x is negligible. Therefore, you can evaluate the integral by replacing
,
∞
499
3
the upper limit of integration, by a number not so large as 10
— say 10 .
Rerun the previous integration problem with this new limit of integration:
Keys:
Display:
Description:
0 š ` 3
New upper limit.
ꢔꢈꢖ_ꢎ
Selects Equation mode; displays
the equation.
{ G
%ºꢈ%ꢅ1.%2ꢎ
X
Integral. (The calculation takes a
minute or two.)
{ )
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
∫ /ꢔ)ꢕꢕꢕꢈꢕꢎ
D–8
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Uncertainty of approximation.
Z
ꢔ)ꢙꢏꢒꢈ.ꢒꢎ
This is the correct answer, but it took a very long time. To understand why,
3
compare the graph of the function between x = 0 and x = 10 , which looks
about the same as that shown in the previous example, with the graph of the
function between x = 0 and x = 10:
f (x)
x
0
10
You can see that this function is "interesting" only at small values of x. At
greater values of x, the function is not interesting, since it decreases smoothly
and gradually in a predictable manner.
The algorithm samples the function with higher densities of sample points until
the disparity between successive approximations becomes sufficiently small.
For a narrow interval in an area where the function is interesting, it takes less
time to reach this critical density.
To achieve the same density of sample points, the total number of sample
points required over the larger interval is much greater than the number
required over the smaller interval. Consequently, several more iterations are
required over the larger interval to achieve an approximation with the same
accuracy, and therefore calculating the integral requires considerably more
time.
Because the calculation time depends on how soon a. certain density of
sample points is achieved in the region where the function is interesting, the
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calculation of the integral of any function will be prolonged if the interval of
integration includes mostly regions where the function is not interesting.
Fortunately, if you must calculate such an integral, you can modify the
problem so that the calculation time is considerably reduced. Two such
techniques are subdividing the interval of integration and transformation of
variables. These methods enable you to change the function or the limits of
integration so that the integrand is better behaved over the intervals) of
integration.
D–10 More about Integration
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E
Messages
The calculator responds to certain conditions or keystrokes by displaying a
message. The £ symbol comes on to call your attention to the message. For
significant conditions, the message remains until you clear it. Pressing
or
†
clears the message; pressing am other key clears the message and
@
executes that key's function.
A running program attempted t select a program
label ( label) while an integration calculation was
∫
∫
∫
ꢋꢄ ꢀꢃ!ꢊ#ꢈ
ꢋꢄ/
running.
A running program attempted to integrate a program
∫
1 ꢋꢄ2
(∫
variable) while another integration
ꢋꢄ G
calculation was running.
A running program attempted to solve a program
while an integration calculation was running.
1 ꢑꢂ#ꢈ2
The catalog of variables (
{
} )
z X
indicates no values stored.
ꢀꢂꢂ #ꢀꢁ /ꢕ
#ꢀꢁ
The calculator is executing a function that might take
ꢃꢀꢂꢃ"ꢂꢀ!ꢊꢄꢆꢎ
a while.
Allows you to verily clearing the equation you are
ꢃꢂꢁ ꢈꢉꢄ@ & ꢄ
ꢃꢂꢁ ꢅꢆꢇ @ & ꢄ
ꢍꢊ#ꢊꢍꢈ ꢌ& ꢕ
editing. (Occurs only in Equation–entry mode.)
in memory.
Allows you to verify clearing all program
(Occurs only in Program–entry mode.)
Attempted to divide by zero. (Includes S if
Y–register contains zero.)
Attempted to enter a program label that already exists
ꢍ"ꢅꢂꢊꢃꢀ!)ꢂꢌꢂꢎ
for another program routine.
Indicates the "top" of equation memory. The memory
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅ
scheme is circular, so
is also the
ꢈꢉꢄ ꢂꢊ ! !ꢑꢅ
"equation" after the last equation in equation
memory.
Messages
E–1
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The calculator is calculating the integral of an
ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ
equation or program. This might take a while.
A running SOLVE or FN operation was interrupted
∫
f
ꢊꢄ!ꢈꢁꢁ"ꢅ!ꢈꢍꢎ
by pressing
or
.
†
Data error:
ꢊꢄ#ꢀꢂꢊꢍ ꢍꢀ!ꢀꢎ
ꢀ
Attempted to calculate combinations or
permutations with r >n, with non–integer r or n, or
12
with n 10 .
≥
ꢀ
Attempted to use a trigonometric or hyperbolic
function with an illegal argument:
°
ꢀ
ꢀ
ꢀ
ꢀ
with x an odd multiple of 90 .
T
or
7 R
with x< –1 or x > 1.
O
L
with x –1; or x 1.
≤
≥
with x < 1.
7 O
A syntax error in the equation was detected during
ꢊꢄ#ꢀꢂꢊꢍ ꢈꢉꢄꢎ
equation evaluation, SOLVE, or ∫ FN.
Attempted a factorial or gamma operation with x as a
ꢊꢄ#ꢀꢂꢊꢍ º7ꢎ
negative integer.
º
ꢊꢄ#ꢀꢂꢊꢍ ¸ ꢎ
Exponentiation error:
th
ꢀ
ꢀ
ꢀ
Attempted to raise 0 to the 0 power or to a
negative power.
Attempted to raise a negative number to a
non–integer power.
Attempted to raise complex number (0 + i 0) to a
number with a negative real part.
Attempted an operation with an indirect address, but
ꢊꢄ#ꢀꢂꢊꢍ 1L2ꢎ
the number in the index register is invalid
i ≥ 34 ≤ i < 1
).
(
or
Attempted to take a logarithm of zero or (0 + i0).
Attempted to take a logarithm of a negative number.
All of user memory has been erased (see page B–3).
ꢂꢑꢆ1ꢕ2ꢎ
ꢂꢑꢆ1ꢄꢈꢆ2ꢎ
ꢇꢈꢇꢑꢁ& ꢃꢂꢈꢀꢁꢎ
The calculator has insufficient memory available to do
ꢇꢈꢇꢑꢁ& ꢋ"ꢂꢂꢎ
the operation (See appendix B).
The condition checked by a test instruction is not true.
(Occurs only when executed from the keyboard.)
ꢄꢑꢎ
E–2
Messages
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Attempted to refer to a nonexistent program label (or
ꢄꢑꢄꢈ%ꢊ !ꢈꢄ!ꢎ
line number) with
Note that the error
,
U U ΠW
ꢄꢑꢄꢈ%ꢊ !ꢈꢄ!
,
, or { }.
ꢋꢄ
can mean
ꢀ
you explicitly (from the keyboard) called a
program label that does not exist; or
ꢀ
the program that you called referred to another
label, which does not exist.
The catalog of programs (
{
} )
z X
indicates no program labels stored.
ꢄꢑ ꢂꢀꢌꢈꢂ ꢎ
ꢅꢆꢇ
SOLVE cannot find the root of the equation using the
current initial guesses (see page C–8). A SOLVE
operation executed in a program does not produce
this error; the same condition causes it instead to skip
the next program line (the line following the
ꢄꢑ ꢁꢑꢑ! ꢋꢄꢍꢎ
instruction
variable).
ꢑꢂ#ꢈ
Warning (displayed momentarily); the magnitude of
a result is too large for the calculator to handle. The
calculator returns 9.99999999999E499 in the
current display format. (See "Range of Numbers and
Overflow" on page 1–12.) This condition sets flag 6.
If flag 5 is set, overflow has the added effect of
halting a running program and leaving the message
in the display until you press a key.
ꢑ#ꢈꢁꢋꢂꢑ$ꢎ
Indicates the "top" of program memory. The memory
ꢅꢁꢆꢇ !ꢑꢅꢎ
scheme is circular, so
is also the "line"
ꢅꢁꢆꢇ !ꢑꢅ
after the last line in program memory.
The calculator is running a program (other than a
SOLVE or ∫ FN routine).
ꢁ"ꢄꢄꢊꢄꢆꢎ
Attempted to execute
variable or ∫
d
ꢈꢂꢈꢃ! ꢋꢄꢎ
ꢑꢂ#ꢈ
variable without a selected program label. This can
happen only the first time that you use SOLVE or FN
ꢋꢄ
∫
after the message
, or it can happen
ꢇꢈꢇꢑꢁ& ꢃꢂꢈꢀꢁ
if the current label no longer exists.
A running program attempted to select a program
ꢑꢂ#ꢈ ꢀꢃ!ꢊ#ꢈꢎ
label (
label) while a SOLVE operation was
ꢋꢄ/
running.
A running program attempted to solve a program
ꢑꢂ#ꢈ1 ꢑꢂ#ꢈ2ꢎ
while a SOLVE operation was running.
A running program attempted to integrate a program
∫
ꢑꢂ#ꢈ1 ꢋꢄ2ꢎ
Messages
E–3
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while a SOLVE operation was running.
The calculator is solving an equation or program for
its root. This might take a while.
ꢑꢂ#ꢊꢄꢆꢎ
Attempted to calculate the square root of a negative
ꢉꢁ! 1ꢄꢈꢆ2ꢎ
!ꢀ! ꢈꢁꢁꢑꢁꢎ
number.
S
tatistics error:
ꢀ
ꢀ
Attempted to do a statistics calculation with n = 0.
ˆ
y
Attempted to calculate s s ,
with n = 1.
,
, m, r, or b
xw
with x–data
ˆ
x y x
ꢀ
ꢀ
Attempted to calculate r,
only (all y–values equal to zero).
or
ˆ
x
ˆ
y
Attempted to calculate
x–values equal.
,
, r, m, or b with all
ˆ
x
The magnitude of the number is too large to be
converted to HEX, OCT, or BIN base; the number
must be in the range
!ꢑꢑ ꢌꢊꢆꢎ
–34,359,738,368 ≤ n ≤34,359,738,367.
A running program attempted an eighth nested
%ꢈꢉ ꢑ#ꢈꢁꢋꢂꢑ$ꢎ
%ꢈꢉ
label. (Up to seven subroutines can be nested.) Since
SOLVE and ∫ FN each uses a level, they can also
generate this error.
The condition checked by a test instruction is true.
(Occurs only when executed frown the keyboard.
&ꢈ ꢎ
Self–Test Messages:
The self–test and the keyboard test passed.
The self test or the keyboard test failed, and
the calculator requires service.
ꢖꢏ ꢊꢊ.ꢑꢚꢎ
ꢖꢏ ꢊꢊ.ꢋꢀꢊꢂ Qꢎ
Copyright message displayed after
ꢃꢑꢅꢁ) ꢐꢅ ꢙꢘ8ꢓꢕꢎꢎ
successfully completing the self test.
E–4
Messages
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F
Operation Index
This section is a quick reference for all functions and operations and their
formulas, where appropriate. The listing is in alphabetical order by the
function's name. This name is the one used in program lines. For example, the
function named FIX n is executed as
{
} n.
z ž
ꢋ%
Nonprogrammable functions have their names in key boxes. For example,
.
a
Non–letter and Greek characters are alphabetized before all the letters;
€
function names preceded by arrows (for example, DEG) are alphabetized
as if the arrow were not there.
ꢀ
The last column, marked , refers to notes at the end of the table.
Name
Keys and Description
Page
ꢀ
+/–
+
–
×
÷
Changes the sign of a number.
™ Addition. Returns y + x.
Subtraction. Returns y – x.
y Multiplication. Returns y × x.
1–10
1–13
1–13
1–13
1–13
6–17
1–2
1–7
6–3
12–6
1–20
6–3
1
1
1
1
1
2
_
„
Division. Returns y x.
p
÷
^
a
Power. Indicates an exponent.
0
Deletes the last digit keyed in; clears
x; clears a menu; erases last function
keyed in an equation; starts equation
editing; deletes a program step.
Displays previous entry in catalog;
moves to previous equation in
equation list; moves program pointer
to previous step.
z —
z ˜
12–19
Displays next entry in catalog; moves 1–20
Operation Index
F–1
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Name
Keys and Description
Page
ꢀ
to next equation in equation list;
moves program pointer to next line
(during program entry); executes the
current program line (not during
program entry).
6–3
12–9
12–19
1/x
10
Reciprocal.
1–12
4–2
1
1
3
x
Common exponential.
z (
Returns 10 raised to the × power.
%
Percent.
4–5
4–5
1
1
1
{ P
Returns (y × x) ÷ 100.
{ S Percent change.
%CHG
Returns (x – y)(100 y).
÷
{ M Returns the approximation
3.14159265359 (12 digits).
6 Accumulates (y, x) into statistics
π
4–3
Σ+
11–2
11–2
11–11
11–11
registers.
z 4
statistics registers.
–
Σ
Removes (y, z) from
Σx
Σx
{ 5 { }
Returns the sum of x–values.
1
1
º
ꢏ
2
{ 5 { }
º
Returns the sum of squares of
x–values.
{ 5 {
Σxy
}
11–11
1
º¸
Returns the sum of products of x–and
y–values.
{ 5 { }
Σy
11–11
11–11
1
1
¸
Returns the sum of y–values.
Σy2
ꢏ
{ 5 { }
¸
Returns the sum of squares of
y–values.
σx
{ 2 {σ }
11–7
1
º
Returns population standard
deviation of x–values:
(x − x)2 ÷ n
∑
i
F–2
Operation Index
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Name
Keys and Description
Page
ꢀ
σy
{ 2 {σ }
Returns population standard
deviation of y–values:
11–7
1
¸
(y − y)2 ÷ n
∑
{ rꢁ
i
θ, r y,x
4–7
€
Polar to rectangular coordinates.
Converts (r, ) to (x, y).
θ
ꢋꢄ G
∫ FN d variable
{ ) { ∫
_} variable
8–2
Integrates the displayed equation or
the program selected by FN=, using
lower limit of the variable of
14–7
integration in the Y–register and
upper limit of the variable if
integration in the X–register.
(
)
{ \ Open parenthesis.
Starts a quantity associated with a
function in an equation.
{ ] Close parenthesis.
Ends a quantity associated with a
function in an equation.
6–7
6–7
2
2
A through Z
ABS
variable or
variable
6–5
4–14
4–4
2
1
1
1
K
H
Value of named variable.
{
[PARTS] {
} Absolute value.
ꢀꢌ
x
z O
Returns
.
ACOS
Arc cosine.
Returns cos –1x.
z 7 z Oꢁ
ACOSH
4–5
Hyperbolic arc cosine.
–1
Returns cosh x.
ALOG
ALL
Common exponential.
6–17
1–16
4–4
2
z (
Returns 10 raised to the specified
power (antilogarithm).
{
}
z ž
ꢀꢂꢂ
Selects display of all significant
digits.
ASIN
Arc sine
1
z L
Operation Index
F–3
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Name
Keys and Description
Page
ꢀ
–1
Returns sin x.
z 7 z Lꢁ
ASINH
4–5
1
Hyperbolic arc sine.
–1
Returns sinh x.
z R Arc tangent.
ATAN
4–4
4–5
1
1
–1
Returns tan x.
z 7 z R
ATANH
Hyperbolic arc tangent.
–1
Returns tanh x.
b
{ }
{ ,
E
11–11
1
Returns the y–intercept of the
y
regression line:
Displays the base–conversion menu.
– mx.
10–1
10–1
z w
BIN
z w {
}
ꢌꢄ
Selects Binary (base 2) mode.
Turns on calculator; clears x; clears
messages and prompts; cancels
menus; cancels catalogs; cancels
equation entry; cancels program
entry; halts execution of an equation;
halts a running program.
1–1
1–3
1–7
1–20
6–3
12–6
12–18
5–5
†
/c
Denominator.
z ‹
Sets denominator limit for displayed
fractions to x. If x = 1, displays
current /c value.
°C
€
Converts ° F to ° C.
4–11
1
z ~
CF n
{
} n
13–12
{ x
Clears flag n (n = 0 through 11).
ꢃꢋ
z b
Displays menu to clear numbers or
parts of memory; clears indicated
variable or program from a MEM
catalog; clears displayed equation.
1–4
1–20
{
}
}
Clears all stored data, equations,
and programs.
Clears all programs (calculator in
Program mode).
1–20
z b
ꢀꢂꢂ
ꢅꢆꢇ
z b {
12–22
F–4
Operation Index
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Name
Keys and Description
Page
ꢀ
{
}
Clears the displayed equation
(calculator in Program mode).
12–6
z b
ꢈꢉꢄ
CL
{ }
11–12
3–3
Σ
z b
Clears statistics registers.
´
CLVARS
z b {
}
#ꢀꢁ
Clears all variables to zero.
CLx
{ }
z b
º
Clears x (the X-register) to zero.
2–2
2–7
12–6
4–11
CM
€
Converts inches to
1
z €
centimeters.
Displays the CMPLX_ prefix for
9–3
9–3
z F
complex functions.
z F _
CMPLX +/–
Complex change
sign.
x
z F „
Returns –(z + i z ).
y
CMPLX +
CMPLX –
Complex addition.
9–3
9–3
Returns (z + i z ) + (z + i z ).
1x
1y
2x
2y
z F „ Complex
subtraction.
1x
z F y Complex
Returns (z + i z ) – (z + i z ).
1y
2x
2y
CMPLX ×
9–3
multiplication.
Returns (z + i z ) × (z + i z ).
1x
1y
2x
2y
z F p Complex division.
CMPLX ÷
9–3
9–3
9–3
Returns (z + i z ) ÷ (z + i z ).
1x
z F 3
reciprocal. Returns 1/(z + i z ).
1y
2x
2y
CMPLX1/x
CMPLXCOS
Complex
x
y
Complex
z F Q
cosine.
Returns cos (z + i z ).
x
z F *
y
x
CMPLXe
9–3
9–3
Complex natural exponential.
+ izy )
e(z
x
Returns
.
z F -
CMPLXLN
Complex natural log.
Returns log, (z + i z ).
x
y
Operation Index
F–5
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
CMPLXSIN
Complex sine.
y
Complex
9–3
z F N
Returns sin (z + i z ).
y
z F T
CMPLXTAN
9–3
9–3
tangent.
Returns tan (z + i z ).
x
y
x
CMPLXy
Complex power.
z F 0
+ iz2y )
Returns (z1x + iz1y)(z
.
2x
Cn,r
[PROB] {
}
4–11
1
{
ꢃQ8T
Combinations of n items taken r at a
time.
Returns n! ÷ (r! (n – r)!).
COS
COSH
DEC
Cosine.
4–4
4–5
1
1
Q
Returns cos x.
z 7 Q Hyperbolic cosine.
Returns cosh x.
{
}
10–1
4–3
z w
Selects Decimal mode.
ꢍꢈꢃ
DEG
{
}
z Ÿ
Selects Degrees angular mode.
z u Radians to degrees.
Returns (360/2π) x.
Displays menu to set the display
format.
ꢍꢆ
DEG
€
4–10
1–15
13–17
1
z ž
DSE variable
variable
z m
Decrement, Skip if Equal or less. For
control number ccccccc.fffii stored in
a variable, subtracts ii (increment
value) from ccccccc (counter value)
and, if the result fff (final value),
≤
skips the next program line.
`
Be gins entry of exponents and adds
"E" to the number being entered.
Indicates that a power of 10 follows.
1–10
1–16
1
ENG n
z ž { } n
ꢈꢄ
Selects Engineering display with n
F–6
Operation Index
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
digits following the first digit (n = 0
through 11).
Separates two numbers keyed in
sequentially; completes equation
entry; evaluates the displayed
equation (and stores result if
appropriate).
1–11
6–4
6–12
š
ENTER
š
2–5
Copies x into the Y–register, lifts y
into the Z–register, lifts z into the
T–register, and loses t.
Activates or cancels (toggles)
Equation–entry mode.
* Natural exponential.
6–3
12–6
4–1
{ G
x
e
1
2
Returns e raised to the x power.
EXP
Natural exponential.
6–17
*
Returns e raised to the specified
power.
{
°F
Converts °C to °F.
4–11
5–1
1
€
z Š
Turn on and off Fraction–display
mode.
FIX n
z ž { } n
1–15
ꢋ%
Selects Fixed display with n decimal
places: 0 ≤ n ≤ 11.
{ x
Displays the menu to set, clear, and
test flags.
13–12
FN = label
label
14–1
14–7
{ V
Selects labeled program as the
current function (used by SOLVE and
∫ FN).
FP
{ [PARTS] { } Fractional part of
4–14
1
ꢋꢅ
x.
{ x
?
FS n
{
} n
13–12
ꢋ @
If flag n (n = 1 through 11) is set,
executes the next program line; if flag
n is clear, skips the next program
line.
GAL
€
Converts liters to gallons. 4–11
1
{ ƒ
Operation Index
F–7
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
GRAD
{
}
4–3
z Ÿ
Sets Grads angular mode.
ꢆꢁ
GTO label
label
13–5
13–16
z U
Sets the program pointer to the
beginning of program label in
program memory.
Sets program pointer to line nn of
12–20
z U Œ
label nn
z U Œ Œ
program label.
Sets program pointer to PRGM TOP. 12–20
HEX
{
}
10–1
z w
ꢐ%
Selects Hexadecimal (base :16)
mode.
Displays the HYP_ prefix for
hyperbolic functions.
z tꢁ
Hours to hours, minutes, seconds.
Converts x from a decimal fraction to
hours–minutes–seconds format.
z sꢁ
Hours, minutes, seconds to hours.
Converts x from
hours–minutes–seconds format to a
decimal fraction.
4–5
4–9
z 7
HMS
€
1
1
HR
€
4–9
i
i or
i
6–5
2
2
K
Value of variable i.
K ’ H ’
H
(i)
6–5
Indirect. Value of variable whose
letter corresponds to the numeric
value stored in variable i.
13–21
IN
Converts centimeters to
4–11
1
{
inches.
z ˆ
€
INPUT variable
variable
12–11
Recalls the variable to the X–register,
displays the variable's name and
value, and halts program execution.
Pressing
(to resume program
f
execution) or
(to execute
z ˜
the current program line) stores your
F–8
Operation Index
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
input in the variable. (Used only in
programs.)
INV
IP
Reciprocal of argument.
6–17
4–14
2
1
3
[PARTS] {
}
{
ꢊꢅ
Integer part of x.
ISG variable
variable
13–17
4–11
z k
Increment, Skip if Greater.
For control number ccccccc.fffii
stored in variable, adds ii (increment
value) to ccccccc (counter value) and,
if the result > fff (final value), skips the
next program line.
KG
€
Converts pounds to
1
1
z }
kilograms.
Converts gallons to liters.
z Žꢁ
4–11
2–8
z ‚
€
LASTx
L
Returns number stored in the LAST X
register.
{ |ꢁ
LB
4–11
12–3
1
€
Converts kilograms to pounds.
LBL label
label
z “
Labels a program with a single letter
for reference by the XEQ, GTO, or
FN= operations. (Used only in
programs.)
LN
Natural logarithm.
e
z + Common logarithm.
4–1
4–1
1
1
-
Returns log x.
LOG
Returns log x.
10
Displays menu for linear regression.
11–4
11–7
{ ,
m
z , { }
Returns the slope of the regression
1
P
2
y
line: [Σ(x –x )(y – )]÷Σ(x –x )
i
j
i
Displays the amount of available
memory and the catalog menu.
1–20
z X
z X {
}
}
Begins catalog of programs.
Begins catalog of variables.
12–21
3–3
ꢅꢆꢇ
{
z X
#ꢀꢁ
Operation Index
F–9
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
Displays menu to set Angular modes
1–14
4–3
z Ÿ
,
•
and the radix ( or ).
n
{ }
z 5
Q
Returns the number of sets of data
points.
11–11
1
OCT
{
}
10–1
1–1
z w
ꢑꢃ
Selects Octal (base 8) mode.
Turns the calculator off.
z ꢀ or
{ ꢀ
[PARTS]
Displays the menu for selecting parts
of numbers.
4–14
4–11
{
Pn,r
[PROB] {
}
1
{
ꢅQ8T
Permutations of n items taken r at a
time. Returns n! (n – r)!.
÷
Activates or cancels (toggles)
12–5
4–11
z d
Program–entry mode.
[PROB]
Displays the menu for probability
functions.
{
PSE
Pause.
12–17
12–18
{ e
Halts program execution briefly to
display x, variable, or equation, then
resumes. (Used only in programs.)
r
{ } Returns the correlation 11–7
coefficient between the x– and
y–values:
1
1
{ ,
T
(x − x)(y − y)
∑
i
i
(x − x)2 × (y − y)2
∑
i
i
RAD
z Ÿ {
}
4–3
ꢁꢍ
Selects Radians angular mode.
RAD
Degrees to radians.
4–10
1–14
z v
Returns (2π/360) x.
€
RADIX ,
{ }
z Ÿ
8
Selects the comma as the radix mark
(decimal point).
RADIX .
{ }
z Ÿ
)
1–14
F–10 Operation Index
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
Selects the period as the radix mark
(decimal point).
RANDOM
[PROB] { }
4–11
1
{
ꢁ
Executes the RANDOM function.
Returns a random number in the
range 0 through 1.
RCL variable
variable
3–1
K
Recall.
Copies variable into the X–register.
RCL+ variable
RCL– variable
RCLx variable
RCL÷ variable
RND
variable
3–5
3–5
3–5
3–5
K ™
Returns x + variable.
variable.
Returns x – variable.
variable.
Returns x × variable.
K „
K y
Round.
K p
Returns x variable.
÷
z I
Round.
4–14
5–8
1
Rounds x to n decimal places in FIX n
display mode; to n + 1 significant
digits in SCI n or ENG n display
mode; or to decimal number closest
to displayed fraction in
Fraction–display mode.
RTN
{ ” Return.
12–3
13–2
Marks the end of a program; the
program pointer returns to the top or
to the calling routine.
¶
R
R
Roll down.
2–3
2–3
9
Moves t to the Z–register, z to the
Y–register, y to the X–register, and x
to the T–register.
µ
Roll up.
{ 8
Moves t to the X–register, z to the
T–register, y to the T–register, and x
to the Y–register.
Displays the standard–deviation
Menu.
11–4
{ 2
Operation Index F–11
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
SCI n
{
} n
1–15
z ž
Selects Scientific display with n
ꢃ
decimal places. (n = 0 through 11.)
[SCRL]
Scroll. Enables and disables scrolling
of equations in Equation and
Program modes.
6–7
12–6
{
SEED
SF n
[PROB] {
}
4–11
{
ꢍ
Restarts the random–number
x
sequence with the seed
} n
.
{
13–12
z x
Sets flag n (n – 0 through 11).
ꢋ
Shows the full mantissa (all 12 digits) 6–20
{
of x (or the number in the current
program line); displays hex
12–22
checksum and decimal byte length
for equations and programs.
N Sine.
SIN
4–4
4–5
1
1
Returns sin x.
z 7 N
SINH
Hyperbolic sine.
Returns sinh x.
SOLVE variable
variable
7–1
14–1
{ œ
Solves the displayed equation or the
program selected by FN=, using
initial estimates in variable and x.
o
f Inserts a blank space character
during equation entry.
6–6
2
SQ
SQRT
STO variable
Square of argument.
< Square root of x.
6–17
1–12
3–1
2
1
z :
variable
Store. Copies x into variable.
H
STO + variable
STO – variable
STO × variable
STO ÷ variable
variable
3–4
3–4
3–4
3–4
H ™
Stores variable + x into variable.
variable
Stores variable – x into variable.
H „
variable
H y
Stores variable × x into variable.
variable
H p
F–12 Operation Index
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
Stores variable ÷ x into variable.
STOP
Run/stop.
12–18
f
Begins program execution at the
current program line; stops a running
program and displays the X–register.
Displays the summation menu.
11–4
11–6
{ 5
sx
{
Uº
}
1
1
{ 2
Returns sample standard deviation of
x–values:
(x − x)2 ÷ (n −1)
∑
{ 2
i
sy
{
U¸
}
11–6
Returns sample standard deviation of
y–values:
(y − y)2 ÷ (n −1)
∑
T
i
TAN
TANH
Tangent. Returns tan x.
4–4
4–5
1
1
z 7 T Hyperbolic tangent.
Returns tanh x.
VIEW variable
variable
3–2
12–14
{ ‰
Displays the labeled contents of
variable without recalling the value to
the stack.
Evaluates the displayed equation.
6–14
13–2
W
XEQ label
label
W
Executes the program identified by
label.
z :
2
x
Square of x.
The x root of y.
4–2
4–2
1
1
th
z .
X
y
x
z / { }
11–4
1
1
º
Returns the mean of x values:
Σ xi ÷ n.
{ ,
ˆ
{ }
11–11
ˆ
x
º
Given a y–value in the X–register,
Operation Index F–13
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Name
Keys and Description
Page
ꢀ
returns the x–estimate based on the
regression line:
= (y – b) ÷ m.
ˆ
x
x!
Factorial (or gamma).
4–11
1
z 1
Returns (x)(x – 1) ... (2)(1), or (x +
1).
z .
Γ
X ROOT
xw
The argument root of
argument .
6–17
11–4
11–4
3–6
2
1
1
2
Returns weighted mean of x values:
(Σy x ) ÷ Σy .
Displays the mean (arithmetic
average) menu.
i i
i
{ /
x<> variable
x<>y
x exchange.
{ Y
Exchanges x with a variable.
Z x exchange y.
Moves x to the Y–register and y to the
X–register.
2–4
?
Displays the "x y" comparison tests
13–8
13–8
z l
menu.
x≠y
z l {≠}
If x y, executes next program line;
if x=y, skips the next program line.
≠
?
x≤y
x<y
x>y
{≤}
13–8
13–8
13–8
13–8
13–8
13–8
z l
If x y, executes next program line;
≤
if x>y, skips next program line,
z l {<}
If x<y, executes next program line;
if x≥y, skips next program line.
?
?
{>}
z l
If x>y, executes next program line;
if x≤y, skips next program line.
?
x≥y
z l {≥}
If x≥y, executes next program line;
if x<y, skips the next program line.
?
x=y
{ }
z l
/
If x=y, executes next program line;
if x≠y, skips next program line.
?
Displays the "x 0" comparison tests
{ n
F–14 Operation Index
File name 32sii-Manual-E-0424
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Name
Keys and Description
Page
ꢀ
menu.
z n
If x≠0, executes next program line;
if x=0, skips the next program line.
?
?
x≠0
x≤0
x<0
x>0
{≠}
13–8
z n {≤}
13–8
13–8
13–8
13–8
13–8
11–4
11–11
If x 0, executes next program line;
≤
if x>0, skips next program line.
?
?
{<}
z n
If x<0, executes next program line;
if x≥0, skips the next program line.
z n {>}
If x>0, executes next program line;
if x≤0, skips the next program line.
?
x 0
{ }
≥
z n ≥
If x 0, executes next program line;
≥
if x<0, skips the next program line.
z n {=}
?
x=0
If x=0, executes next program line;
if x≠0, skips next program lire:
y
{ }
1
1
z /
¸
Returns the mean of y values.
Σy ÷ n.
i
z ,
ˆ
{ }
ˆ
y
¸
Given an x–value in the X–register,
returns the y–estimate based on the
ˆ
y
regression line:
z q
= m x + b.
y,x θ,r
Rectangular to polar
4–7
4–2
€
coordinates. Converts (x, y) to (r, θ).
x
y
Power.
1
0
th
Returns y raised to the x power.
Notes:
1. Function can be used in equations.
2. Function appears only in equations.
Operation Index F–15
File name 32sii-Manual-E-0424
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Index
ALL format. See display format
in equations, 6-6
Special characters
£, 1-21
in programs, 12-6
Setting, 1-17
@. See backspace key
alpha characters, 1-2
¤ annunciator, 1-1, A-2
angles
ƒ „ annunciators
between vectors, 15-1
converting format, 4-11
converting units, 4-11
implied units, 4-3, A-2
binary numbers, 10-7
equations, 6-8, 12-7, 12-16
_. See equation-entry cursor
¾. See digit-entry cursor
¡ annunciators, 1-2
angular mode, 4-3, A-2, B-5
annunciators
alpha, 1-2
ž annunciator
menus, 1-5
scrolling, 6-8, 12-7, 12-16
ST annunciator
in catalogs, 3-4, 5-4
in fractions, 3-4, 5-2, 5-3
Π(in fractions), 1-19, 5-1
). See integration
_, 1-11
battery, 1-1, A-2
descriptions, 1-8
flags, 13-11
list of, 1-9
low-power, 1-1, A-2
shift keys, 1-2
answers to questions, A-1
arithmetic
binary, 10-3
% functions, 4-6
∫ FN. See integration
π, 4-3, A-2
complex-number, 9-4
general procedure, 1-14
hexadecimal, 10-3
intermediate results, 2-13
long calculations, 2-13
octal, 10-3
A
order of calculation, 2-16
stack operation, 2-5, 9-2
absolute value (real number),4-15
assignment equations, 6-11, 6-12,
6-13, 7-1
addressing
indirect, 13-19, 13-20,.13-21
asymptotes of functions, C-9
Index–1
File name 32sii-Manual-E-0424
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converting to, 10-1
range of, 10-6
A..Z annunciator, 1-2, 3-2, 6-5
scrolling, 10-7
typing, 10-1
viewing all digits, 3-4, 10-7
B
borrower (finance), 17-1
backspace key
branching, 13-2, 13-15, 14-6
canceling VIEW, 3-4
clearing messages, 1-3, E-1
clearing X-register, 2-2, 2-8
deleting program lines, 12-20
equation entry, 1-3, 6-9
leaving menus, 1-3, 1-8
operation, 1-3
C
†
adjusting contrast, 1-1
canceling prompts, 1-3, 6-16,
12-14
program entry, 12-7
starts editing, 6-10, 12-7, 12-20
canceling VIEW, 3-4
clearing messages, .1-3, .E-1
clearing X-register, 2-2, 2-8
interrupting programs, 12-19
leaving catalogs, 1-3, 3-4
leaving Equation mode, 6-4, 6-5
leaving menus, 1-3, 1-8
leaving Program mode, 12-6, 12-7
on and off, 1-1
balance (finance), 17-1
base
affects display, 10-5
arithmetic, 10-3
converting, 10-1
default, B-5
programs, 12-25
setting, 10-1, 14-10
operation, 1-3
stopping integration, 8-2, 14-7
stopping SOLVE, 7-7, 14-1
BASE menu, 10-1
base mode
default, B-5
calculator
equations, 6-6, 6-13, 12-25
fractions, 5-2
programming, 12-25
setting, 12-25, 14-10
adjusting contrast, 1-11
default settings, B-5
environmental limits, A-2
questions about, A-1
repair service, A-7
resetting, A-4, B-3
self-test, A-5
shorting contacts, A-4
testing operation, A-4, A-5
turning on and off, 1-1
warranty, A-6
batteries, 1-1, A-2
Bessel function, 8-3
best-fit regression, 11-8, 16-1
BIN annunciator, 10-1
binary numbers. See numbers
arithmetic, 10-3
Index–2
File name 32sii-Manual-E-0424
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polynomial roots, 15-22
viewing, 9-2
cash flows, 17-1
catalogs
conditional tests, 13-6, 13-7, 13-8,
13-11, 13-16, 14-6
leaving, 1-3
program, 1-21, 12-22
using, 1-21
constant (filling stack), 2-7
Continuous Memory, 1-1
contrast adjustment, 1-1
conversion functions, 4-8
conversions
variable, 1-21, 3-4
chain calculations, 2-13
change-percentage function, 4-6
changing sign of numbers, 1-11,
1-14, 9-3
angle format, 4-11
angle units, 4-11
checksums
equations, 6-21, 12-7, 12-24
programs, 12-22, 12-23
coordinates, 4-8, 9-6, 15-1
length units, 4-12
mass units, 4-12
%CHG arguments, 4-7
number bases, 10-1
temperature units, 4-12
time format, 4-11
clearing
equations, 6-10
general information, 1-3
memory, 1-22, A-1
messages, 1-21
volume units, 4-12
coordinates
numbers, 1-11, 1-13
programs, 1-22, 12-23
statistics registers, 11-2, 11-13
variables, 1-22, 3-4, 3-5
X-register, 2-2, 2-7
converting, 4-5, 4-8, 15-1
transforming, 15-34
correlation coefficient, 11-8, 16-1
cosine (trig), 4-4, 9-3
cross product, 15-1
clearing memory, A-4, B-4
CLEAR menu, 1-4
F, 9-1, 9-3
cubic equations, 15-22
curve fitting, 11-8, 16-1
/c value, 5-6, B-5, B-8
combinations, 4-13
commas (in numbers), 1-16, A-1
D
comparison tests, 13-7 complex
numbers
Decimal mode. See base mode
decimal point,, 1-16, A-1
coordinate systems, 9-6
entering, 9-1
on stack, 9-2
degrees
angle units, 4-3, A-2
converting to radians, 4-11
operations, 9-1, 9-3
Index–3
File name 32sii-Manual-E-0424
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duplicating numbers, 2-6
ending equations, 6-5, 6-9, 6-10,
12-6
evaluating equations, 6-12, 6-13
separating numbers, 1-13, 1-15,
2-6
denominators
controlling, 5-6, 13-9, 13-13
range of, 1-19, 5-1, 5-3
setting maxim urn, 5-5
digit-entry cursor
backspacing, 1-3, 6-9, 12-7
in equations, 6-6
in programs, 12-7
meaning, 1-12
stack operation, 2-6
EQN annunciator
in equation list, 6-5, 6-8
in Program mode, 12-6
discontinuities of functions, C-6
EQN LIST TOP, 6-8, E-2
display
equality equations, 6-11, 6-12, 7-1
adjusting contrast, 1-1
annunciators, 1-8
function names in, 4-15
X-register shown, 2-2
equation-entry cursor
backspacing, 1-3, 6-9, 12-21
operation, 6-6
display format
equation list
affects integration, 8-2, 8-6, 8-8
affects numbers, 1-16
affects rounding, 4-15
default, B-5
periods and commas in, 1-16, A-1
setting, 1-16, A-1
adding to, 6-5
displaying, 6-8
editing, 6-10
EQN annunciator, 6-5
in Equation mode, 6-4
operation summary, 6-4
DISP menu, 1-16
"do if true", 13-6, 14-6
dot product, 15-1
DSE, 13-16
Equation mode
backspacing, 1-3, 6-9
during program entry, 12-6
leaving, 1-3, 6-4
shows equation list, 6-4
starting, 6-4, 6-8
E
` (exponent), 1-12
equations
and fractions, 5-10
as applications, 17-1
E in numbers, 1-11, 1-17, A-1
base mode, 6-6, 6-13, 12-25
checksums, 6-21, 12-7, 12-24, B-2
compared to RPN, 6-18, 12-4
controlling evaluation, 13-10
deleting, 1-4, 6-10
ENG format, 1-17. See also display
format
š
clearing stack, 2-6
copying viewed variable, 12-15
Index–4
File name 32sii-Manual-E-0424
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deleting in programs, 12-7, 12-20
displaying, 6-8
displaying in programs, 12-15,
12-18, 13-10
syntax, 6-16, 6-20, 12-15
TVM equation, 17-1
types of, 6-11
uses, 6-1
editing, 1-3, 6-9, 6-10
editing the programs, 12-7, 12-20
entering, 6-5, 6-9
entering in programs, 12-6
evaluating, 6-12, 6-13, 6-14, 7-6,
12-4, 13-10
functions, 6-6, 6-17, F-1
in programs, 12-4, 12-6, 12-7,
12-24, 13-10
variables in, 6-5, 7-1
with (i), 13-24
error messages, E-1
errors
clearing, 1-3
correcting, 2-9, E-1
estimation (statistical), 11-8, 16-1
executing programs, 12-10
integrating, 8-2
exponential curve fitting, 16-1
exponential functions, 1-12, 4-2, 9-3
exponents of ten, 1-11, 1-12
lengths, 6-21, 12-7, H-2
list of. See equation list
long, 6-8
memory usage, 12-22, B-2
messages in, 12-15
expression equations, 6-11, 6-12,
7-1
multiple roots, 7-8
no root, 7-7
no size, limit, 6,5
F
factorial function, 4-12
Š
numbers in, 6-6
numeric value of, 6-12, 6-13, 6-14,
7-1, 7-6, 12-4
operation summary, 6-4
parentheses, 6-6, 6-7, 6-16
polynomial, 15-22
not programmable, 5-10, 13-9,
13-13
toggles display mode, 1-20, 5-1,
A-2
precedence of operators, 6-16
prompt for values, 6-13, 6-15
prompting in programs, 13-10,
14-2, 14-8
toggles flag, 13-9
financial calculations, 17-1
FIX format, 1-16. See also display
format
roots, 7-1
flags
scrolling, 6-8, 12-7, 12-16
simultaneous, 15-13
solving, 7-2, C-1
annunciators, 13-11
clearing, 13-11
stack usage, 6-13
storing variable value, 6-13
default states, 13-8, B-5
equation evaluation, 13-10
Index–5
File name 32sii-Manual-E-0424
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equation prompting, 13-10
fraction display, 5-6, 13-9
meanings, 13-8
operations, 13-11
overflow, 13-9
setting, 13-11
testing, 13-8, 13-11
unassigned, 13-9
typing, 1-19, 5-1
functions
complex-number, 9-3
in equations, 6-6, 6-17, F-1
in programs, 12-7 list of, F-1
memory usage, 12-22, B-2
names in display, 4-15, 12-7
nonprogrammable, 12-24
one-number, 1-14, 2-9, 9-3
real-number, 4-1
FLAGS menu, 13-11
flow diagrams, 13-2
∫ FN. See integration
two-number, 1-15, 2-10, 9-3
future balance (finance), 17-1
FN=
in programs, 14-5, 14-9
integrating programs, 14-7
solving programs, 14-1
G
fractional-part function, 4-15
gamma function, 4-12
Fraction-display mode
affects rounding, 5-9
affects VIEW, 12-15
go to. See GTO
grads (angle units), 4-3, A-2
Grandma Hinkle, 11-7
grouped standard deviation, 16-19
setting, 1-20, 5-1., A-2
showing hidden digits, 3-3
fractions
U
accuracy indicator, 5-2, 5-3
and equations, 5-10
and programs, 5-10, 12-15
base 10 only, 5-2
finds PRGM TOP, 12-6, 12-21,
13-5
finds program labels, 12-10,
12-21, 13-5
calculating with, 5-1
denominators, 1-19, 5-5, 5-6, 13-9,
13-3
displaying, 1-20, 5-1, 5-2, 5-5, A-2
flags, 5-6, 13-9 formats, 5-6
not statistics registers, 5-2
reducing, 5-3, 5-6
rounding, 5-9
round-off, 5-4, 5-9
setting format, 5-6, 13-9, 13-13
showing integer digits, 3-3, 5-5
finds program lines, 12-20, 12-21,
13-5
GTO, 13-4, 13-16
guesses (for SOLVE), 7-2, 7-6, 7-7,
7-10, 14-5
H
help about calculator, A-1
hexadecimal numbers. See hex
Index–6
File name 32sii-Manual-E-0424
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limits of, 8-2, 14-7, D-7
memory usage, 8-2, 12-22, B-2,
B-3
numbers
HEX annunciator, 10-1
hex numbers. See numbers arithmetic,
10-3
purpose, 8-1
restrictions, 14-10
results on stack, 8-2, 8-7
resuming, 14-7
stopping, 8-2, 14-7
subintervals, D-7, D-9
time required, 8-6, D-7
transforming variables, D-9
uncertainty of result, 8-2, 8-6, 8-7,
D-2
converting to, 10-1
range of, 10-6
typing, 10-1
Horner's method, 12-26
humidity limits for calculator, A-2
hyperbolic functions, 4-6
I
using, 8-2
variable of, 8-2
i, 3-8, 13-19
intercept (curve-fit), 11-8, 16-1
interest (finance), 17-3
(i), 3-8, 13-19, 13-20, 13-24
imaginary part (complex numbers),
9-1, 9-2
intermediate results, 2-13
inverse function, 1-14, 9-3
inverse hyperbolic functions, 4-6 .
inverse-normal distribution, 16-12
inverse trigonometric functions, 4-4
ISG, 13-16
indirect addressing, 13-19, 13-20,
13-21
INPUT
always prompts, 13-10
entering program data, 12-12
in integration programs, 14-8
in SOLVE programs, 14-2
responding to, 12-14
K
keys
showing hidden digits, 12-14
integer-part function, 4-15
alpha, 1-2
letters, 1-2
shifted, 1-2
top-row actions, 6-8, 12-7
integration
accuracy, 8-2, 8-6, 8-7, D-2
base mode, .12-25, 14-10
difficult functions, D-2, D-7
display format, 8-2, 8-6, 8-8
evaluating programs, 14-7
how it works, D-1
L
LASTx function, 2-9
LAST X register, 2-9, B-8
in programs, 14-9
interrupting, B-3
Index–7
File name 32sii-Manual-E-0424
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variable catalog, 1-21, 3-4
memory
lender (finance), 17-1
length conversions, 4-12
letter keys, 1-2
amount available, 1-21, B-2
clearing, 1-4, 1-22, A-1, A-4, B-1,
11-4
clearing equations, 6-10
clearing programs, 1-22, 12-6,
12-23
clearing statistics registers, 11-2,
11-13
clearing variables, 1-22, 3-5
contents, 1-21
limits of integration, 8-2, 14-7
linear regression (estimation), 11-8,
16-1
linear-regression menu, 11-8
logarithmic curve fitting, 16-1
logarithmic functions, 4-2, 9-3
loop counter, 13-16, 13-17, 13-21
looping, 13-15, 13-16
deallocating, B-3
equations, B-2
full, A-1
Łukasiewicz, 2-1
integration usage, 8-2
maintained while off, 1-1
programs, 12-21, 12-22, B-3
size, 1-21, B-1
M
mantissa, 1-12, 1-18
mass conversions, 4-12
math
stack, 2-1
statistics registers, 11-13
usage, 12-22, B-1, B-2
variables, 3-5
complex-number, 9-1, 9-4
general procedure, 1-14
intermediate results, 2-13
long calculations, 2-13
order of calculation, 2-16
real-number, 4-1
MEMORY CLEAR, A-4, B-4, E-3
MEMORY FULL, B-1, E-3
menu keys, 1-5
stack operation, 2-5, 9-2
menus
matrix inversion, 15-13
maximum of function, C-9
mean menu, 11-4
example of using, 1-7
general operation, 1-5
leaving, 1-3, 1-8
list of, 1-6
means (statistics)
messages
calculating, 11-4
normal distribution, 16-12
clearing, 1-3, 1-21
displaying, 12-15, 12-18
in equations, 12-15
responding to, 1-21, E-1
summary of, E-1
X
program catalog, 1-21, 12-22
reviews memory, 1-21
Index–8
File name 32sii-Manual-E-0424
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negative, 1-11, 9-3, 10-5
order in calculations, 1-15
periods and commas in, 1-16, A-1
precision, 1-16, C-16
prime, 17-7
range of, 1-13, 10-6
real, 4-1, 8-1
recalling, 3-2
reusing, 2-6, 2-11
rounding, 4-15
minimum of function, C-9
modes. See angular mode, base
mode, Equation mode,
Fraction-display mode,
Program-entry mode
MODES menu
angular mode, 4-4
setting radix, 1-1.6
money (finance), 17-1
showing all digits, 1-18, 10-8
storing, 3-2
truncating, 10-5
N
negative numbers, 1-11, 9-3, 10-5
nested routines, 13-3, 14-10
normal distribution, 16-12
typing, 1-11, 1-12, 10-1
O
numbers. See binary numbers, hex
numbers, octal numbers,
variables
octal numbers. See numbers
arithmetic, 10-3
converting to, 10-1
range of, 10-6
typing, 10-1
bases, 10-1, 12-25
changing sign of, 1-11, 1-14, 9-3
clearing, 1-3, 1-4, 1-11, 1-13
complex, 9-1
decimal places, 1-16
display format, 1-16, 10-5
doing arithmetic, 1-14
editing, 1-3, 1-11, 1-13
E in, 1-11, 1-12, A-1
exchanging, 2-4
OCT annunciator, 10-1
ꢀ, 1-1
one-variable statistics, 11-2
overflow
flags, 13-9, E-4
result of calculation, 1-13, 10-3,
10-6
setting response, 13-9, E-4
testing occurrence, 13-9
finding parts of, 4-15
fractions in, 1-19, 5-1
in equations, 6-0i
in programs, 12-6
P
internal representation, 1-16, 10-5
large and small, 1-11, 1-13
limitations, 1-11
mantissa, 1-12
memory usage, 12-22, B-2
π, 4-3, A-2
parentheses
in arithmetic, 2-13
Index–9
File name 32sii-Manual-E-0424
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in equations, 6-6, 6-7, 6-16
memory usage, 12-22
clearing, 12-6
duplicate, 12-6
entering, 12-3, 12-6
executing, 12-10
indirect addressing, 13-19, 13-20,
13-21
moving to, 12-10, 12-21
purpose, 12-3
PARTS menu, 4-15
pause. See PSE
payment (finance), 17-1
percentage functions, 4-6
periods (in numbers), 1-16, A-1
permutations, 4-13
typing name, 1-2
viewing, 12-22
polar-to-rectangular coordinate
conversion, 4-8, 9-6, 15-1
program lines. See programs
program names. See program labels
poles of functions, C-6
program pointer, 12-6, 12-10, 12-11,
12-19, 12-21, B-5
polynomials, 12-26, 15-22
population standard deviations, 11-7
power annunciator, 1-1, A-2
power curve fitting, 16-1
programs. See program labels
base mode, 12-25
branching, 13-2, 13-4, 13-6,
13-15
power functions, 1-12, 4-2, 9-4
calculations in, 12-13
calling routines, 13-2, 13-3
catalog of, 1-21, 12-22
checksums, 12-22, 12-23, B-3
clearing, 12-6, 12-22, 12-23
clearing all, 12-6, 12-23
comparison test, 13-7
conditional tests, 13-6, 13-7, 13-8,
13-11, 13-16, 14-6
data input, 12-5, 12-12
data output, 12-5, 12-12, 12-14,
12-18
deleting, 1-22
deleting all, 1-4
precedence (equation operators),
6-16
precision (numbers), 1-16, 1-18,
C-16
present value, See financial
calculations
PRGM TOP, 12-4, 12-6, 12-21, E-4
prime number generator, 17-7
probability
functions, 4-12
normal distribution, 16-12
PROB menu, 4-13
deleting equations, 12-7, 12-20
deleting lines, 12-20
designing, 12-3, 13-1
editing, 1-3, 12-7, 12-20
editing equations, 12-7, 12-20
program catalog, 1-21, 12-22
Program-entry mode, 1-3, 12-6
program labels
branching to, 13-2, 13-4, 13-15
checksums, 12-23
Index–10
File name 32sii-Manual-E-0424
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entering, 12-5
using SOLVE, .14-5
variables in, 12-12, 1.4-1, 14-7
equation evaluation, 13-10
equation prompting, 13-10
equations in, 12-4, 12-6
errors in, 12-19
executing, 12-10
flags, 13-8, 13-11
for integration, 14-7
for SOLVE, 14-1, C-1
fractions with, 5-10, 12-15, 13-9
functions not allowed, 12-24
indirect addressing, 13-19, 13-20,
13-21
prompts
affect stack, 6-16, 12-13
clearing, 1-3, 6-16, 12-14
equations, 6-15
INPUT, 12-12, 12-14, 14-2, 14-8
programmed equations, 13-10,
14-2, 14-8
responding to, 6-15, 12-14
showing hidden digits, 6-16,
12-14
PSE
inserting lines, 12-6, 12-20
interrupting, 12-19
pausing programs, 12-12, 12-19,
14-9
lengths, 12-22, 12-23, B-3
line numbers, 12-3, 12-20, 12-21
loop counter, 13-16, 13-17
looping, 13-15, 13-16
memory usage, 12-22, B-2
messages in, 12-15, 1.2-18
moving through, 12-11
not stopping, 12-18
preventing program stops, 12-18,
13-10
Q
quadratic equations, 15-22
questions, A-1
numbers in, 12-6
pausing, 12-19
R
prompting for data, 12-12
purpose, 12-1
resuming, 1.2-15
return at end, 12-4
routines, 13-1
RPN operations, 12-4
running, 12-10, 12-22
showing long number, 12-6
stepping through, 12-10
stopping, 12-14, 12-16, 12-19
techniques, 13-1
R¶ and Rµ, 2-3
radians
angle units, 4-3, A-2
converting to degrees, 4-11
radix mark, 1-16, A-1
random numbers, 4-13, B-5
RCL, 3-2, 12-13
RCL arithmetic, 3-6, B-8
real numbers
integration with, 8-1
operations, 4-1
SOLVE with, 14-2
testing, 12-10
using integration, 14-9
Index–11
File name 32sii-Manual-E-0424
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12-4
real part (complex numbers), 9-1, 9-2
recall arithmetic, 3-6, B-8
in programs, 12-4 origins, 2-1
f
rectangular-to-polar coordinate
conversion, 4-8, 9-6, 15-1
ending prompts, 6-13, 6-15, 7-2,
12-14
interrupting programs, 12-19
resuming programs, 12-15,
12-16, 12-19
running programs, 12-22 stopping
integration, 8-2, 14-7
stopping SOLVE, 7-7, 14-1
regression (linear), 11-8, 16-1
repair service, A-7
resetting the calculator, A-4, B-3
return (program). See programs
Reverse Polish Notation. See RPN
rolling the stack, 2-3
running programs, 12-10, 12-22
root functions, 4-2
roots. See SOLVE
checking, 7-6, C-3
in programs, 14-5
multiple, 7-8
S
sample standard deviations, 11-6
SCI format. See display format
in programs, 12-6
none found, 7-7, C-9
of equations, 7-1
of programs, 14-1
polynomial, 15-22
quadratic, 15-22
setting, 1-17
[SCRL], 6-8, 12-7
scrolling
binary numbers, 1.0-7
equations, 6-8, 12-7, 12-16
rounding
fractions, 5-9, 12-18
numbers, 4-15
round-off
seed (random number), 4-13
self-test (calculator), A-5
service, A-7
fractions, 5-4, 5-9
integration, 8-6
SOLVE, C-16
shift keys, 1-2
statistics, 11-11
trig functions, 4-4
equation checksums, 6-21, R-2
equation lengths, 6-21, B-2
fraction digits, 3-3, 5-5
number digits, 1-18, 12-6
program checksums, 12-22, 12-23,
B-3
routines
calling, 13-2
nesting, 13-3, 14-10
parts of programs, 13-1
program lengths, 12-23, B-3
prompt digits, 6-16, 10-8, 12-14
RPN
compared to equations, 6-18,
Index–12
File name 32sii-Manual-E-0424
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variable digits, 3-3, 3-4, 10-8,
12-15
o, 6-6, 6-18
square function, 1-14, 4-2
sign conventions (finance), 17-1
square-root function, 1-14 stack. See
stack lift
sign (of numbers), 1-11, 1-14, 9-3,
10-5
affected by prompts, 6-16, 12-13
complex numbers, 9-2
simultaneous equations, 15-13
sine (trig), 4-4, 9-3, A-2
effect of
, 2-6
š
equation usage, 6-13
exchanging with variables, 3-8
exchanging X and Y, 2-4
filling with constant, 2-7
long calculations, 2-13
operation, 2-1, 2-5, 9-2
program calculations, 12-13
program input, 12-12
program output, 12-12
purpose, 2-1, 2-2
registers, 2-1
reviewing, 2-3
rolling, 2-3
separate from variables, 3-2
size limit, 2-4, 9-2
single-step execution, 12-10
slope (curve-fit), 11-8, 16-1
SOLVE
asymptotes, C-9
base mode, 12-25, 14-10
checking results, 7-6, C-3
discontinuity, C-6
evaluating equations, 7-1, 7-6
evaluating programs, 14-1
flat regions, C-9
how it works, 7-6, C-1
initial guesses, 7-2, 7-6, 7-7, 7-10,
14-5
in programs, 14-5
interrupting, 8-3
unaffected by VIEW, 12-15
memory usage, 12-22, B-2, B-3
minimum or maximum, C-9
multiple roots, 7-8
no restrictions, 14-10
no root found, 7-7, 14-6, C-9
pole, C-6
stack lift. See stack
default state, B-5
disabling, B-6
enabling, B-6
not affecting, B-7
operation, 2-5
purpose, 7-1
standard-deviation menu, 11-6, 11-7
real numbers, 14-2
results on stack, 7-2, 7-6, C-3
resuming, 14-1
round-off, C-16
stopping, 7-2, 7-7
standard deviations
calculating, 11-6, 11-7
grouped data, 16-19
normal distribution, 1.6-12
statistical data. See statistics registers
clearing, 1-4, 11-2
underflow, C-16
using, 7-2
correcting, 11-2
Index–13
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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entering, 11-1
initializing, 11-2
T
tangent (trig), 4-4, 9-3, A-2
memory usage, 12-22, B-2
one-variable, 11-2
precision, 11-11
sums of variables, 11-12
two-variable, 11-2
temperatures
converting units, 4-12
limits for calculator, A-2
testing the calculator, .A-4, A-5
test menus, 13-7
statistics
calculating, 11-4
time formats, 4-11
curve fitting, 11-8, 16-1
distributions, 16-12
grouped data, 16-19
one-variable data, 11-2
operations, 11-1
time value of money, 17-1
transforming coordinates, 15-34
T-register, 2-5, 2-7
trigonometric functions, 4-4, 9-3
troubleshooting, A-4, A-5
turning on and off, 1-1
TVM, 17-1
two-variable data, 11-2
statistics menus, 11-1, 1.1-4
statistics registers- See statistical data
accessing, 11-14
twos complement, 10-3, 10-5
two-variable statistics, 11-2
clearing, 1-4, 11-2, 11-13
contain summations, 11-1, 11-12,
11-14
correcting data, 11-2
initializing, 11-2
memory, 11-13
memory usage, 12-22, B-2
no fractions, 5-2
viewing, 11-12
U
uncertainty (integration), 8-2, 8-6,
8-7
underflow, C-16
units conversions, 4-12
STO, 3-2, 12-12
STO arithmetic, 3-5
STOP, 12-19
V
storage arithmetic, 3-5
variable catalog, 1-21, 3-4
subroutines. See routines sums of
statistical variables, 11-12
variables
arithmetic inside, 3-5
catalog of, 1-21, 3-4
clearing, 1-22, 3-4, 3-5
clearing all, 1-4, 3-5
syntax (equations), 6-16, 6-20,
12-15
Index–14
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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clearing while viewing, 12-15
default, B-5
no stack effect, 12-15
stopping programs, 12-14
exchanging with X, 3-8
indirect addressing, 13-19, 13-20
in equations, 6-5, 7-1
in programs, 12-12, 14-1, 14-7
memory usage:, 12-22, B-2
names, 3-1
number storage, 3-1
of integration, 8-2, 14-7
polynomials, 12-26
volume conversions, 4-12
w
warranty, A-6
weight conversions, 4-12
weighted means, 11-4 windows
(binary numbers), 10-7
program input, 12-13
program output, 12-14, 12-18
recalling, 3-2, 3-4
separate from stack, 3-2
showing all digits, 3-3, 3-4, 10-8,
12-15
solving for, 7-2, 14-1, 14-5, C-1
storing, 3-2
storing from equation, 6-13
typing name, 1-2
X
W
evaluating equations, 6-12, 6-14
running programs 12-10, 12-22
X-register
affected by prompts, 6-16
arithmetic with variables, 3-5
clearing, 1-4, 2-2, 2-7
clearing in programs, 12-7
displayed, 2-2
during programs pause, 12-19
exchanging with variables, 3-8
exchanging with Y, 2-4
not clearing, 2-5 part of stack, 2-1
testing, 13-7
viewing, 3-3, 12-14, 12-18
vectors
application program, 15-1
coordinate conversions, 4-10, 9-7,
15-1
operations, 15-1
VIEW
unaffected by VIEW, 12-15
displaying program data, 12-14,
12-18, 14-5
X ROOT arguments, 6-18
displaying variables, 3-3, 10-8
Index–15
File name 32sii-Manual-E-0424
Printed Date : 2003/4/24
Size : 17.7 x 25.2 cm
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Batteries are delivered with this product,
when empty do not throw them away but
correct as small chemical waste.
Bij dit produkt zijn batterijen. Wanneer
deze leeg zijn, moet u ze niet weggooien
maar inleveren aIs KCA.
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