HP (Hewlett Packard) 32SII User Manual

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

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

Contents

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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

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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

Contents

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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

Contents

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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

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

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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

Contents

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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

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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

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

†

™

„.

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

{

example, to turn the calculator off, press and release the

shift key, then

{

press

†

^Pressingz ^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.

ꢀ

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.

ꢀ

Equation–entry mode:

Erases the character immediately to the left of " "

(the equation–entry cursor). If a number entry in

your equation is complete,

erases the entire

number. If the number is not complete,

erases

the character immediately to the left of "_" (the

number–entry cursor. "_" changes back to " "

when number entry is complete.

also clears error messages, and deletes the

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

Name

Description

Chapter

Numeric Functions

PARTS

ꢊꢅ ꢋꢅ ꢀꢌ

Number–altering functions: integer part,

fractional part, and absolute value.

PROB

L.R.

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·

Arithmetic mean of statistical x– and y–values;

weighted mean of statistical x–values.

s,σ

Uº U¸ º ¸ꢎ

Sample standard deviation, population

standard deviation.

Q º ¸ º^ꢏ¸^ꢏº¸

SUMS

BASE

Statistical data summations.

ꢍꢈꢃ ꢐ% ꢑꢃ ꢌꢄ

Base conversions (decimal, hexadecimal,

octal, and binary).

Programming Instructions

FLAGS

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)

Name

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.

)

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

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

at a time. Refer to

in the table on page 1–4.

z b

ꢀ

Pressing

cancels any other menu.

†

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:

The

and

keys are

z ˜

z —

active for stepping through a list.

When in Fraction–display mode

(press

), only one of the

z Š

" " 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

neither part of "

"' is on, the

exact value of the fraction is being

displayed.

Left shift is active.

Right shift is active.

PRGM

Program–entry is active. Blinks while

program is running.

EQN

Equation–entry mode is active, or the

calculator is evaluating an expression

or executing an equation.

0 1 2 3

Indicates which flags are set (flags 4

through 11 have no annunciator.

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.

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.

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 (

) to

see the leading or trailing characters.

A..Z

The alphabetic keys are active.

Attention! Indicates a special condition

or an error.

Battery power is low.

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

and delete the last digit, or press

to clear the whole number.

†

Getting Started

1–9

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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_

ꢕ)ꢕꢕꢕꢔꢎ

.000042

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.

1–10 Getting Started

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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

) ꢏ ꢏꢈ.ꢖꢒ_

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).

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

123

A completed number.

ꢔꢏꢖ)ꢕꢕꢕꢕꢎ

ꢔꢏꢘ)ꢕꢕꢕꢕꢎ

Another completed number.

™

If digit entry is not terminated (if the cursor is present),

backspaces to

erase the last digit. If digit entry is terminated (no cursor),

acts like

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

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

2. Press the function key. (For a shifted function, press the appropriate

shift key first.)

{

For example, calculate 1/32 and

and change its sign.

Then square the last result

148.84

Keys:

Display:

Description:

Operand.

ꢖꢏ_

Reciprocal of 32.

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

{

1. Key in the first number.

2. Press

to separate the first number from the second.

3. Key in the second number. (Do not press

4. Press the function key. (For a shifted function, press the appropriate shift

key first.)

Type in both cumbers (separate them by pressing

Note

before pressing a function key.

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For example:

To calculate:

Press:

Display:

123 + 3

12 – 3

š ™

ꢔꢗ)ꢕꢕꢕꢕꢎ

ꢓ)ꢕꢕꢕꢕꢎ

š „

12 3

š y

ꢖ )ꢕꢕꢎ

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

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 { }.

)

For example, the number one million looks like:

ꢀ

if you press { } or

ꢔ8ꢕꢕꢕ8ꢕꢕꢕ8ꢕꢕꢕꢕ

)

ꢀ

if you press { }.

ꢔ)ꢕꢕꢕ)ꢕꢕꢕ8ꢕꢕꢕꢕ

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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

)

ꢋꢊ%

the number of decimal places to be displayed. For 10 or 11 places, press

0 or

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

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

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" 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

)

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

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

(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

1.3

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.

ꢕ)ꢕꢔꢘꢔꢎ

(hold)

Shows full precision until you release

ꢔꢘꢕꢓꢒꢕꢔꢘꢕꢓꢒꢕꢎ

{

Fractions

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

separates the integer part of the number from its fractional part.)

2. Key in the fraction numerator and press again. The second

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

twice for fraction entry.

For example, to enter the fractional number 12 / , press these keys:

Keys:

Display:

Description:

Enters the integer part of the number.

ꢔꢏ_ꢎ

The

key is interpreted in the normal

ꢔꢏ)_

manner.

Enters the numerator of the fraction (the

number is still displayed in decimal

form).

ꢔꢏ)ꢖ_ꢎ

The calculator interprets the second Œ

as a fraction and separates the

numerator from denominator.

Appends the denominator of the

fraction.

ꢔꢏ)ꢖ+_ꢎ

ꢔꢏ)ꢖ+ꢙ_ꢎ

Terminates digit entry; displays

the number in the current display format.

ꢔꢏ)ꢖꢘꢗꢕꢎ

If the number you enter has no integer part (for example, / ), just start the

number without an integer.

Keys:

Display:

Description:

Œ Œ

Enters no integer part. (3

Œ Œ

ꢕ ꢖ+ꢙꢎ

also works.)

Terminates digit entry; displays the

number in the current display format

(FIX 4).

ꢕ)ꢖꢘꢗꢕꢎ

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Displaying Fractions

^PresszꢁŠ to switch between Fraction–display mode and the current

decimal display mode.

Keys:

Display:

Description:

Œ Œ

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

ꢔꢏ)ꢖꢘꢗꢕꢎ

z Š

ꢔꢏ ꢖ+ꢙꢎ

Now add 3/4 to the number in the X–register (12 / ):

Keys:

Display:

Description:

Œ Œ

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

†

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

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. ^Pressz 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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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

logician Jan ukasiewicz (1878–1956).

While conventional algebraic notation places the operators between the

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

2–1

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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

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

z b

{ } only when the X–register is displayed.

{ } when the X–register is displayed and digit entry is

also acts like

z b

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

"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}

back to where they started:

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.

R (Roll Up)

The (roll up) key has a similar function to

except that it "rolls" the

{ 8

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.

The Automatic Memory Stack

2–3

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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

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

The

function is used primarily for two purposes:

ꢀ

To view the contents of the Y–register and then return them to y (press

twice).

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 ⁸y ⁹Z p

The keystrokes to calculate this expression from left–to–right are:

š y p

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

The Automatic Memory Stack

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3 + 4 – 9

–2

™

ꢃꢁ

„

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)

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

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

2–5

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5 + 6

1 lost

2 lost

ꢄꢁ

ꢅꢁ

™

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

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

. 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

number to itself, press

š ™

Filling the to with a Constant

The re

plicating effect of

together with the replicating effect of stack

drop (from T into Z) allows you t fill the stack with a numeric constant for

calculations.

2–6

The Automatic Memory Stack

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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

1.5

100

1.5

150

1.5

225

1.5

šꢁ

1.5

337.5

100

yꢁ

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.)}

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

The Automatic Memory Stack

2–7

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cancel that display and shows the X–register.

†

ꢀ

When viewing an equation,

equation to allow for editing.

displays the cursor at the end the

During equation entry,

backspaces over the displayed equation,

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:

ꢆ

ꢇ

†ꢁ

ꢂ

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.

2–8

The Automatic Memory Stack

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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

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:

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

), you can correct it by using

and inverse of the

.ꢁ

z Ž

two–number function (

„ꢁ ™ 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:

The Automatic Memory Stack

2–9

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

Suppose you made a mistake while calculating

16 × 19 = 304.

There are three kinds of mistakes you could have made:

Wring

Calculation:

Mistake:

Correction:

Wrong function

„

z Ž ™ꢁ

z Ž y

Wrong first number 16

Wrong second

z Ž y

z Ž p

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

2–10 The Automatic Memory Stack

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

96.704

52.3947

š X

149.0987

52.3947

™

LAST X

52.3947

149.0987

52.3947

z Ž X

2.8457

LAST X

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.

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:

The Automatic Memory Stack 2–11

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

To Rigel Centaurus: 4.3 yr (9.5 10 m/yr).

To Sirius: 8.7 yr × (9.5 × 10 m/yr).

Keys:

Display:

Description:

4.3

Light–years to Rigel Centaurus.

Speed of light, c.

ꢒ)ꢖꢕꢕꢕꢎ

^9.5` ¹⁵

ꢓ)ꢗꢈꢔꢗꢎ

Meters to R. Centaurus.

Retrieves c.

ꢒ)ꢕꢙꢗꢕꢈꢔ ꢎ

ꢓ)ꢗꢕꢕꢕꢈꢔꢗꢎ

ꢙ)ꢏ ꢗꢕꢈꢔ ꢎ

8.7

z Ž

Meters to Sirius.

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:

š ™

Calculates the intermediate result first.

ꢔꢗ)ꢕꢕꢕꢕꢎ

2–12 The Automatic Memory Stack

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Size : 17.7 x 25.2 cm

You don't need to press

to save this intermediate result before

proceeding; since it is a calculated result, it is saved automatically.

Keys:

Display:

Description:

Pressing the function key produces the

ꢔꢕꢗ)ꢕꢕꢕꢕꢎ

answer. This result can be used in

further calculations.

Now study the following examples. Remember that you need to press

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:

Calculates (3 + 10) first.

Puts 2 before 13 so the division is

correct: 2 ÷ 13.

™

ꢔꢖ)ꢕꢕꢕꢕꢎ

ꢕ)ꢔꢗꢖꢙꢎ

²Z p

Calculate 4 ÷ [(14 + (7 × 3) – 2] :

Keys:

Display:

Description:

š y

Calculates (7 3).

ꢏꢔ)ꢕꢕꢕꢕꢎ

ꢖꢖ)ꢕꢕꢕꢕꢎ

™ „

Calculates denominator.

Puts 4 before 33 in preparation for

division.

⁴Z

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.

The Automatic Memory Stack 2–13

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Size : 17.7 x 25.2 cm

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:

š ™

First adds (3+4)

ꢘ)ꢕꢕꢕꢕꢎ

ꢔꢔ)ꢕꢕꢕꢕꢎ

ꢘꢘ)ꢕꢕꢕꢕꢎ

š ™

Then adds (5+6)

Then multiplies the intermediate

answers together for the final

answer.

Exercises

Calculate:

(16.3805x5)

= 181.0000

0.05

Solution:

16.3805

š y <

.05

Calculate:

[(2+ 3)× (4+ 5)] + [(6+ 7)× (8+ 9) = 21.5743

Solution:

y < ™

š ™ š ™ y < š ™ š ™

Calculate:

(10 – 5) ÷ [(17 – 12) × 4] = 0.2500

Solution:

„ y

š „ Z p

„ y p

š „

2–14 The Automatic Memory Stack

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Size : 17.7 x 25.2 cm

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

™ „ Z p

If you work the problem from left–to–right, press

š š y ™ „ p

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

operands for nomcommutaiive functions (

and

„

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

Saves 4 and 14 as intermediate

numbers in the stack.

ꢎ

ꢔꢒ)ꢕꢕꢕꢕꢎ

The Automatic Memory Stack 2–15

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Size : 17.7 x 25.2 cm

At this point the stack is full with

numbers for this calculation.

Intermediate result.

ꢖ_ꢎ

™

ꢏꢔ)ꢕꢕꢕꢕꢎ

ꢖꢗ)ꢕꢕꢕꢕꢎ

ꢖꢖ)ꢕꢕꢕꢕꢎ

ꢕ)ꢔꢏꢔꢏꢎ

Intermediate result.

„

Final result.

More Exercises

Practice using RPN by working through the following problems:

Calculate:

(14 + 12) × (18 – 12) ÷ (9 – 7) = 78.0000

A Solution:

„ y š „ p

™

Calculate:

23 – (13 × 9) + 1/7 = 412.1429

A Solution:

²³z : ¹³š ⁹y „ ⁷3 ™

Calculate:

(5.4× 0.8)÷ (12.5− 0.7³) = 0.5961

Solution:

5.4

š 0

12.5

Z „ pꢁ<

5.4 š ^.8y ^12.5š .7 š ³0 „ 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

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

5.2

8.33

7.46

0.32

3.15

„

y z Ž

„

y p

š 2.75 „ ^4.3y ^1.71š ^2.01y „ p <ꢁ

The Automatic Memory Stack 2–17

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Size : 17.7 x 25.2 cm

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.

To store a copy of a displayed number (X–register) to a variable:

Press letter–key.

To recall a copy of a number from a variable to the display:

^PressK ^{letter–key.}

Storing Data into Variables

3–1

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

Example: Storing Numbers.

Store Avogadro's number (approximately 6.0225 × 10 ) in A.

Keys:

Display:

Description:

^6.0225` ²³

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

†

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

{

)

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

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

To cancel the VIEW display, press

once.

†

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 {

} 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}.

z X

to move the list and display the desired

z ˜ z —

variable. (Note the

annunciator, indicating that the left–shifted

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< ^and6 ^{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

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

3–3

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Size : 17.7 x 25.2 cm

Store zero in it: Press 0

variable.

To clear selected variables:

1. Press } and use

{

z ˜ z —

to display the

z X

#ꢀꢁ

variable.

^{2. Press}z b^.

3. Press

to cancel the catalog.

†

To clear all variables at once:

^Pressz 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

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

Results: 15–3

thatis, A–x

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

K ™

(one instruction) uses half as much memory as

(two instructions).

™

Results: 3÷12,

thatis, x÷A

0.25

K p ꢈ

Storing Data into Variables

3–5

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Size : 17.7 x 25.2 cm

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:

Stores the assumed values into the

variable.

ꢔ)ꢕꢕꢕꢕꢎ

ꢏ)ꢕꢕꢕꢕꢎ

ꢖ)ꢕꢕꢕꢕꢎ

ꢎ

Add 1 to D, E, And F.

H ™

H ™ H

ꢎ

™

ꢔ)ꢕꢕꢕꢕꢎ

ꢍ/ꢏ)ꢕꢕꢕꢕꢎ

ꢈ/ꢖ)ꢕꢕꢕꢕꢎ

ꢋ/ꢒ)ꢕꢕꢕꢕꢎ

ꢔ)ꢕꢕꢕꢕꢎ

Displays the current value of D.

{ ‰

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:

Calculates 3 D.

K p

ꢔ)ꢗꢕꢕꢕꢎ

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

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

Example:

Keys:

Display:

Description:

¹²H ^A

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

{ 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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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

Real–Number Functions

4–1

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

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

.(For y > 0, x can be any rational number; for y < 0, x must be are integer;

for y = 0, x must be positive.)

To Calculate:

Press:

Result:

z :

ꢏꢏꢗ)ꢕꢕꢕꢕꢎ

ꢔ8ꢕꢕꢕ8ꢕꢕꢕ)ꢕꢕꢕꢕꢎ

ꢖꢗ)ꢕꢕꢕꢕꢎ

z (

š 0

–1.4

1.4

_ 0

ꢕ)ꢖꢘꢙꢓꢎ

(–1.4)

1.4

_ š 0

.ꢏ)ꢘꢒꢒꢕꢎ

To calculate a root x of a number y (the x root of y), key in y

x, then

press

. For y<0, x must be an integer.

z .

To Calculate:

Press:

Result:

125

_ š z .

.ꢗ)ꢕꢕꢕꢕꢎ

ꢗ)ꢕꢕꢕꢕꢎ

−125

š z .

³125

4–2

Real–Number Functions

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

.37893

1.4

_ 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

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

Trigonometric Functions

With x in the display:

To Calculate:

Press:

Sine of x.

Cosine of x.

Tangent of x.

Arc sine of x.

Arc cosine of x.

Arc tangent of x.

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

significant digits).

Keys:

Display:

Description:

{

ꢁꢍ

}

Sets Radians mode; RAD

annunciator on.

z Ÿ

ꢎ

Œ Œ š

5 ÷ 7 in decimal format.

ꢕ)ꢘꢔꢒꢖꢎ

.ꢕ) ꢏꢖꢗꢎ

Cos (5/7) .

{ M y Q

{

ꢍꢆ

}

Switches to Degrees mode (no

annunciator).

z Ÿ

128.57

Calculates cos 128.57°, which is

the same as cos (5/7)π.

.ꢕ) ꢏꢖꢗꢎ

4–4

Real–Number Functions

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

Programming Note:

Equations using inverse trigonometric functions to determine an angle , often

look something like this:

= arctan (y x).

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

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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Size : 17.7 x 25.2 cm

To Calculate

Press:

x% of y

š { P

Percentage change from y to x. (y≠ 0) y

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

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

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.

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

mode. A calculated result for θ will be between –180° and 180°, between –π

and π radians, or between –200 and 200 grads.

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

z q

(rectangular–to–polar)

(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.

Real–Number Functions

4–7

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y,x θ, r

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.

30^o

Keys:

Display:

Description:

{

ꢍꢆ

}

Sets Degrees mode.

z Ÿ

ꢎ

Calculates x.

{

ꢙ) ꢕꢖꢎ

Displays y.

ꢗ)ꢕꢕꢕꢕꢎ

4 š ³z q

Calculates hypotenuse (r).

Displays θ.

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

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.

36.5^o

77.8 ohms

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

ꢏ)ꢗꢒꢕꢔꢎ

Displays y, ohms reactance, X .

.ꢒ )ꢏꢘꢘꢏꢎ

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

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

Œ Œ

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

(kilograms)

z }

{ |

ꢕ)ꢒꢗꢖ

1 kg

(pounds)

ꢏ)ꢏꢕꢒ

32 ºF

100 ºC

1 in

ºC

ºF

(°C)

z ~

ꢕ)ꢕꢕꢕꢕ

100

(°F)

{

ꢏꢔꢏ)ꢕꢕꢕꢕ

(centimeters)

z €

ꢏ)ꢗꢒꢕꢕ

100 cm

1 gal

1 l

100

(inches)

{

z ‚

{ ƒ

ꢖꢓ)ꢖꢘꢕꢔ

(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

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

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:

Twenty–four people grouped six

at a time.

[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.

_ꢎ

ꢖ8ꢕꢕꢖ)ꢕꢕꢕꢕꢎ

[PROB] { , }

ꢃQ T

{

Brings total number of

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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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

the numerator and denominator. To enter 2 / , press 2

Œ Œ

8. To

enter / , press

Œ Œ

8 or 5

Œ Œ

ꢀ

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.

ꢔ ꢔ+ꢏꢎ

ꢔ ꢖ+ꢒꢎ

ꢔ)ꢘꢗꢕꢕꢎ

ꢔ ꢖ+ꢒꢎ

Œ Œ š

Enters 1 / .

Displays x as a decimal number.

z Š

Displays x as a fraction.

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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Size : 17.7 x 25.2 cm

ꢀ

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

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

2 /

ꢏ ꢖ+ꢙꢎ

ꢔꢒ ꢔꢗ+ꢖꢏꢎ

ꢒ ꢔ+ꢏꢎ

ꢓ ꢖ+ꢗꢎ

14 /

6 /

2.83333333333

.183105468750

(Illegal entry)

ꢏ ꢗ+

ꢕ ꢘ+ꢖꢙꢏꢖ

8192

12345

12345678

16 /

(Illegal entry)

16384

Accuracy Indicators

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.

ꢀ

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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This diagram shows how the displayed fraction relates to nearby values —

means the exact numerator is "a little above" the displayed numerator, and

means the exact numerator is "a little below".

0 7/16

6.5

7.5

(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

force all fractions to have 5 as the denominator, then / is displayed as

ꢕ

3.3333

because the exact fraction is approximately

/ , "a little above"

ꢖ+ꢗ

/ . Similarly, – / is displayed as

5 because the true numerator is

.ꢕ ꢖ+

"a little above" 3.

^{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

example, if you enter 2 / , you see

, even though that's the exact

ꢏ ꢏ+ꢖ

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*

Calculates e .

...ꢕꢒ ꢙꢙꢙ+ꢖꢔꢏꢗꢎ

Shows all decimal digits.

ꢔꢏꢕꢏ ꢕꢒ)ꢏꢙꢒꢔ ꢎ

{

H ^A

Stores value in A.

...ꢕꢒ ꢙꢙꢙ+ꢖꢔꢏꢗꢎ

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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Size : 17.7 x 25.2 cm

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:

—

Most precise

Clear

Set

Factors of denominator

Fixed denominator

Clear

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

†

ꢋ @

to clear the

response.

&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)

Most precise

2 77/100

2 10/13

2 3/4

Factors of

denominator

(2.7699)

2 1051/1365

2 3153/4095

Fixed

denominator

(2.7699)

(2.7500)

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 ^ꢀ

2.5

2 1/2

2 /

2.9999

2 2/3

2 7/11

Most precise

Factors of

denominator

2 1/2

T2 11/16

S2 5/8

Fixed

denominator

2 0/16

2 8/16

2 11/16

2 16/16

2 10/16

ꢀ For a /F value of 16.

Example:

Suppose a stock has a current value of 48 / . If it goes down 2 / , what

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{ ‹

Sets up fraction format for /

ꢎ

increments.

48 Œ 1 Œ 4 š

Enters the starting value.

ꢒꢙ ꢔ+ꢒꢎ

ꢒꢗ ꢗ+ꢙꢎ

Sꢖꢙ ꢖ+ꢒꢎ

Œ Œ Œ „

Subtracts the change.

Finds the 85–percent value to the

{ P

nearest / .

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:

Suppose you have a 56 / –inch space that you want to divide into six equal

sections. How wide is each section, assuming you can conveniently measure

/ –inch increments What's the cumulative roundoff error

Keys:

Display:

Description:

Sets up fraction format for

{ ‹

ꢎ

/ –inch increments. (Flags 8

and 9 should be the same as for

the previous example.)

56 Œ 3 Œ ⁴H ^D

Stores the distance in D.

ꢎ

ꢗ ꢖ+ꢒꢎ

Sꢓ ꢘ+ꢔ ꢎ

The sections are a bit wider than

9 / inches.

Rounds the width to this value.

Width of six sections.

z I

ꢓ ꢘ+ꢔ ꢎ

ꢗ ꢗ+ꢙꢎ

.ꢕ ꢔ+ꢙꢎ

.ꢕ)ꢔꢏꢗꢕꢎ

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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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

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

calculates the volume of 16 inches of

{ M y

z : y

2 / –inch diameter pipe (78.5398 cubic inches). However, by storing the

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

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Begins a new equation, turning on

¾ꢎ

the " " equation–entry cursor.

turns on the A..Z

annunciator so you can enter a

variable name.

V types and moves the

{ c

#/¾ꢎ

cursor to the right.

Digit entry uses the "_" digit–entry

cursor.

.25

#/ꢕ)ꢏꢗ_ꢎ

ends the number and restores

y { M y

^{#/ꢕ)ꢏꢗº}π^º¾ꢎ

the " " cursor.

K 0

types .

^{/ꢕ)ꢏꢗº}π^ºꢍ:ꢏ_ꢎ

scrolls o f the left side of the

y K

^ꢕ)ꢏꢗºπ^{ºꢍ:ꢏºꢂ¾ꢎ}

display.

Terminates and displays the

^{#/ꢕ)ꢏꢗº}π^{ºꢍ:ꢏºꢎ}

equation.

shows that part of the

€

equation doesn't fit in the display,

and above means you can

press

to see characters in that

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

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.

2 Œ 1 Œ 2

Enters 2 / inches as a fraction.

ꢍ@ ꢏ ꢔ+ꢏꢎ

Stores D, prompts for L; value is

current value of L.

ꢂ@valueꢎ

Stores L; calculates V in cubic inches

and stores the result in V.

#/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

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

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

{ )

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 —

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.

4. Press

to terminate the equation and see it in the display. The

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).

When you press

, the A..Z annunciator shows that you can press a

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

Œ _

more digits. Don't use

for subtraction.

When you start entering the number, the cursor changes from " " to "_" to

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

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,

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

ꢁ/¾ꢎ

Enters a number, changing the

cursor to "_".

ꢁ/ ꢏ_ꢎ

y K y

Enters infix operators.

Enters a prefix function with a left

parenthesis.

ꢁ/ꢏºꢃº¾ꢎ

ꢁ/ꢏºꢃºꢃꢑ 1¾ꢎ

K K

{ ]

Enters the argument and right

parenthesis. This final parenthesis

is optional.

ºꢃºꢃꢑ 1!.ꢀ2¾ꢎ

Terminates the equation and

displays it.

ꢁ/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

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

means scrolling is turned on.

2. Press

to scroll the equation one character at a time, showing

^{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

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

Entering and Evaluating Equations

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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.

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

number, the calculator switches back to the " " equation–entry cursor.

If the " " equation–entry cursor is on, pressing

deletes the entire

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 " "

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.

4. Press

(or

) to save the edited equation in the equation list,

†

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 —

ꢈꢉꢄ ꢂꢊ ! !ꢑꢅ

ꢈꢉꢄ ꢂꢊ !

!ꢑꢅ

Example: Editing an Equation.

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

ºꢃºꢃꢑ 1!.ꢀ2¾ꢎ

shows the " " cursor at the end of

the equation.

Deletes the right parenthesis.

ꢏºꢃºꢃꢑ 1!.ꢀ¾ꢎ

Shows the end of edited equation in

/ꢏºꢃºꢃꢑ 1!.ꢀꢎ

š 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.

ꢀ

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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ꢀ

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

actions differ only in how they evaluate assignment equations:

ꢀ

returns the value of the equation, regardless of the type: equation.

returns the value of the equation—unless it's an assignment–type

equation. For an assignment equation,

returns the value f the

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)

x + y – r

f(x) ^ꢀ

0.5 × b × h ^ꢀ

y – f(x)

Assignment: y = f(x)

Example: A = 0.5 × b x h

A – 0.5 × b × h

Expression: f(x)

Example: x + 1

f(x)

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

. The equation prompts for a value for each

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

ꢀ

If you want, a different value, type the value and press

. (Also see

"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.)

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,

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.

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)

#/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.

Stores D, prompts for L, whose

current value, 16.0000.

ꢂ@ꢔ )ꢕꢕꢕꢕꢎ

ꢎ

y f

1000

Stores L in millimeters; calculates V

in cubic: millimeters, stores the

#/ꢔꢓ8ꢏꢒꢏ8ꢏꢗꢗ)ꢕꢕꢎ

result in V, and displays V.

` 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

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.

ꢍ@ꢖꢗ)ꢕꢕꢕꢕꢎ

35.5

fꢁ

ꢂ@ꢏꢕ8ꢕꢕꢕ)ꢕꢕꢕꢕꢎ

Keeps the same L; calculates the

.ꢗꢗꢖ8ꢘꢕꢗ)ꢘꢕꢗꢔꢎ

value of the equation—the

imbalance between the left and

right sides.

` 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

To change the number, type the new number and press

.This

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

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

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

ꢊꢄ1%-ꢔ2, 1%-ꢔ2ꢎ

.ꢀꢎ

Unary Minus (

)

Power (

)

%: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

a × (b ) = c

ꢀºꢌ:ꢖ/ꢃꢎ

(a × b) = c

1ꢀºꢌ2:ꢖ/ꢃ

a + (b c) = 12

ꢀ-ꢌªꢃ/ꢔꢏ

(a + b) ÷ c = 12

1ꢀ-ꢌ2ªꢃ/ꢔꢏꢎ

0ꢃꢐꢆ1!-ꢔꢏ ꢀ. 2:ꢏꢎ

[%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

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.

LOG

EXP

ALOG

RND

SQRT

INV

SIN

SINH

ABS

COS

COSH

TAN

TANH

ASIN

ASINH

ACOS

ACOSH

%CHG

ATAN

ATANH

XROOT

DEG

€

RAD

€

HMS

€

Cn,r

Pn,r

°C

€

°F

€

GAL

RANDOM

–

x w

σ x

σ y

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

to is equivalent to

%ꢁ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

EXP

ALOG

INV

1/x

X ROOT

Example: Perimeter of a Trapezoid.

The following equation calculates the perimeter of a trapezoid. This is how

the equation might appear in a book:

Perimeter = a + b + h (

)

sinθ sinφ

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

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

^by3^.)

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:

cos(π /n)

sin(π/n)

n d ²

Area =

You can specify this equation as

Entering and Evaluating Equations 6–19

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ꢀ/ꢕ)ꢏꢗºꢄºꢍ:ꢏºꢃꢑ 1 ªꢄ2ª ꢊꢄ1 ªꢄ2ꢎ

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:

.25

{ G K {

y K y K 0 y Q

{ M p K { ] p N { M p K { ]

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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Solving Equations

In chapter 6 you saw how you can use

to find the value of the

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

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

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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

ꢀ

If you want a different clue, type or calculate the value and press

(For details, see "Responding to Equation Prompts" in chapter 6.)

You can half a running calculation b pressing or

†

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:

d = v t + / g t

where d is the distance, v is the initial velocity, t is the time, and g is the

acceleration due to gravity.

Type in the equation:

Keys:

Display:

Description:

Clears memory.

z b

ꢎ

{ } { }

ꢀꢂꢂ &

Selects Equation mode.

{ G

ꢈꢉꢄ ꢂꢊ ! !ꢑꢅꢎ

7–2

Solving Equations

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or current equationꢎ

K ^D{ c

Starts the equation.

ꢍ/#º!-¾ꢎ

K y K

™

y K 0

y K

!-ꢕ)ꢗºꢆº!: ꢏ_ꢎ

Terminates the equation and

ꢍ/#º!-ꢕ)ꢗºꢆº!ꢎ

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

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

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

!@valueꢎ

ꢆ@valueꢎ

9.8

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

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.

Solving Equations

7–3

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Retains 9.8 in G; prompts for T.

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

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:

{ G K y

Selects Equation mode and

starts the equation.

ꢅº¾ꢎ

K { cꢁ

K ^Nyꢁ

ꢎ

K yꢁK

ꢅº#/ꢄºꢁº!¾ꢎ

ꢅº#/ꢄºꢁº!ꢎ

Terminates and displays the

equation.

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:

Solves for P; prompts for V.

{ œ

²f

#@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

^.0821f

7–4

Solving Equations

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Calculates T (Kelvins).

!@ꢏꢓꢘ)ꢔꢕꢕꢕꢎ

273.1™

Stores 297.1 in T; solves for P in

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

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

šꢁ

273.1

™

!@ꢏꢓꢔ)ꢔꢕꢕꢕꢎ

ꢑꢂ#ꢊꢄꢆꢎ

ꢄ/ꢕ)ꢕꢔꢕꢗꢎ

ꢕ)ꢏꢓꢏꢓꢎ

Stores 291.1 in T; solves for N.

Calculates mass in grams, N × 28.

Calculates density in grams per

liter.

K ^Vp

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 —

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

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

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 † ^ora ^{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

Solving Equations

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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 † ^orf^{. 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

d = v t + / gt

can have two solutions for t. You can direct the answer to the only

meaningful one (t > 0) by entering appropriate guesses.

he example using this equation earlier in this chapter didn't require you

Solving Equations

7–7

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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

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

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.

40 2H

80 2H

7–8

Solving Equations

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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

K { c

#/¾ꢎ

{ \

„ꢁ

ꢎ

K { ]

#/1ꢒꢕ.ꢐ2¾ꢎ

„

ꢕ.ꢐ2º1ꢏꢕ.ꢐ2¾ꢎ

y { \

K { ]

y y K

º1ꢏꢕ.ꢐ2ºꢒºꢐ¾ꢎ

#/1ꢒꢕ.ꢐ2º1ꢏꢕꢎ

Terminates and displays the

equation.

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

H 20

ꢏꢕ_ꢎ

Solving Equations

7–9

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Displays current equation.

Solves for H; prompts for V.

Stores 7500 in V; solves for H.

{ G

#/1ꢒꢕ.ꢐ2ºꢏꢕꢎ

#@valueꢎ

{ œ

7500

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.

ꢔꢗ)ꢕꢕꢕꢕꢎ

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

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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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

I = f(x)dx

∫

f (x)

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

8–1

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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

then key in the upper limit.

3. Display the equation: Press

and, if necessary, scroll through

{ G

the equation list (press

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

†

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

Integrating Equations

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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

J₀=

∫

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

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.

ꢃꢑ 1%º ꢊꢄ1!2ꢎ

Checksum and length.

ꢃꢚ/ꢋꢓꢖꢌ ꢕꢔꢏ)ꢕꢎ

{ꢁ

Integrating Equations

8–3

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

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

š { M

ꢖ)ꢔꢒꢔ ꢎ

Displays the function.

Prompts for the variable of

integration.

{ G

{ )

ꢃꢑ 1%º ꢊꢄ1!2ꢎ

∫ ꢋꢄ G_ꢎ

Prompts for value of X.

x = 2. Starts integrating;

%@value

ꢎ

ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ

calculates result for

^πf(t)

∫ /ꢕ)ꢘꢕꢖꢒꢎ

∫

The final result for

{ M p

ꢕ)ꢏꢏꢖꢓꢎ

J (2).

Now calculate J (3) with the same limits of integration. You must respecify the

limits of integration (0, ) since they were pushed off the stack by the

subsequent division by π.

Keys:

Display:

Description:

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.

Prompts for value of X.

x = 3. Starts integrating and

calculates the result for

%@ꢏ)ꢕꢕꢕꢕꢎ

ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ

∫ /.ꢕ)ꢙꢘꢕꢎ

^πf(t)

∫

The final result for

J (3).

{ M p

–0.260

8–4

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

sinx

S(t) = (

)dx

∫

Find Si (2).

Enter the expression that defines the integrand's function:

sinx

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

Starts the equation.

N K

{ ]

ꢊꢄ1%¾ꢎ

The closing right parenthesis is

required in this case.

ꢊꢄ1%2¾ꢎ

p K

{

†

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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Size : 17.7 x 25.2 cm

Enters limits of integration (lower

first).

ꢏ_ꢎ

Displays the current equation.

Calculates the result for Si(2).

{ G

ꢊꢄ1%2ª%ꢎ

ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ

∫ /ꢔ) ꢕꢗꢒꢎ

{ )

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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Size : 17.7 x 25.2 cm

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

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

{ )

The uncertainty of the

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

{ )

ꢊꢄ!ꢈꢆꢁꢀ!ꢊꢄꢆꢎ

∫ /ꢔ) ꢕꢗꢒꢈꢕꢎ

Note that the uncertainty is about

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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Operations with Comb Numbers

The HP 32SII can use complex numbers in the form

+ .

It has operations for complex arithmetic (+, –, ×, ÷), complex trigonometry (sin,

z₁^z

cos, tan), and the mathematics functions –z, 1/z,

ln z, and e . (where

z and

are complex numbers).

To enter a complex number:

1. Type the imaginary part.

2. Press

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

š š š z F ™

^{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 :

Operations with Comb Numbers

9–1

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Z₁

Z₂

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.

Z₁

Z₂

Complex function

imaginary part

real part

Complex input

z or z and z

Complex result, z

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 ).

(For two–number operations, the first complex number, z , is replicated in the

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

2. Select the complex function.

Functions for One Complex Number, z

To Calculate:

Press:

z F _

z F 3

z F -

z F *

z F N

z F Q

z F T

Change sign,–

Inverse, 1/

Natural log, ln

Natural antilog,

Sin

Cos

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

z₁^z

š x š. (For

², key in the base part, z , first.)

2. Enter the second complex number, z , by keying in y

x . (For

z₁^z

², key in the exponent, z , second.)

3. Select the arithmetic operation:

Arithmetic With Two Complex Numbers, z₁and z₂

To Calculate:

Press:

Addition, z + z

z F ™

z F „

z F y

z F p

z F 0

Subtraction, z – z

Multiplication, z × z

Division, z ÷ z

z₁^z

Power function,

Examples:

Here are some examples of trigonometry and arithmetic with complex

numbers:

Evaluate sin (2 + i 3)

Keys:

Display:

Description:

Real part of result.

ꢎ

z F N

ꢓ)ꢔꢗꢒꢗꢎ

.ꢒ)ꢔ ꢙꢓꢎ

Result is 9.1545 – i 4.1689.

Evaluate the expression

(z + z ),

where z = 23 + i 13, z = –2 + i z = 4 – i

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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[1 (z + z )]

Keys:

Display:

Description:

³_ š ⁴zꢁ

Add z + z ; displays real

š _ šꢁ

ꢎ

part.

ꢎ

F ™

ꢏ)ꢕꢕꢕꢕꢎ

ꢕ)ꢏꢗꢕꢕꢎ

ꢎ

1 ÷ (z +z ).

z F 3

z ÷ (z +z ).

z F y

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:

Œ Œ _ š

Enters imaginary part of first

complex number as a

fraction.

.ꢕ)ꢒꢕꢕꢕꢎ

Enters real part of first

complex number.

ꢒ)ꢕꢕꢕꢕꢎ

Œ Œ _ š

Enters imaginary part of

second complex number as a

fraction.

.ꢕ)

ꢘꢎ

³z F y

Completes entry of second

number and then multiplies

ꢔꢔ)ꢘꢖꢖꢖꢎ

the two

complex

numbers.

Result is 11.7333 – i

3.8667.

.ꢖ)ꢙ ꢘꢎ

⁻², where z = (1 + i ). Use

to evaluate z ;

–2

e^z

Evaluate

enter –2 as –2 + i 0.

z F 0

Keys:

Display:

Description:

Operations with Comb Numbers

9–5

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Size : 17.7 x 25.2 cm

š š

Intermediate result of

ꢎ

–2

ꢎ

(1 + i )

š _ zꢁ

ꢕ)ꢕꢕꢕꢕꢎ

F 0

z F *

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)

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.

62^o

185 lb

170 lb

143^o

100 lb

261 ^o

Keys:

Display:

Description:

{

ꢍꢆ

}

Sets Degrees mode.

z Ÿ

ꢎ

{ r

185

Enters L and converts it to

ꢎ

rectangular form.

ꢙ )ꢙꢗꢏꢏꢎ

143

170

Eaters and converts L .

{

.ꢔꢖꢗ)ꢘ ꢙꢕꢎ

Adds vectors.

z F ™

.ꢒꢙ)ꢓꢔꢗꢙꢎ

.ꢔꢗ) ꢒꢖꢒꢎ

261 š ¹⁰⁰{

Enters and converts L .

z F ™

z q

Adds L + L + L .

. ꢒ)ꢗꢗꢓꢏꢎ

ꢔꢘꢙ)ꢓꢖꢘꢏꢎ

Converts vector hack to polar

form; displays r.

Displays θ.

ꢔꢔꢔ)ꢔꢒꢙꢓꢎ

Operations with Comb Numbers

9–7

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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 ^andꢆ^{, 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

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.

Keys:

Display:

Description:

125.99

Converts just the integer part (125)

ꢘꢍꢎ

Base Conversions and Arithmetic 10–1

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{

}

of the decimal number to base 16

and displays this value.

Base 8.

ꢐ%

{

}

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

digits (the maximum display) long.

Keys:

Display:

Description:

{HX}

Use the

ꢏꢒꢋꢋ_ꢎ

key to type "F".

z w

24FF

{

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

<^.

Displays the rest of the number.

ꢔꢕꢎ

The full number is

10010011111111 .

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

™ „ y

base. The only function keys that are actually deactivated outside of Decimal

mode are and . However, you should realize

< * - 0 3

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

Keys:

Display:

Description:

{

}

Sets base 16; HEX

annunciator on.

Result.

z w

ꢐ%

ꢎ

12F

E9A

™

ꢋꢃꢓꢎ

7760 – 4326 =

{

ꢑꢃ

}

Sets base 8: OCT

ꢘꢘꢔꢔꢎ

z w

annunciator on. Converts

displayed number to octal.

Result.

7760

4326

„

ꢖꢒꢖꢏꢎ

100 – 5 =

100 š ⁵p

Integer part of result.

ꢔꢒꢎ

5A0 + 1001100 =

z w ^{} 5A0

ꢐ%

Set base 16; HEX

ꢗꢀꢕ_ꢎ

Base Conversions and Arithmetic 10–3

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Size : 17.7 x 25.2 cm

annunciator on.

Changes to base 2; BIN

z w ^{} 1001100

ꢌꢄ

ꢔꢕꢕꢔꢔꢕꢕ_ꢎ

annunciator on. This

terminates digit entry, so no

is needed between

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:

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

annunciator

€

(or both) appears, indicating in which direction the additional digits lie. Press

^{the indicated key (}< ^or6 ^{) 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

10–6 Base Conversions and Arithmetic

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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

ꢔꢏꢖꢒꢗ ꢘꢔꢏꢖꢒꢗꢎ

ꢀ/...ꢒꢗ ꢘꢔꢏꢖꢒꢗꢎ

ꢔꢏꢖꢒꢗ ꢘꢔꢏꢖꢒꢗ

Drops leftmost three digit's.

Shows all digits.

{ ‰

(hold)

{

}

Restores Decimal mode.

ꢔꢔ8ꢏꢔꢓ8ꢒꢘꢖ8 ꢖꢘ)ꢕꢎ

z w

ꢍꢈꢃ

Base Conversions and Arithmetic 10–7

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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.

ꢀ

Linear regression and linear estimation (x and y).

Weighted mean (x weighted by y).

A Summation statistics: n, Σx, Σy, Σx , Σy , and Σxy.

s ,

SUMS

x , y

L.R.

y x y xy

x w

Entering Statistical Data

One– and two–variable statistical data are entered (or deleted) in similar

^{fashion using the}6 ^(orz 4 ^{) key. Data values are accumulated as}

summation statistics in six statistic's registers (28 through 33), whose names

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

Statistical Operations 11–1

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Size : 17.7 x 25.2 cm

Entering One–Variable Data

1. Press

{Σ} to clear existing statistical data.

z b

2. Key in each x–value and press

3. The display shows n, the number of statistical data values now

accumulated.

Pressing

actually enters two variables into the statistics registers because

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.

1. Press

{ } to clear existing statistical data.

z b Σ

2. Key in the y–value first and press

^{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

11–2 Statistical Operations

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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

If the incorrect values were the ones just entered, press

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

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

ꢔ)ꢕꢕꢕꢕꢎ

ꢏ)ꢕꢕꢕꢕꢎ

400

z Ž

ꢒꢕꢕ)ꢕꢕꢕꢕꢎ

twice to check y.)

Deletes the last data pair.

Reenters the last data pair.

z 4

ꢔ)ꢕꢕꢕꢕꢎ

ꢏ)ꢕꢕꢕꢕꢎ

Statistical Operations 11–3

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Deletes the first data pair.

Reenters the first data pair.

There is still a. total of two

data pairs in the statistics

registers.

z 4

ꢔ)ꢕꢕꢕꢕꢎ

ꢏ)ꢕꢕꢕꢕꢎ

Statistical Calculations

Once you have entered your data, you can use the functions in the statistics

menus.

Statistics Menus

Key

Description

L.R.

The linear–regression menu: linear

{ ,

ˆ ˆ

estimation { } { } and curve–fitting { }

º ¸

{ } { }. See ''Linear Regression'' later in

P E

this chapter.

z /

z 2

The mean menu: { } { } {

}. See

º ¸ º·

"Mean" below.

s,σ

The standard–deviation menu: { } {

}

Uº U¸

{σ } {σ }. See "Sample Standard

Deviation" and "Population Standard

Deviation" later in this chapter.

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.

Press

{ } for the mean of the y–values.

{ /

{

} for the weighted mean of the x–values using the

º·

11–4 Statistical Operations

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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

ꢔ)ꢕꢕꢕꢕꢎ

ꢎ

^9.256 ¹⁰6 ^12.5

Enters the remaining data;

six data points accumulated.

Calculates the mean time to

complete the process.

8.5

)ꢕꢕꢕꢕꢎ

ꢔꢔ)ꢏꢓꢔꢘꢎ

{ }

{ /

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.

6ꢁ

ꢔ)ꢕꢕꢕꢕꢎ

ꢏ)ꢕꢕꢕꢕꢎ

ꢖ)ꢕꢕꢕꢕꢎ

6ꢁ

4.7

Statistical Operations 11–5

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Size : 17.7 x 25.2 cm

1000

4.1

}

Four data pairs accumulated.

Calculates the mean price

weighted for the quantity

purchased.

šꢁ

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

{

} for the standard deviation of x–values.

{ 2

Uº

} for the standard deviation of y–values.

U¸

The {σ } and {σ } keys in this menu are described in the next section,

"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:

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

ꢔ)ꢕꢕꢕꢕꢎ

ꢎ

12.5

Enters the remaining data; six

data points entered.

8.5

)ꢕꢕꢕꢕꢎ

ꢏ)ꢗꢙꢕꢙꢎ

{

}

Calculates the standard deviation

time.

{ 2

11–6 Statistical Operations

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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

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

173

174

ꢎ

180

ꢏ)ꢕꢕꢕꢕꢎ

ꢒ)ꢕꢕꢕꢕꢎ

ꢖ) ꢖꢔꢗꢎ

{ }

Four data points accumulated.

Calculates the population

standard deviation.

{ 2 σ

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

{ }

Estimates (predicts) x for a given hypothetical value of

y, based on the line calculated to fit the data.

{ }

Estimates (predicts) y for a given hypothetical value of

x, based on the line calculated to fit the data.

{ }

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.

{ }

Slope of the calculated line.

{ }

y–intercept of the calculated line.

ꢀ

To find an estimated value for x (or y), key in a given hypothetical value

for y (or x), then press

{ } (or

{ }).

{ ,

^{To find the values that define the line that best fits your data, press}{

followed by { }, { }, or { }.

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

11–8 Statistical Operations

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data.

4.63 š ⁰6

Enters data; displays n.

ꢔ)ꢕꢕꢕꢕꢎ

ꢏ)ꢕꢕꢕꢕꢎ

ꢖ)ꢕꢕꢕꢕꢎ

ꢒ)ꢕꢕꢕꢕꢎ

ꢗ)ꢕꢕꢕꢕꢎ

5.78

6.61

7.21

7.78

Five data pairs entered.

Displays linear–regression

menu.

ˆ ˆ

º ¸

{ ,

T P E

{ }

Correction coefficient; data

closely approximate a straight

line.

ꢕ)ꢓꢙꢙꢕꢎ

{ }

Slope of the line.

y–intercept.

z ,

ꢕ)ꢕꢖꢙꢘꢎ

ꢒ)ꢙꢗ ꢕꢎ

z , ^{{ }}

Statistical Operations 11–9

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8.50

7.50

6.50

5.50

4.50

(70, y)

r = 0.9880

m = 0.0387

b = 4.8560

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:

Enters hypothetical x–value.

The predicted yield in tons per

hectare.

ꢘꢕ_ꢎ

ꢘ)ꢗ ꢔꢗꢎ

{ }

{ ,

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:

11–10 Statistical Operations

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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

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

0, and 1; then add 7777000 back to x and

7777000 × m. To calculate , be sure to supply an x–value that is less

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

^Executingz 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.

Press { } to recall the sum of the x–values.

Press { } to recall the sum of the y–values.

Statistical Operations 11–11

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ꢏ

ꢀ

Press { }, { }, and { } to recall the sums of the squares and the sum

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

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

š 6

ꢔ)ꢕꢕꢕꢕꢎ

ꢏ)ꢕꢕꢕꢕ

4 š ³6

{

z X

#ꢀꢁ

}

´º¸/ꢔꢒ)ꢕꢕꢕꢕ

Σxy register.

´¸^ꢏ/ꢏꢕ)ꢕꢕꢕꢕ

´º^ꢏ/ꢔꢕ)ꢕꢕꢕꢕ

´¸/ )ꢕꢕꢕꢕ

´º/ꢒ)ꢕꢕꢕꢕ

Views y register.

z —

†

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

. The

{ }.

11–12 Statistical Operations

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If not enough calculator memory is available to hold the statistics registers

when you first press (or ), the calculator displays

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

Number

Description

Number of accumulated data pairs.

Sum of accumulated x–values.

Σx

Sum of accumulated y–values.

Sum of squares of accumulated x–values.

Sum of squares of accumulated y–values.

Σxy

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.

Statistical Operations 11–13

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Part 2

Programming

Statistics Programs

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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

formula A = π r and press

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 º^ꢏꢎ

ꢕꢕꢏ ꢎ

ꢕꢕꢖ ºꢎ

Simple Programming 12–1

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This very simple program assumes that the value for the radius is in the X–

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

ꢕꢕꢔ º^ꢏ

ꢕꢕꢏ π

ꢕꢕꢖ º

z :

{ M

(Radius)

Area = x

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

The answer!

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.

12–2 Simple Programming

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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

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

to activate Program–entry mode.

z d

2. Press

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 — ^orz ˜ ^{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

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

€

for this program line. You can use

and

to scroll the display. Press

[SCRL] to turn off

and to use the top–row keys to enter program

{

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

equation. It deletes the rightmost function or variable if an equation is

being entered (" " cursor present).

ꢀ

{

} deletes a program lime if it contains an equation.

z b

ꢈꢉꢄ

To program a function to clear the K–register, use

{ }.

z b

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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

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 º^ꢏꢎ

ꢀꢕꢖ πꢎ

ꢀꢕꢒ ºꢎ

ꢀꢕꢗ ꢁ!ꢄꢎ

}

Enters the three program lines.

z :

{ M

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

z “

ꢈꢕꢔ ꢂꢌꢂ ꢈꢎ

Stores radius in variable R.

Selects Equation–entry mode;

enters the equation; returns to

Program–entry mode.

ꢈꢕꢏ !ꢑ ꢁꢎ

ꢎ

{ G { Mꢁ

0 š

y K

ꢎ

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)

^PressW ^{label to execute the program labeled with that letter. If there is}

only one program in memory, you can also execute it by pressing

Œ Œ f ^{(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

executing .A or E.

Keys:

Display:

Description:

Enters the radius, then starts

program A. The resulting area is

displayed.

ꢁ"ꢄꢄꢊꢄꢆꢎ

ꢘꢙ)ꢗꢖꢓꢙꢎ

2.5

Calculates area of the second

circle using program E.

Calculates area of the third

circle.

ꢔꢓ) ꢖꢗꢕꢎ

ꢔꢏꢒ)ꢕꢏꢗꢔꢎ

{ 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

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:

Moves program counter to label A.

z U

ꢗ)ꢕꢕꢕꢕꢎ

ꢀꢕꢔ ꢂꢌꢂ ꢀꢎ

ꢗ)ꢕꢕꢕꢕꢎ

ꢀꢕꢏ º^ꢏꢎ

(hold)

z ˜

(release)

z ˜

(release)

z ˜

(release)

z ˜

(release)

z ˜

(release)

Squares input.

ꢏꢗ)ꢕꢕꢕꢕꢎ

ꢀ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

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 º^ꢏꢎ

ꢀ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

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

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

To change the number, type the new number and press

, This

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

press 2

š 0 f

ꢀ

To calculate with the displayed number, press

typing another number.

before

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.)

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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

ꢀ/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.

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}< ^and6 ^{keys to see the rest.)}

ꢀ

Pressing

X–register.

Pressing

(or

) erases the VIEW display and shows the

†

clears the contents of the displayed variable.

z b

^Pressf ^{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

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

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

and

to scroll the display. You can press

[SCRL] to turn off

{

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:

V = πR H

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

Labels program.

z ˆ

Instructions to prompt for radius

and height.

z ˆ

{ G {

Calculates the volume.

ꢎ

0 ²y K ^H

M y K

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Keys:

Display:

Description:

ꢃꢕꢒ πºꢁ:ꢏºꢐꢎ

Checksum and length of equation.

ꢃꢚ/ꢓꢔꢓꢒ ꢕꢔꢏ)ꢕꢎ

{

Store the volume in V.

ꢃꢕꢗ !ꢑ #ꢎ

ꢎ

yꢁ{ M

Calculates the surface area.

{ G

ꢎ

yꢁK y

{ \ K

ꢎ

™ K {

ꢎ

] š

{

ꢃꢕ ꢏºπºꢁºꢁ1ꢎ

Checksum and length of equation.

ꢃꢚ/ꢀꢓꢔꢔ ꢕꢔꢙ)ꢕꢎ

Stores the surface area in S.

ꢃꢕꢘ !ꢑ ꢎ

ꢎ

{

{ x

ꢋ

}

Sets flag 10 to display equations.

ꢃꢕꢙ ꢋ ꢔꢕꢎ

ꢎ

{ G K

Displays message in equations.

o ™

K K

ꢎ

o K

ꢎ

K ^RK ^E

ꢎ

K š

ꢃꢕꢓ #ꢑꢂ - ꢀꢁꢎ

{ }

ꢃꢋ ꢎ

Clears flag 10.

{ x

ꢃꢔꢕ ꢃꢋ ꢔꢕꢎ

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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Now find the volume and surface area–of a cylinder with a radius of 2 /

cm and a height of 8 cm.

Keys:

Display:

Description:

Starts executing C; prompts for R.

ꢁ@valueꢎ

(It displays whatever value

happens to be in R.)

Enters 2 / as a fraction. Prompts

Œ Œ

ꢐ@valueꢎ

for H.

Message displayed.

#ꢑꢂ - ꢀꢁꢈꢀꢎ

#/ꢔꢗꢘ)ꢕꢘꢓ ꢎ

/ꢔ ꢒ)ꢓꢖꢖ ꢎ

Volume in cm .

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)

ꢀ

^Pressingf ^{(run/stop) during program entry inserts a STOP instruction.}

This will halt a running program until you resume it by pressing

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

†

(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

(

label).

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.)

12–18 Simple Programming

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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

) 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

label nn moves the program pointer there.)

U Œ

2. Delete the line you want to change—if it contains an equation, press

{

}; otherwise, press

. The pointer then moves to the

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

†

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

anything in the equation.

3. Press

as required to delete the function or number you want to change,

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then enter the desired corrections.

4. Press

to end the equation.

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 ˜

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.

—

ꢀ

^PressU Œ Œ to 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

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

while the label is

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.

Simple Programming 12–21

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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

prompts you for confirmation. Press { }.

ꢃꢂ ꢅꢆꢇ @ & ꢄ

4. Press

to cancel program entry.

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:

1. ^Pressz 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

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).

Simple Programming 12–23

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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

ꢀꢕꢓ ꢐꢈ%

PRGM

ꢀꢔꢕ

HEX

ꢀꢕꢓ ꢐꢈ%

PRGM

ꢀꢔꢕ ꢔꢖ

ꢍꢎ

12–24 Simple Programming

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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

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:

Ax + 13x + Cx +D x + E

(Ax + Bx + Cx + D ) x + E

((Ax + Bx + C ) x + D )x + E

(((Ax + B )x + C ) x + D )x + E

Example:

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

z “

Fills the stack with x.

z ˆ

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ꢅꢕꢒ ꢈꢄ!ꢈꢁꢎ

ꢅꢕꢗ ꢈꢄ!ꢈꢁꢎ

ꢅꢕ ꢗꢎ

5x.

ꢅꢕꢘ ºꢎ

ꢅꢕꢙ ꢏꢎ

5x + 2.

™

ꢅꢕꢓ -ꢎ

(5x + 2)x.

ꢅꢔꢕ ºꢎ

(5x + 2)x .

ꢅꢔꢔ ºꢎ

(5x + 2)x .

ꢅꢔꢏ ºꢎ

{ ”

z X

ꢅꢔꢖ ꢁ!ꢄꢎ

{

}

ꢅꢆꢇ ꢂꢌꢂ ꢅ ꢕꢔꢓ)ꢗꢎ

Displays label P, which takes 19.5

bytes.

Checksum and length.

† †

ꢃꢚ/ꢘꢋꢌꢒ ꢕꢔꢓ)ꢗꢎ

Cancels program entry.

ꢎ

Now evaluate this polynomial x = 7.

Keys:

Display:

Description:

Prompts for x.

Result.

%@valueꢎ

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

Simple Programming 12–27

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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).

Programming Techniques 13–1

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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:

Program execution branches from this line to

€

ꢀꢕꢗ ꢆ!ꢑ ꢌ

ꢌꢕꢔ ꢂꢌꢂ ꢌ

the line marked

("from 1").

Program execution branches from a line

marked

("to 1") to this line.

€

13–2 Programming Techniques

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Starts here.

ꢍꢕꢔ ꢂꢌꢂ ꢍ

ꢍꢕꢏ ꢊꢄꢅ"! %

ꢍꢕꢖ %ꢈꢉ ꢉ

ꢍꢕꢒ !ꢑ ꢉ

ꢍꢕꢗ #ꢊꢈ$ ꢉ

ꢍꢕ ꢆ!ꢑ ꢍ

Calls subroutine Q.

Return here.

€

Starts D again.

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$

Programming Techniques 13–3

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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 º^ꢏ

A .

€

ꢔꢔ %ꢈꢉ ꢉ

ꢔꢏ %ꢈꢉ ꢉ

ꢔꢖ %ꢈꢉ ꢉ

ꢔꢒ ꢉꢁ!

ꢔꢗ ꢁ!ꢄ

A + B .

€

A + B + C

A²+ B²+ C²+ D²

Returns to main routine.

135 Nested subroutine

ꢉꢕꢔ ꢂꢌꢂ ꢉ

ꢉꢕꢏ º65¸

ꢉꢕꢖ º^ꢏ

Adds x .

ꢉꢕꢒ -

246

Returns to subroutine S.

ꢉꢕꢗ ꢁ!ꢄ

13–4 Programming Techniques

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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:

Programming Techniques 13–5

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ꢕꢔ ꢂꢌꢂ

Can start here.

Branches to Z.

Can start here.

€

ꢕꢗ ꢆ!ꢑ '

ꢂꢕꢔ ꢂꢌꢂ ꢂ

Branches to Z.

Can start here.

€

ꢂꢕꢗ ꢆ!ꢑ '

ꢈꢕꢔ ꢂꢌꢂ ꢈ

Branches to Z.

Branch to here.

€

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.

ꢀ

z U Œ Œ

ꢅꢁꢆꢇ !ꢑꢅ

To a line number:

label nn (nn < 100). For example,

z U Œ

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,

z U

13–6 Programming Techniques

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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.

Skip next if false.

€

ꢀꢕꢗ º/ꢕ@

ꢀꢕ ꢆ!ꢑ ꢌ

ꢀꢕꢘ ꢂꢄ

€

ꢀ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.

Programming Techniques 13–7

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Tests of Comparison (x?y, x?0)

There are 12 comparisons available for programming. Pressing

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

&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

13–8 Programming Techniques

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Size : 17.7 x 25.2 cm

!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 &

ꢀ

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

Programming Techniques 13–9

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Size : 17.7 x 25.2 cm

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

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.

. Pressing any other key executes

†

13–10 Programming Techniques

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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

you keyed in, or the displayed (current) value of the variable if

your sole response to the prompt.

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.)

Programming Techniques 13–11

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Size : 17.7 x 25.2 cm

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

flag 11.

FLAGS Menu

Menu Key

Description

{

} n

Set flag. Set flag n.

ꢋ

} n

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.

13–12 Programming Techniques

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Size : 17.7 x 25.2 cm

ꢀ

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.

ꢂꢕꢖ ꢋ ꢕ

Clears flag 1, the indicator for In Y.

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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Size : 17.7 x 25.2 cm

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

(the

key) for each space character.

3. Press

to insert the message in the current program line and end

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.

13–14 Programming Techniques

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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:

Executes label F; prompts for a

fractional number (V).

2.53

#@valueꢎ

Stores 2.53 in V; prompts for

denominator (D).

ꢍ@valueꢎ

Stores 16 as the /c value. Displays

message, then the decimal

number.

ꢍꢈꢃꢊꢇꢀꢂꢎ

ꢏ)ꢗꢖꢕꢕꢎ

Message indicates the fraction

ꢇꢑ ! ꢅꢁꢈꢃꢊ ꢈꢎ

format (denominator is no greater

than 16), then shows the fraction.

dꢏ ꢙ+ꢔꢗꢎ

indicates that the numerator is

"a little below" 8..

Message indicates the fraction

ꢋꢀꢃ!ꢑꢁ ꢍꢈꢄꢑꢇꢎ

Programming Techniques 13–15

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Keys:

Display:

Description:

format (denominator is factor of

16), then shows the fraction.

Message indicates the fraction

cꢏ ꢔ+ꢏꢎ

ꢋꢊ%ꢈꢍ ꢍꢈꢄꢑꢇꢎ

format (denominator is 16), then

cꢏ ꢙ+ꢔ ꢎ

shows the fraction.

Stops the program and clears flag

f † {

ꢏ)ꢗꢖꢕꢕꢎ

{

ꢃꢋ

}

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

O for the new–to–old system).

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.

13–16 Programming Techniques

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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

(increment; skip if greater than). or

(decrement; skip if less

{ 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

Programming Techniques 13–17

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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.

13–18 Programming Techniques

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€

$ꢕꢔ ꢂꢌꢂ $

€

$ꢕꢓ ꢍ ꢈ ꢀ

$ꢔꢕ ꢆ!ꢑ $

$ꢔꢔ %ꢈꢉ %

If current value >

final value,

If current value ≤

final value, exit

loop.

continue loop.

$ꢕꢔ ꢂꢌꢂ $

€

$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

Programming Techniques 13–19

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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

‘

’

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

13–20 Programming Techniques

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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

If i contains:

Then (i) will address:

variable A or label A

variable Z or label Z

variable i

n register

x register

y register

x register

y register

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.

Programming Techniques 13–21

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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

13–22 Programming Techniques

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and line Y08 calls a different subroutine to compute

increased by 6:

after i has been

&ꢕ

&ꢕꢘ !ꢑ- L

&ꢕꢙ %ꢈꢉ1L2

If i hold:

Then XEQ(i) calls:

To:

LBL A

Compute

model.

Compute

model.

for straight–line

LBL B

LBL C

for logarithmic

Compute for exponential

model.

Compute

model.

LBL D

LBL G

for power model.

for straight–line

LBL H

LBL I

LBL J

Compute

model.

Compute

model.

for logarithmic

for exponential

for power model.

Compute

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

ꢀꢕꢔ ꢂꢌꢂ ꢀ

ꢀꢕꢏ ꢔ)ꢕꢔꢏ

Programming Techniques 13–23

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

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

13–24 Programming Techniques

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

Programming Techniques 13–25

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

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

{ ‰

. The current best estimate of the root

to view it without disturbing the

†

stack. To resume the calculation, press

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).

Solving and Integrating Programs 14–1

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

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

P = Pressure (atmospheres or N/m ).

V = Volume (liters).

N = Number of moles of gas.

14–2 Solving and Integrating Programs

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Size : 17.7 x 25.2 cm

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:

Selects "G"—the program. SOLVE

evaluates to find the value of the

{ Vꢁ

ꢎ

Solving and Integrating Programs 14–3

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Size : 17.7 x 25.2 cm

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

ꢄ@valueꢎ

ꢁ@valueꢎ

!@valueꢎ

ꢎ

.005

.0821

24 273.1

šꢁ

™

!@ꢏꢓꢘ)ꢔꢕꢕꢕꢎ

ꢑꢂ#ꢊꢄꢆꢎ

ꢅ/ꢕ)ꢕ ꢔꢕꢎ

Stores 297.1 in T; solves for P.

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

Labels the program.

z “

ꢐꢕꢔ ꢂꢌꢂ ꢐꢎ

{ }

ꢋ ꢐꢕꢏ ꢋ ꢔꢔꢎ

Enables equation prompting.

{ x

Evaluates the equation, clearing

flag 11. (Checksum and length:

13E3 015.0).

{ G

K ^Py

ꢎ

K {

K ^Ny

ꢎ

K y

K ^Tš

{ ”

†

ꢐꢕꢖ ꢅº#/ꢄºꢁºꢎ

Ends the program.

Cancels Program–entry mode.

ꢐꢕꢒ ꢁ!ꢄꢎ

ꢕ)ꢕ ꢔꢕꢎ

Checksum and length of program: 8AD6 19.5

14–4 Solving and Integrating Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Now calculate the change in pressure of the carbon dioxide if its temperature

drops by 10 °C from the previous example.

Keys:

Display:

Description:

Stores previous pressure.

ꢕ)ꢕ ꢔꢕꢎ

Enters the limits of integration

(lower limit first).

{ V

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

#@ꢏ)ꢕꢕꢕꢕꢎ

ꢄ@ꢕ)ꢕꢕꢗꢕꢎ

ꢁ@ꢕ)ꢕꢙꢏꢔꢎ

!@ꢏꢓꢘ)ꢔꢕꢕꢕꢎ

!@ꢏꢙꢘ)ꢔꢕꢕꢕꢎ

ꢑꢂ#ꢊꢄꢆꢎ

„

Stores 287.1 in T; solves for new P.

ꢅ/ꢕ)ꢕꢗꢙꢓꢎ

.ꢕ)ꢕꢕꢏꢔꢎ

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

Solving and Integrating Programs 14–5

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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.

Program Lines:

Description:

Setup for X.

%ꢕꢔ ꢂꢌꢂ %

Index for X.

%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

)

RTN

ꢋꢔꢕ

14–6 Solving and Integrating Programs

File name 32sii-Manual-E-0424

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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

then

key in the upper limit.

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

†

However, no information about the integration is available until the

calculation finishes normally. To resume the calculation, press again.

Pressing

while an integration calculation is running cancels the ∫ FN

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

sin x

S_i(t) = (

)dx

∫

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

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

{

}

ꢁꢍ ꢎ

Selects Radians mode.

z Ÿ

Selects label S as the integrand.

Enters lower and upper limits of

integration.

{ V

ꢎ

ꢏ_ꢎ

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

_M^De⁻⁽

)

2dD.

S 2π

∫

Solving and Integrating Programs 14–9

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The ^{((D−M)÷S) ÷2}function is calculated by the routine labeled F. Other

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

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

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

File name 32sii-Manual-E-0424

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Size : 17.7 x 25.2 cm

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.

Mathematics Programs 15–1

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

This program uses the following equations. Coordinate conversion:

X²+ Y²+ Z²

X = R sin(P) cos(T)

R =

Y = R sin(P) sin(T)

T = arctan (Y/X)

Z = R cos(P)

P = arctan

X²+ Y²

Vector addition and subtraction:

v + v = (X + U)i + (Y + V)j + (Z + W)k

v – v = (U – X)i + (V – Y)j + (W – Z)k

Cross product:

v × v = (YW – ZV )i + (ZU – XW)j + (XV – YU)k

Dot Product:

D = XU + YV + ZW

Angle between vectors (γ):

R₁× R₂

G = arccos

where

v = X i + Y j + Z k

and

v =U i + V j + W k

The vector displayed by the input routines (LBL P and LBL R) is V .

Program Listing:

15–2 Mathematics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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).

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.

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).

ꢅꢕꢓ !ꢑ 'ꢎ

ꢅꢔꢕ ꢁ ꢎ

ꢅꢔꢔ 8T ¸8ºꢎ

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

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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.

ꢃꢕꢘ ꢁꢃꢂ 'ꢎ

ꢃꢕꢙ ꢁꢃꢂº #ꢎ

ꢃꢕꢓ ꢁꢃꢂ %ꢎ

ꢃꢔꢕ ꢁꢃꢂº $ꢎ

ꢃꢔꢔ .ꢎ

Calculates (ZU – WX), which is the Y component.

ꢃꢔꢏ ꢁꢃꢂ %ꢎ

ꢃꢔꢖ ꢁꢃꢂº "ꢎ

ꢃꢔꢒ ꢁꢃꢂ &ꢎ

Mathematics Programs 15–5

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Program Listing:

Program Lines:

Description

ꢃꢔꢗ ꢁꢃꢂº #ꢎ

ꢃꢔ .ꢎ

Stores (XV – YU), which is the Z component.

ꢃꢔꢘ !ꢑ 'ꢎ

ꢃꢔꢙ ꢁ ꢎ

Stores Y component.

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

ꢍꢔꢙ º65¸ꢎ

ꢍꢔꢘ ¸8º

8Tꢎ

ꢍꢔꢓ ꢁ ꢎ

Calculates the magnitude of the U, V, W vector.

ꢍꢏꢔ º65¸ꢎ

ꢍꢏꢕ ¸8º

8Tꢎ

15–6 Mathematics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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.

Mathematics Programs 15–7

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

3. Key in R and press

, key in T and press

, then key in P and press

Continue at step 5.

4. Key in X and press

, key in Y and press

, and key in Z and press

^{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

b. Subtract vector one from vector two by pressing

c. Compute the cross product by pressing

^{d. Compute the dot product by pressing}W ^{D and the angle between}

vectors by pressing

7. Optional: to review v ^{in polar form, press}W ^{P, then press}f

repeatedly to see the individual elements.

8. Optional: to review v in rectangular form, press

R, then press

repeatedly to see the individual elements.

9. If you added, subtracted, or computed the cross product, v has been

replaced by the result, v is not altered. To continue calculations based on

the result, remember to press

E before keying in a new vector.

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 .

The radius, the angle in the x–y plane (θ), and the angle

from the Z axis of v (U).

The dot product

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

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

rectangular to polar conversion capability to find the total distance and the

direction to the transmitter.

(y)

7.3

Transmitter

15.7

Antenna

(x)

Keys:

Display:

Description:

z Ÿ {

}

Sets Degrees mode.

ꢎ

ꢍꢆ

%@valueꢎ

7.3

Starts rectangular input/display

routine.

fꢁ

&@valueꢎ

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

Calculates P, the angle from the

z-axis.

Mathematics Programs 15–9

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

T = 215

P = 17

F = 23

1.07m

T = 80

P = 74

125

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

²¹⁵f

ꢁ@ꢔꢘ)ꢕꢕꢕ

W ^E

Enters vector by copying it into v .

ꢁ@ꢔꢘ)ꢕꢕꢕ

Sets radius of v , equal to 23.

!@.ꢔꢒꢗ)ꢕꢕꢕꢕꢎ

ꢅ@ꢔꢘ)ꢕꢕꢕꢕꢎ

Sets T equal to 80.

15–10 Mathematics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Sets P equal to 74.

ꢁ@ꢏꢖ)ꢕꢕꢕꢕꢎ

ꢁ@ꢏꢓ)ꢒꢘꢒꢔꢎ

Adds the vectors and displays the

resultant R.

Displays T of resultant vector.

Displays P of resultant vector.

Enters resultant vector.

!@ꢓꢕ)ꢘꢕꢖꢏꢎ

ꢅ@ꢖꢓ)ꢓꢒꢒꢗꢎ

ꢁ@ꢏꢓ)ꢒꢘꢒꢔꢎ

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

Sets R equal to 1.07.

Sets T equal to 125.

!@ꢓꢕ)ꢘꢕꢖꢏꢎ

ꢅ@ꢖꢓ)ꢓꢒꢒꢗꢎ

ꢁ@ꢔ)ꢕꢘꢕꢕꢎ

ꢁ@ꢔꢙ)ꢕꢏꢕꢓꢎ

Sets P equal to 63.

Calculates cross product and

displays R of result.

Displays T of cross product.

Displays P of cross product.

Displays rectangular form of cross

product.

!@ꢗꢗ)ꢖꢘꢔꢓꢎ

ꢅ@ꢔꢏꢒ)ꢖꢒꢔꢏꢎ

%@ꢙ)ꢒꢗꢗꢒꢎ

&@ꢔꢏ)ꢏꢒꢖꢓꢎ

'@.ꢔꢕ)ꢔ ꢕꢎ

The dot product can be used to resolve the force (still in v ) along the axis of

the lever.

Keys:

Display:

Description:

W ^P

Starts polar input routine.

Defines the radius as one unit

vector.

ꢁ@ꢔꢙ)ꢕꢏꢕꢓꢎ

!@ꢗꢗ)ꢖꢘꢔꢓꢎ

125

Sets T equal to 125.

Sets P equal to 63.

ꢅ@ꢔꢏꢒ)ꢖꢒꢔꢏꢎ

ꢁ@ꢔ)ꢕꢕꢕꢕꢎ

Mathematics Programs 15–11

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Calculates dot product.

ꢍ/ꢏꢒ)ꢔꢙꢙꢏꢎ

ꢆ/ꢖꢒ)ꢙꢒꢓꢕꢎ

Calculates angle between

resultant force vector and lever.

Gets back to input routine.

ꢁ@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







   

   

B E H Y = K

   



   

C F I Z

   

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







   

   

′

B E H K = Y

   

   

 ′

′    

C F I L

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

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

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.

ꢊꢕꢓ !ꢑ %ꢎ

ꢊꢔꢕ ꢁꢃꢂ ꢃꢎ

ꢊꢔꢔ ꢁꢃꢂº ꢍꢎ

ꢊꢔꢏ ꢁꢃꢂ ꢀꢎ

ꢊꢔꢖ ꢁꢃꢂº ꢋꢎ

ꢊꢔꢒ .ꢎ

Calculates F' × determinant = CD – AF.

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

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,

ꢊꢖꢖ !ꢑ ꢀꢎ

ꢊꢖꢒ ꢁꢃꢂ ꢃꢎ

ꢊꢖꢗ ꢁꢃꢂº ꢐꢎ

ꢊꢖ ꢁꢃꢂ ꢌꢎ

ꢊꢖꢘ ꢁꢃꢂº ꢊꢎ

ꢊꢖꢙ .ꢎ

Calculates B' × determinant = CH – BI.

ꢊꢖꢓ ꢁꢃꢂ ꢌꢎ

ꢊꢒꢕ ꢁꢃꢂº ꢋꢎ

ꢊꢒꢔ ꢁꢃꢂ ꢃꢎ

ꢊꢒꢏ ꢁꢃꢂº ꢈꢎ

ꢊꢒꢖ .ꢎ

Calculates C' × determinant = BF – CE.

ꢊꢒꢒ !ꢑ ꢃꢎ

ꢊꢒꢗ ꢁ ꢎ

Stores B'.

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

Program Lines:

Description

ꢊꢗꢏ ꢁꢃꢂ ꢍꢎ

ꢊꢗꢖ ꢁꢃꢂº ꢐꢎ

ꢊꢗꢒ ꢁꢃꢂ ꢈꢎ

ꢊꢗꢗ ꢁꢃꢂº ꢆꢎ

ꢊꢗ .ꢎ

Calculates G', × determinant = DH – EG.

ꢊꢗꢘ !ꢑ ꢆꢎ

ꢊꢗꢙ ꢁ ꢎ

Stores D'.

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

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

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

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

4. Optional: press

D to compute determinant of 3 3 system.

5. Press

^{6. Optional: press}W ^{A and repeatedly press}f ^{to review the values of}

I to compute inverse of 3 × 3 matrix.

the inverted matrix.

7. ^PressW ^{M to multiply the inverted matrix by the column vector and to}

see the value of X . Press

to see the value of Y, then press

again

to see the value of Z.

8. For a new case, go back to step 2.

Variables Used:

A through I

J through L

X through Z

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

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:

Starts input routine.

ꢀ@valueꢎ

Sets first coefficient, A, equal to

23.

ꢌ@valueꢎ

Sets B equal to 8.

ꢃ@valueꢎ

ꢍ@valueꢎ

ꢈ@valueꢎ

Sets C equal to 4.

Sets D equal to 15.

Continues entry for E through L.

Returns to first coefficient entered.

Calculates the inverse and displays

the determinant.

ꢀ@ꢏꢖ)ꢕꢕꢕꢕꢎ

ꢒ8ꢗꢓꢙ)ꢕꢕꢕꢕꢎ

Multiplies by column vector to

compute X.

%/ꢕ)ꢓꢖꢕ ꢎ

Calculates and displays Y.

Calculates and displays Z.

Begins review of the inverted

matrix.

¸/ꢕ)ꢘꢓꢒꢖꢎ

'/.ꢕ)ꢔꢖ ꢒꢎ

ꢀ@ꢕ)ꢕꢒꢙꢖꢎ

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.

Mathematics Programs 15–19

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Displays next value.

Inverts inverse to produce original

matrix.

ꢆ@.ꢕ)ꢕ ꢕꢏꢎ

ꢐ@ꢕ)ꢕꢗꢓ ꢎ

ꢊ@ꢕ)ꢕꢏꢙꢓꢎ

ꢕ)ꢕꢕꢕꢏꢎ

W ^I

Begins review of inverted matrix.

Displays next value, ...... and so

on.

ꢀ@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–1

x + a x

+ ... + a x + a = 0

n–1

where n = 2, 3, 4, or 5. The coefficient of the highest–order term (a ) is

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:

y + b y + b y + b = 0

where b = – a

b = a a – 4a

3 1

15–20 Mathematics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

b = a (4a – a ) – a .

Let y be the largest real root of the above cubic. Then the fourth–order

polynomial is reduced to two quadratic polynomials:

x + (J + L)× + (K + M) = 0

x + (J – L)x + (K – M) = 0

where J = a /2

K = y /2

J²− a₂+ y₀

L =

× (the sign of JK – a /2)

K²− a₂

M =

Roots of the fourth degree polynomial are found by solving these two

quadratic polynomials.

A quadratic equation x + a x + a = 0 is solved by the formula

a₁

x_1,2= −

( )²− a₀

If the discriminant d = (a /2) – a ≥ 0, the roots are real; if d < 0, the roots

u iv = −(a 2) i − d

are complex, being

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

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 .

15–22 Mathematics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Program Lines:

Description

ꢉꢕꢖ ꢏꢎ

a /2.

ꢉꢕꢒ ªꢎ

–a /2.

ꢉꢕꢗ -+.ꢎ

ꢉꢕ ꢈꢄ!ꢈꢁꢎ

ꢉꢕꢘ ꢈꢄ!ꢈꢁꢎ

ꢉꢕꢙ !ꢑ ꢋꢎ

ꢉꢕꢓ º^ꢏꢎ

Saves – a /2.

Stores real part if complex root.

(a /2) .

ꢉꢔꢕ ꢁ ꢎ

a .

(a /2) – a .

ꢉꢔꢔ .ꢎ

Initializes flag 0.

Discriminant (d) < 0

Sets flag 0 if d < 0 (complex roots).

ꢉꢔꢏ ꢃꢋ ꢕꢎ

ꢉꢔꢖ º6ꢕ@ꢎ

ꢉꢔꢒ ꢋ ꢕꢎ

ꢉꢔꢗ ꢀꢌ ꢎ

ꢉꢔ ꢉꢁ!ꢎ

Stores imaginary part if complex root.

ꢉꢔꢘ !ꢑ ꢆꢎ

ꢉꢔꢙ ꢋ @ꢎ

ꢉꢔꢓ ꢁ!ꢄꢎ

Complex roots

Returns if complex roots.

ꢉꢏꢕ !ꢑ. ꢋꢎCalculates – a /2 –

ꢉꢏꢔ ꢁ ꢎ

ꢉꢏꢏ !ꢑ- ꢆꢎCalculates – a /2 +

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

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

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.

ꢈꢕꢘ !ꢑ ꢌꢎ

ꢈꢕꢙ ꢁ ꢎ

ꢈꢕꢓ !ꢑ ꢃꢎ

ꢈꢔꢕ ꢁꢃꢂ ꢈꢎ

ꢈꢔꢔ ꢁꢃꢂ- %ꢎ

ꢈꢔꢏ !ꢑ ꢍꢎ

ꢈꢔꢖ #ꢊꢈ$ %ꢎ

Calculates a .

Stores a .

Displays real root of fifth–order polynomial.

Checksum and length: 0FE9 019.5

Starts fourth–order solution routine.

ꢍꢕꢔ ꢂꢌꢂ ꢍꢎ

ꢍꢕꢏ ꢒꢎ

4a .

ꢍꢕꢖ ꢁꢃꢂº ꢃꢎ

ꢍꢕꢒ ꢁꢃꢂ ꢍꢎ

ꢍꢕꢗ º^ꢏꢎ

a .

4a – a .

ꢍꢕ .ꢎ

a (4a – a ).

ꢍꢕꢘ ꢁꢃꢂº ꢀꢎ

ꢍꢕꢙ ꢁꢃꢂ ꢌꢎ

ꢍꢕꢓ º^ꢏꢎ

a .

b =a (4a – a ) – a .

ꢍꢔꢕ .ꢎ

Stores b .

a .

ꢍꢔꢔ !ꢑ ꢈꢎ

ꢍꢔꢏ ꢁꢃꢂ ꢃꢎ

15–24 Mathematics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Program Lines:

Description

b = –a .

ꢍꢔꢖ -+.ꢎ

ꢍꢔꢒ !ꢑ ꢆꢎ

ꢍꢔꢗ ꢁꢃꢂ ꢍꢎ

ꢍꢔ ꢁꢃꢂº ꢌꢎ

ꢍꢔꢘ ꢒꢎ

Stores b .

a .

a a .

3 1

4a .

b = a a – 4a .

ꢍꢔꢙ ꢁꢃꢂº ꢀꢎ

ꢍꢔꢓ .ꢎ

3 1

Stores b .

ꢍꢏꢕ !ꢑ ꢋꢎ

ꢍꢏꢔ ꢒꢎ

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

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

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

Mathematics Programs 15–25

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Program Lines:

Description

K = y /2

ꢋꢕꢒ !ꢑª ꢈꢎ

ꢋꢕꢗ ꢓꢎ

ꢋꢕ ꢔꢕ^%ꢎ

ꢋꢕꢘ ꢔ+ºꢎ

ꢋꢕꢙ ꢁꢃꢂ ꢈꢎ

ꢋꢕꢓ ºꢏꢎ

–9

Creates 10 as a lower bound for M

K .

M = K –a .

ꢋꢔꢕ ꢁꢃꢂ. ꢀꢎ

ꢋꢔꢔ º6¸@ꢎ

ꢋꢔꢏ ꢃꢂºꢎ

ꢋꢔꢖ ꢉꢁ!ꢎ M = K²− a₀

–9

If M < 10 , use 0 for M .

Stores M.

ꢋꢔꢒ !ꢑ ꢀꢎ

ꢋꢔꢗ ꢁꢃꢂ ꢍꢎ

JK.

ꢋꢔ ꢁꢃꢂº ꢈꢎ

a .

ꢋꢔꢘ ꢁꢃꢂ ꢌꢎ

ꢋꢔꢙ ꢏꢎ

a /2

ꢋꢔꢓ ªꢎ

JK – a /2.

ꢋꢏꢕ .ꢎ

ꢋꢏꢔ º/ꢕ@ꢎ

ꢋꢏꢏ ꢔꢎ

Use 1 if JK – a /2 = 0

Stores 1 or JK – a /2.

ꢋꢏꢖ !ꢑ ꢌꢎ

ꢋꢏꢒ ꢀꢌ ꢎ

Calculates sign of C.

ꢋꢏꢗ !ꢑª ꢌꢎ

ꢋꢏ ꢁꢃꢂ ꢍꢎ

ꢋꢏꢘ º^ꢏꢎ

J -– a .

ꢋꢏꢙ ꢁꢃꢂ. ꢃꢎ

ꢋꢏꢓ ꢁꢃꢂ- ꢈꢎ

ꢋꢖꢕ ꢁꢃꢂ- ꢈꢎ

ꢋꢖꢔ ꢉꢁ!ꢎ C =

J -– a +y .

J²− a₂+ y₀

Stores C with proper sign.

J + L.

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

Program Lines:

Description

polynomial.

J – L.

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

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

polynomials. (If a is 0, then 0 is a real root. Reduce the polynomial by one

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

2. Key in the program routines; press

when done.

†

3. Press

P to start the polynomial root finder.

4. Key in F, the order of the polynomial, and press

5. At each prompt, key in the coefficient and press

. You're not

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

Order

Constant

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,

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.

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.

Mathematics Programs 15–29

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Example 1:

Find the roots of x – x – 101x +101x + 100x – 100 = 0.

Keys:

Display:

Description:

Starts the polynomial root finder;

prompts for order.

ꢋ@valueꢎ

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.

ꢈ@valueꢎ

ꢍ@value

_ f

¹⁰¹f

ꢃ@value

101

ꢌ@value

100

ꢀ@valueꢎ

%/ꢔ)ꢕꢕꢕꢕꢎ

100

_ f

Calculates the second root.

Displays the third root.

%/ꢔ)ꢕꢕꢕꢕꢎ

%/ꢔꢕ)ꢕꢕꢕꢕꢎ

%/.ꢔꢕ)ꢕꢕꢕꢕꢎ

L/.ꢔ)ꢕꢕꢕꢕꢎ

Displays the fourth root.

Displays the fifth root.

Example 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:

Starts the polynomial root finder;

prompts for order.

ꢋ@valueꢎ

Stores 4 its F; prompts for D.

Stores –8/4 in D; prompts for C.

ꢍ@valueꢎ

ꢎ

p f

_ š

ꢃ@value

ꢎ

Store –13/4 in C. prompts for B.

_ š

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

Stores –10/4 in B; prompts for A.

Stores 22/4 in A; calculates the

first root.

ꢀ@value

p f

%/ꢕ)ꢙꢙꢏꢕꢎ

Calculates the second root.

Displays the real part of the third

root.

%/ꢖ)ꢔꢔꢙꢕꢎ

%/.ꢔ)ꢕꢕꢕꢕꢎ

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:

x + x – 6 = 0

Keys:

Display:

Description:

Starts the polynomial root finder;

prompts for order.

ꢋ@valueꢎ

Stores 2 its F; prompts for B.

Stores 4 its B; prompts for A.

Stores –6 its A; calculates the first

root.

ꢋ@valueꢎ

%/.ꢖ)ꢕꢕꢕꢕꢎ

Calculates the second root.

%/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

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

The inverse transformation is accomplished with the formulas below.

x = u cos – v sin + m

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

Old coordinate

system

[0, 0]

[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

ꢍꢕꢏ ꢊꢄꢅ"! ꢇꢎ

x–coordinate.

Prompts for and stores N, the new origin's

ꢍꢕꢖ ꢊꢄꢅ"! ꢄꢎ

y–coordinate.

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

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

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

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

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

5. Key in the rotation angle T and press

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.

8. Key in X and press

9. Key in Y, press

, and see the x–coordinate, U, in the new system.

10.Press

and see the y–coordinate, V, in the new system.

11. For another old–to–new transformation, press

and go to step 8. For a

new–to–old transformation, continue with step 12.

12.Press O to start the new–to–old transformation routine.

Mathematics Programs 15–35

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

to see X.

15. Press

to see Y.

16. For another new–to–old transformation, press

and go to step 13. For

an old–to–new transformation, go to step 7.

Variables Used:

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.

Remark:

For translation only, key in zero for T. For rotation only, key in zero for M and

Example:

For the coordinate stems shorn below, convert points P , P and P ,which are

currently in the (X, Y) system, to points in the (X', Y') system. Convert point P' ,

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

(6, 8)

( 9, 7)

_ _

( 5, 4)

(M, N)

P' (2.7, 3.6)

(M, N) = (7, 4)

T = 27 ^o

Keys:

Display:

Description:

{ }

ꢍꢆ ꢎ

Sets Degrees mode since T is given

in degrees.

z Ÿ

Starts the routine that defines the

transformation.

ꢇ@valueꢎ

Store 7 in M.

ꢄ@value

Store –4 in N.

_ f

!@value

Stores 27 in T.

ꢇ@ꢘ)ꢕꢕꢕꢕꢎ

Starts the old–to–new routine.

%@valueꢎ

Mathematics Programs 15–37

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Stores –9 in X.

_ f

&@valueꢎ

Stores 7 in Y and calculates U.

Calculates V.

"/.ꢓ)ꢏ ꢏꢏꢎ

#/ꢔꢘ)ꢕ ꢒꢓꢎ

%@.ꢓ)ꢕꢕꢕꢕꢎ

Resumes the old–to–new routine

for next problem.

Stores –5 in X.

_ f

&@ꢘ)ꢕꢕꢕꢕꢎ

"/.ꢔꢕ) ꢓꢏꢔꢎ

#/ꢗ)ꢒꢒꢘꢓꢎ

%@.ꢗ)ꢕꢕꢕꢎ

Stores –4 in Y.

_ f

Calculates V.

Resumes the old–to–new routine

for next problem.

Stores 6 in X .

&@.ꢒ)ꢕꢕꢕꢎ

"/ꢒ)ꢗꢗ ꢓꢎ

#/ꢔꢔ)ꢔꢒ ꢔꢎ

"@ꢒ)ꢗꢗ ꢓꢎ

#@ꢔꢔ)ꢔꢒ ꢔꢎ

%/ꢔꢔ)ꢕꢒꢕꢔꢎ

&/.ꢗ)ꢓꢙꢔꢙꢎ

Stores 8 in Y and calculates U.

Calculates V.

Starts the new–to–old routine.

Stores 2.7 in U.

^2.7f

3.6

Stores –3.6 in V and calculates X.

Calculates Y.

_ f

15–38 Mathematics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

estimates

Regression" in chapter 11.)

and

. (For definitions of these values, see "Linear

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

Exponential Curve Fit

Straight Line Fit

y = B Mx

y = Be

Logarithmic Curve Fit

Power Curve Fit

y = B + MIn x

y = Bx

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.

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

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

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

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.

Calls subroutine to compute

&ꢕꢖ %ꢈꢉ1L2ꢎ

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

&ꢕꢙ %ꢈꢉ1L2ꢎ

Stores

in X for next loop.

&ꢕꢓ !ꢑ %ꢎ

&ꢔꢕ ꢆ!ꢑ &ꢎ

Loops for another estimate.

Checksum and length: BA07 015.

This subroutine calculates

model.

for the straight–line

ꢀꢕꢔ ꢂꢌꢂ ꢀꢎ

ꢀꢕꢏ ꢁꢃꢂ ꢇꢎ

ꢀꢕꢖ ꢁꢃꢂº %ꢎ

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

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

Program Lines:

Description

Restores index value to its original value.

ꢆꢕꢏ !ꢑ. Lꢎ

ꢆꢕꢖ ꢁꢃꢂ &ꢎ

ꢆꢕꢒ ꢁꢃꢂ. ꢌꢎ

Calculates

=(Y – B) ÷ M.

ꢆꢕꢗ ꢁꢃꢂª ꢇꢎ

Returns to the calling routine.

Checksum and length: 0D3F 009.0

ꢆꢕ ꢁ!ꢄꢎ

This subroutine calculates

model.

for the logarithmic

ꢌꢕꢔ ꢂꢌꢂ ꢌꢎ

ꢌꢕꢏ ꢁꢃꢂ %ꢎ

ꢌꢕꢖ ꢂꢄꢎ

ꢌꢕꢒ ꢁꢃꢂº ꢇꢎ

Calculates

= 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

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

Returns to the calling routine.

ꢐꢕꢘ ꢁ!ꢄꢎ

Checksum and length: B00D 010.5

This subroutine calculates

model.

for the exponential

ꢃꢕꢔ ꢂꢌꢂ ꢃꢎ

ꢃꢕꢏ ꢁꢃꢂ ꢇꢎ

ꢃꢕꢖ ꢁꢃꢂº %ꢎ

ꢃꢕꢒ H^%ꢎ

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

Program Lines:

Description

This subroutine calculates

for the exponential

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.

ꢊꢕ ꢁꢃꢂª ꢇꢎ

ꢊꢕꢘ ꢁ!ꢄꢎ

Checksum and length: 7D3B 010.5

This subroutine calculates

for the power model.

ꢍꢕꢔ ꢂꢌꢂ ꢍꢎ

ꢍꢕꢏ ꢁꢃꢂ %ꢎ

ꢍꢕꢖ ꢁꢃꢂ ꢇꢎ

ꢍꢕꢒ ¸^%ꢎ

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.

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)

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

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:

ꢀ

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

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)

and press

U to undo (remove) the last data pair. If you discover that

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

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.

to see the

value prompt for the

–estimation routine.

^%@ˆ

based on x, key in x at the value prompt,

to see

(

based on y, press

until you see the

value prompt, key in y, then press

to see

(

x ^%@

12. For more estimations, go to step 10 or 11.

13. For a new case, go to step 2.

Variables Used:

Regression coefficient (y–intercept of a straight line);

16–8 Statistics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

also used for scratch.

Regression coefficient (slope of a straight line).

Correlation coefficient; also used for scratch.

The x–value of a data pair when entering data; the

hypothetical x when projecting ; or

(x–estimate) when given a hypothetical y.

The y–value of a data pair when entering data; the

hypothetical y when projecting

; or

(y–estimate) when given a hypothetical x.

Index variable used to indirectly address the correct

–, –projection equation.

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.

40.5

38.6

102

37.9

1.00

36.2

97.5

35.1

95.5

34.6

104.5

Keys:

Display:

Description:

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.

%@ꢔ)ꢕꢕꢕꢕꢎ

40.5

&@valueꢎ

104.5

38.6

%@ꢏ)ꢕꢕꢕꢎ

&@ꢔꢕꢒ)ꢗꢕꢕꢕꢎ

%@ꢖ)ꢕꢕꢕꢎ

¹⁰²f

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.

&@ꢔꢕꢏ)ꢕꢕꢕꢕꢎ

Statistics Programs 16–9

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Retrieves

prompt.

%@ꢒ)ꢕꢕꢕꢕꢎ

%@ꢖ)ꢕꢕꢕꢕꢎ

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.

37.9

100

&@ꢔꢕꢏ)ꢕꢕꢕꢕꢎ

%@ꢒ)ꢕꢕꢕꢕꢎ

&@ꢔꢕꢕ)ꢕꢕꢕꢕꢎ

%@ꢗ)ꢕꢕꢕꢕꢎ

&@ꢓꢘ)ꢗꢕꢕꢕꢎ

%@ )ꢕꢕꢕꢕꢎ

&@ꢓꢗ)ꢗꢕꢕꢕꢎ

%@ꢘ)ꢕꢕꢕꢕꢎ

ꢁ/ꢕ)ꢓꢓꢗꢗꢎ

36.2

97.5

35.1

95.5

34.6

Calculates regression coefficient B.

Calculates regression coefficient

ꢌ/ꢖꢖ)ꢗꢏꢘꢔꢎ

ꢇ/ꢔ)ꢘ ꢕꢔꢎ

Prompts for hypothetical x–value.

%@ꢘ)ꢕꢕꢕꢕꢎ

&@ꢓꢙ) ꢗꢏ ꢎ

%@ꢖꢙ)ꢖꢖꢖ ꢎ

Stores 37 in X and calculates

¹⁰¹f

Stores 101 in Y and calculates

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

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Logarithmic

W ^L

Exponential

W ^E

Power

W ^P

0.9959

8.9730

0.6640

98.6845

38.3151

To start:

0.9965

0.9945

–139.0088

65.8446

98.7508

38.2857

51.1312

0.0177

Y ( when X=37)

98.5870

38.3628

X ( when Y=101)

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.

"Upper tail"

area

Q [x]

Statistics Programs 16–11

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

^xe^{−((x−x)÷σ ) ÷2}dx

σ 2π

Q(x) = 0.5−

∫

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.

ꢍꢕꢒ !ꢑ ꢉꢎ

Displays Q(X).

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

Size : 17.7 x 25.2 cm

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

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

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

ꢋꢕꢏ ꢁꢃꢂ ꢍꢎ

ꢋꢕꢖ ꢁꢃꢂ. ꢇꢎ

ꢋꢕꢒ ꢁꢃꢂª ꢎ

ꢋꢕꢗ º^ꢏꢎ

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

16–14 Statistics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

3. After the prompt for M, key in the population mean and press

. (If the

mean is zero, just press

4. After the prompt for S, key in the population standard deviation and press

. (If the standard deviation is 1, just press

)

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

displayed.

8. To calculate Q(X) for a new X with the same mean and standard deviation,

press and go to step 7.

9. To calculate X given Q(X), press

10.After the prompt, key in the value of Q(X) and press

. The result, X, is

displayed.

11. To calculate X for a new Q(X) with the same mean and standard deviation,

press and go to step 10.

Variables Used:

Dummy variable of integration.

Population mean, default value zero.

Probability corresponding to the upper–tail area.

Population standard deviation, default value of 1.

Variable used temporarily to pass the value S ×

inverse program.

to the

2π

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

Size : 17.7 x 25.2 cm

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

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

Accepts the default value of 1 for

ꢔ)ꢕꢕꢕꢕꢎ

%@valueꢎ

ꢉ/ꢕ)ꢕꢕꢔꢒꢎ

Starts the distribution program and

prompts for X.

Enters 3 for X and starts

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.

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

Keys:

Display:

Description:

16–16 Statistics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Resumes program.

%@ꢖ)ꢕꢕꢕꢕꢎ

ꢉ/ꢕ)ꢕꢏꢏꢘꢎ

Enters X–value of 2 and calculates

Q(X).

10000

Multiplies by the population for the

revised estimate.

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

15.3

Starts the distribution program and

prompts for X.

%@valueꢎ

Enters 90 for X and calculates

Q(X).

ꢉ/ꢕ)ꢕꢔꢔꢔꢎ

Thus, we would expect that only about 1 percent of the students would do

better than score 90.

Keys:

Display:

Description:

Starts the inverse routine.

Stores 0.1 (10 percent) in Q(X)

and calculates X.

ꢉ@ꢕ)ꢕꢔꢔꢔꢎ

%/ꢘꢒ) ꢕꢘꢙꢎ

0.01

Resumes the inverse routine.

ꢉ@ꢕ)ꢔꢕꢕꢕꢎ

Statistics Programs 16–17

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

0.8

Stores 0.8 (100 percent minus 20

%/ꢒꢏ)ꢔꢏꢖꢏꢎ

percent) in Q(X) and calculates X.

Grouped Standard Deviation

The standard deviation of grouped data, S , is the standard deviation of

data points x , x , ... , x , occurring at positive integer frequencies f1, f2, ... ,

fn.

( xf )²

x²−

∑

i i

∑

S_xg=

( f )−1

∑

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

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

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Program Lines:

Description

Updates

in register 28.

∑

ꢋꢕꢗ !ꢑ-1L2ꢎ

_{ꢋꢕ ꢁꢃꢂº %ꢎ}x_if_i

ꢋꢕꢘ ꢏꢓꢎ

Stores index for register 29.

ꢋꢕꢙ !ꢑ Lꢎ

ꢋꢕꢓ ꢁ ꢎ

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.

∑

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

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 .

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

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.

3. Key in x –value (data point) and press

4. Key in f –value (frequency) and press

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

have pressed

in step 4, press

U and then press

again.

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

display the grouped standard deviation.

8. Press to display the weighted mean of the grouped data.

9. ^{To add additional data points, press}f ^{and continue at step 3.}

To start a new problem, start at step 2.

Variables Used:

Data point.

Data–point frequency.

Data–pair counter.

Grouped standard deviation.

Weighted mean.

16–20 Statistics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Index variable used to indirectly address the correct

statistics register.

Summation f .

Summation Σx f .

i i

Summation x f .

i i

Example:

Enter the following data and calculate the grouped standard deviation.

Group

115

Keys:

Display:

Description:

Prompts for the first x .

%@valueꢎ

ꢋ@valueꢎ

ꢄ/ꢔ)ꢕꢕꢕꢕꢎ

⁵f

Stores 5 in X; prompts for first f .

Stores 17 in F; displays the

counter.

Prompts for the second x .

%@ꢗ)ꢕꢕꢕꢕꢎ

ꢋ@ꢔꢘ)ꢕꢕꢕꢕꢎ

ꢄ/ꢏ)ꢕꢕꢕꢕꢎ

%@ꢙ)ꢕꢕꢕꢕꢎ

ꢋ@ꢏ )ꢕꢕꢕꢕꢎ

ꢄ/ꢖ)ꢕꢕꢕꢕꢎ

Prompts for second f .

Displays the counter.

Prompts for the third x .

Prompts for the third f .

Displays the counter.

You erred by entering 14 instead of 13 for x . Undo your error by executing

routine U:

W ^U

Removes the erroneous data;

displays the revised counter.

ꢄ/ꢏ)ꢕꢕꢕꢕꢎ

Prompts for new third x .

%@ꢔꢒ)ꢕꢕꢕꢕꢎ

ꢋ@ꢖꢘ)ꢕꢕꢕꢕꢎ

Prompts for the new third f .

Statistics Programs 16–21

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Keys:

Display:

Description:

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 .

¹⁵f

Prompts for the fourth f .

Displays the counter.

Prompts for the fifth x .

Prompts for the fifth f .

Displays the counter.

Prompts for the sixth x .

Prompts for the sixth f .

115

Displays the counter.

Calculates and displays the

grouped standard deviation

(sx) of the six data points.

Calculates and displays

†

ꢇ/ꢏꢖ)ꢒꢕꢙꢒꢎ

ꢏꢖ)ꢒꢕꢙꢒꢎ

weighted mean ( ).

Clears VIEW.

16–22 Statistics Programs

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

+ F(1+ (I 100))^−N+ B = 0









I 100

Balance, B

Payments, P

N 1

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

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

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

y { \ ™

K p

100

{ ] 0

„ K ^N{ ]

p K ™ K y

{ \ ™ K

100

{ ]

0 „ K ^N

™ K

Terminates the equation.

ꢅºꢔꢕꢕº1ꢔ.1ꢔ-ꢎ

(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

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

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

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

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.

c. Press

B to calculate initial balance of a loan or savings

{ œ

account.

d. Press

P to calculate periodic payment.

{ œ

loan.

F to calculate future value or balance of a

e. Press

{ œ

4. Key in the values of the four known variables as they are prompted for;

^pressf ^{after each value.}

5. When you press the last

, the value of the unknown variable is

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

Variables Used:

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%.)

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

(

Displays the leftmost part of the

TVM equation.

{ G z

Rºꢔꢕꢕº1ꢔ.1ꢔ-ꢎ

as needed )

Selects P; prompts for I.

Converts your annual interest rate

input to the equivalent monthly

rate.

{ œ

ꢊ@valueꢎ

10.5

ꢊ@ꢕ)ꢙꢙꢎ

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

Stores 36 in N; prompts for F.

Stores 0 in F; prompts for D.

Calculates B, the beginning loan

balance.

ꢋ@valueꢎ

ꢌ@valueꢎ

7250

ꢌ@ꢗ8ꢘꢗꢕ)ꢕꢕꢎ

1500 „

Stores 5750 in B; calculates

monthly payment, P.

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.

Selects I; prompts for P.

Rounds the payment to two

decimal places.

{ œ

z I

R@.ꢔꢙ )ꢙꢓꢎ

Calculates new payment.

Stores –176,89 in P; prompts for

™

R@.ꢔꢘ )ꢙꢓꢎ

ꢄ@ꢖ )ꢕꢕꢎ

Retains 36 in N; prompts for F.

Retains 0 in F; prompts for B.

Retains 5750 in B; calculates

monthly interest rate.

ꢋ@ꢕ)ꢕꢕꢎ

ꢌ@ꢗ8ꢘꢗꢕ)ꢕꢕꢎ

ꢑꢂ#ꢊꢄꢆꢎ

ꢊ/ꢕ)ꢗ ꢎ

)ꢘꢗꢎ

Calculates annual interest, rate.

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

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.

Selects F; prompts for P.

{ œ

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

ꢄ@ꢖ )ꢕꢕꢎ

ꢌ@ꢗ8ꢘꢗꢕ)ꢕꢕꢎ

ꢑꢂ#ꢊꢄꢆꢎ

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

LBL Y

VIEW Prime

LBL Z

Note: x is the

value in the

X-register.

P + 2

→

LBL P

Start

→

LBL X

FP [

]

→

yes

x = 0

yes

D >

√

+ 2

→

Miscellaneous Programs and Equations 17–7

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

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.

4. To see the next prime number, press

Variables Used:

Prime value and potential prime values.

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

789

Calculates next prime number after

ꢅ/ꢘꢓꢘ)ꢕꢕꢕꢕꢎ

ꢅ/ꢙꢕꢓ)ꢕꢕꢕꢕꢎ

789.

Calculates next prime number after

797.

17–10 Miscellaneous Programs and Equations

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

Part 3

Appendixes and Reference

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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

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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

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

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 (

{ }, { }, or { } ).

ꢍꢆ ꢁꢍ ꢆꢁ

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

Support, Batteries, and Service

File name 32sii-Manual-E-0424

Printed Date : 2003/4/24

Size : 17.7 x 25.2 cm

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< ^and6^{, 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.

A–4

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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.

†

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

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.

†

You can halt this factory test by pressing any key.

Support, Batteries, and Service

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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

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.

A–6

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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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Size : 17.7 x 25.2 cm

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.

A–8

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File name 32sii-Manual-E-0424

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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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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

B–1

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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

z —

˜ ) until you see the desired equation.

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

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

User Memory and the Stack

File name 32sii-Manual-E-0424

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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

key.

†

2. Press and hold down

3. Press

. (You will be pressing three keys simultaneously). When you

release all three keys, the display shows

is successful.

if the operation

ꢇꢈꢇꢑꢁ& ꢃꢂꢈꢀꢁ