HP (Hewlett Packard) HP 35S Scientific Calculator 35s User Manual

HP 35s scientific calculator  
user's guide  
H
HP part number F2215AA-90001  
Edition 1  
Contents  
1. Getting Started............................................................1-1  
Alpha Keys........................................................................ 1-3  
The Display and Annunciators.................................................1-12  
Periods and Commas in Numbers () () .......................... 1-23  
Contents  
1
Complex number display format (, , ·‚)....................1-24  
2. RPN: The Automatic Memory Stack..............................2-1  
3. Storing Data into Variables .........................................3-1  
2
Contents  
4. Real–Number Functions...............................................4-1  
Entering π.......................................................................... 4-3  
Contents  
3
5. Fractions.....................................................................5-1  
6. Entering and Evaluating Equations...............................6-1  
4
Contents  
7. Solving Equations........................................................7-1  
8. Integrating Equations ..................................................8-1  
Integrating Equations ( FN) .................................................... 8-2  
9. Operations with Complex Numbers .............................9-1  
10.Vector Arithmetic ......................................................10-1  
Contents  
5
Vectors in Programs................................................................... 10-7  
11.Base Conversions and Arithmetic and Logic................11-1  
12.Statistical Operations ................................................12-1  
6
Contents  
13.Simple Programming.................................................13-1  
Contents  
7
14.Programming Techniques ..........................................14-1  
Tests of Comparison (x?y, x?0)...........................................14-7  
The Variables "I" and "J" ................................................... 14-20  
Program Control with (I)/(J) ................................................ 14-23  
Equations with (I)/(J) .......................................................... 14-23  
8
Contents  
15.Solving and Integrating Programs..............................15-1  
16.Statistics Programs....................................................16-1  
17.Miscellaneous Programs and Equations......................17-1  
A. Support, Batteries, and Service ................................... A-1  
Service...................................................................................A-8  
Contents  
9
B. User Memory and the Stack.........................................B-1  
C. ALG: Summary ...........................................................C-1  
Doing Two argument Arithmetic in ALG ..................................... C-2  
Hyperbolic functions................................................................. C-6  
D. More about Solving.................................................... D-1  
10  
Contents  
Part 1  
Basic Operation  
   
1
Getting Started  
Watch for this symbol in the margin. It identifies examples or  
keystrokes that are shown in RPN mode and must be  
performed differently in ALG mode.  
Appendix C explains how to use your calculator in ALG mode.  
v
Important Preliminaries  
Turning the Calculator On and Off  
To turn the calculator on, press . ON is printed on the bottom of the key.  
To turn the calculator off, press . That is, press and release the shift  
key, then press (which has OFF printed in yellow above it). Since the calculator  
has Continuous Memory, turning it off does not affect any information you've stored.  
To conserve energy, the calculator turns itself off after 10 minutes of inactivity. If you  
see the low–power indicator ( ) in the display, replace the batteries as soon as  
possible. See appendix A for instructions.  
Adjusting Display Contrast  
Display contrast depends on lighting, viewing angle, and the contrast setting. To  
increase or decrease the contrast, hold down the key and press or .  
1-1  
       
Highlights of the Keyboard and Display  
Shifted Keys  
Each key has three functions: one printed on its face, a left–shifted function  
(yellow), and a right–shifted function (blue). The shifted function names are printed  
in yellow above and in blue on the bottom of 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 .  
1-2  
   
Pressing 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  
Left-shifted  
function  
Right-shifted  
function  
Letter for alphabetic  
key  
Most keys display a letter in their bottom right corner, 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 13.  
Cursor Keys  
Each of the four cursor direction keys is marked with an arrow. In this text we will  
use the graphics Õ, Ö, × and Øto refer to these keys.  
1-3  
   
Backspacing and Clearing  
Among the first things you need to know are how to clear an entry, correct a  
number, and clear the entire display to start over.  
Keys for Clearing  
Key  
Description  
Backspace.  
If an expression is in the process of being entered, erases the  
character to the left of the entry cursor ( _ ). Otherwise, with a  
completed expression or the result of a calculation in line 2,   
replaces that result with a zero. also clears error messages  
and exits menus. behaves similarly when the calculator is in  
program-entry and equation-entry modes, as discussed below:  
Equation–entry mode:  
If an equation is in the process of being entered or edited,  
erases the character immediately to the left of the insert  
cursor; otherwise, if the equation has been entered (no insert  
cursor present), deletes the entire equation.  
Program-entry mode:  
If a program line is in the process of being entered or  
edited, erases the character to the left of the insert  
cursor; otherwise, if the program line has been entered,   
deletes the entire line.  
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).  
1-4  
 
Keys for Clearing (continued)  
Description  
Key  
The CLEAR menu (      )  
contains options for clearing x (the number in the X-register), all  
direct variables, all of memory, all statistical data, all stacks and  
indirect variables.  
If you press (), a new menu    is  
displayed so you can verify your decision before erasing  
everything in memory.  
During program entry,  is replaced by . If you press  
(), a new menu     is displayed, so you  
can verify your decision before erasing all your programs.  
During equation entry,  is replaced by . If you press  
(), the     menu is displayed, so you can  
verify your decision before erasing all your equations.  
When you select (), the command is pasted into the  
command line with three placeholders. You must enter a 3-digit  
number in the placeholder blanks. Then all the indirect variables  
whose addresses are greater than the address entered are  
erased. For example: CLVAR056 erases all indirect variables  
whose address is greater than 56.  
1-5  
Using Menus  
There is a lot more power to the HP 35s than what you see on the keyboard. This is  
because 16 of the keys are menu keys. There are 16 menus in all, which provide  
many more functions, or more options for more functions.  
HP 35s Menus  
Menu  
Name  
Menu  
Description  
Chapter  
Numeric Functions  
L.R.  
12  
12  
ˆ
ˆ     
Linear regression: curve fitting and linear estimation.  
y
x ,  
    
Arithmetic mean of statistical x– and y–values;  
weighted mean of statistical x–values.  
  σσ  
Sample standard deviation, population standard  
deviation.  
s,σ  
12  
4
Menu to access the values of 41 physics constants—  
refer to  
CONST  
"Physics constants" on page 4–8.  
SUMS  
BASE  
   
       
12  
12  
Statistical data summations.  
     
Base conversions (decimal, hexadecimal, octal, and  
binary).  
INTG  
4,C  
11  
 ÷      
Sign value, integer division, remainder from division,  
greatest integer, fractional part, integer part  
LOGIC  
       
Logic operators  
1-6  
 
Programming Instructions  
    
FLAGS  
x?y  
14  
14  
14  
Functions to set, clear, and test flags.  
≠ ≤ < > ≥ =  
Comparison tests of the X–and Y–registers.  
≠ ≤ < > ≥ =  
x?0  
Comparison tests of the X–register and zero.  
Other functions  
MEM  
   
1, 3, 12  
Memory status (bytes of memory available); catalog  
of variables; catalog of programs (program labels).  
      
MODE  
4, 1  
1
Angular modes and operation mode  
DISPLAY  
         
   
Fixed, scientific, engineering, full floating point  
numerical display formats; radix symbol options (. or  
,); complex number display format (in RPN mode,  
only xiy and rθa are available)  
RR ꢁ  
C
     
Functions to review the stack in ALG mode –X–, Y–,  
Z–, T–registers  
Functions to clear different portions of memory—refer  
CLEAR  
1, 3,  
6, 12  
to   
in the table on page 1–5.  
To use a menu function:  
1. Press a menu key to display a set of menu items.  
2. Press Õ Ö × Øto move the underline to the item you want to select.  
3. Press while the item is underlined.  
With numbered menu items, you can either press while the item is  
underlined, or just enter the number of the item.  
1-7  
Some menus, like the CONST and SUMS, have more than one page. Entering these  
menus turns on the or annunciator. In these menus, use the Õand Ö  
cursor keys to navigate to an item on the current menu page; use the Øand ×  
keys to access the next and previous pages in the menu.  
Example:  
In this example, we use the DISPLAY menu to fix the display of numbers to 4 decimal  
places and then compute 6÷7. The example closes using the DISPLAY menu to return  
to full floating point display of numbers.  
Keys:  
Display:  
Description:  
Initial display  
Enter the DISPLAY menu  
8  
   
   
   
or   
The Fix command is pasted to line 2  
Fix to 4 decimal places  
  
  
  
  
  
Perform the division  
Return to full precision  
8  
  
Menus help you execute dozens of functions by guiding you to them. You don’t have  
to remember the names of all the functions built into the calculator nor search  
through the functions printed on the 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:  
1-8  
 
Pressing backs out of the 2–level CLEAR or MEM menu, one level at a  
time. Refer to in the table on page 1–5.  
Pressing or cancels any other menu.  
Keys: Display:  
  
_  
  
  
  
  
8  
_  
or   
Pressing another menu key replaces the old menu with the new one.  
Keys:  
Display:  
  
_  
  
  
  
  
  
8  
  
  
  
  
RPN and ALG Modes  
The calculator can be set to perform arithmetic operations in either RPN (Reverse  
Polish Notation) or ALG (Algebraic) mode.  
In Reverse Polish Notation (RPN) mode, the intermediate results of calculations are  
stored automatically; hence, you do not have to use parentheses.  
In Algebraic mode (ALG), you perform arithmetic operations using the standard  
order of operations.  
To select RPN mode:  
Press 9{() to set the calculator to RPN mode. When the calculator  
is in RPN mode, the RPN annunciator is on.  
1-9  
 
To select ALG mode:  
Press 9{() to set the calculator to ALG mode. When the calculator  
is in ALG mode, the ALG annunciator is on.  
Example:  
Suppose you want to calculate 1 + 2 = 3.  
In RPN mode, you enter the first number, press the key, enter the second  
number, and finally press the arithmetic operator key: .  
In ALG mode, you enter the first number, press , enter the second number, and  
finally press the key.  
RPN mode  
ALG mode  
1 2   
1 2   
In ALG mode, the results and the calculations are displayed. In RPN mode, only the  
results are displayed, not the calculations.  
You can choose either ALG (Algebraic) or RPN (Reverse Polish  
Notation) mode for your calculations. Throughout the manual, the  
Note  
v“ in the margin indicates that the examples or keystrokes in RPN  
mode must be performed differently in ALG mode. Appendix C  
explains how to use your calculator in ALG mode.  
Undo key  
The Undo Key  
The operation of the Undo key depends on the calculator context, but serves largely  
to recover from the deletion of an entry rather than to undo any arbitrary operation.  
See The Last X Register in Chapter 2 for details on recalling the entry in line 2 of the  
display after a numeric function is executed. Press : immediately after  
using or to recover:  
an entry that you deleted  
an equation deleted while in equation mode  
a program line deleted while in program mode  
In addition, you can use Undo to recover the value of a register just cleared using  
the CLEAR menu. The Undo operation must immediately follow the delete operation;  
any intervening operations will keep Undo from retrieving the deleted object. In  
addition to retrieving an entire entry after its deletion, Undo can also be used while  
editing an entry. Press : while editing to recover:  
a digit in an expression that you just deleted using   
an expression you were editing but cleared using   
a character in an equation or program that you just deleted using (while  
in equation or program mode)  
Please note also that the Undo operation is limited by the amount of available  
memory.  
 
The Display and Annunciators  
First Line  
Second Line  
The display comprises two lines and annunciators.  
Entries with more than 14 characters will scroll to the left. During input, the entry is  
displayed in the first line in ALG mode and the second line in RPN mode. Every  
calculation is displayed in up to 14 digits, including an sign (exponent), and  
exponent value up to three digits.  
Annunciators  
The symbols on the display, shown in the above figure, are called annunciators.  
Each one has a special significance when it appears in the display.  
 
HP 35s Annunciators  
Meaning  
Annunciator  
Chapter  
The "(Busy)" annunciator appears while  
an operation, equation, or program is  
executing.  
When in Fraction–display mode (press   
5
), only one of the "" or "" halves  
of the "ꢄꢅ"' 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 "ꢄꢅ" is on, the  
exact value of the fraction is being  
displayed.  
Left shift is active.  
1
1
Right shift is active.  
RPN  
Reverse Polish Notation mode is active.  
Algebraic mode is active.  
1, 2  
1, C  
13  
6
ALG  
PRGM  
EQN  
Program–entry is active.  
Equation–entry mode is active, or the  
calculator is evaluating an expression or  
executing an equation.  
0 1 2 3 4  
RAD or GRAD  
HEX OCT BIN  
HYP  
Indicates which flags are set (flags 5  
through 11 have no annunciator).  
Radians or Grad angular mode is set. DEG  
mode (default) has no annunciator.  
Indicates the active number base. DEC  
(base 10, default) has no annunciator.  
Hyperbolic function is active.  
14  
4
11  
4, C  
HP 35s Annunciators (continued)  
Meaning  
Annunciator  
Chapter  
There are more characters to the left or right in  
the display of the entry in line 1 or line 2. Both  
of these annunciators may appear  
1, 6  
,ꢆ  
simultaneously, indicating that there are  
characters to the left and right in the display of  
an entry. Entries in line 1 with missing  
characters will show an ellipsis (…) to indicate  
missing characters. In RPN mode, use the Õ  
and Ökeys to scroll through an entry and  
see the leading and trailing characters. In ALG  
mode, use Õand Öto see the  
rest of the characters.  
1, 6, 13  
The Øand ×keys are active for stepping  
through an equation list, a catalog of  
variables, lines of a program, menu pages, or  
programs in the program catalog.  
,ꢃ  
A..Z  
The alphabetic keys are active.  
3
1
Attention! Indicates a special condition or an  
error.  
Battery power is low.  
A
Keying in Numbers  
The minimum and maximum values that the calculator can handle are  
499  
9.99999999999 . If the result of a calculation is beyond this range, the error  
message “” appears momentarily along with the annunciator. The  
overflow message is then replaced with the value closest to the overflow boundary  
that the calculator can display. The smallest numbers the calculator can distinguish  
-499  
from zero are 10  
. If you enter a number between these values, the calculator  
will display 0 upon entry. Likewise, if the result of calculation lies between these two  
values, the result will be displayed as zero. Entering numbers beyond the maximum  
range above will result in an error message “ ”; clearing the error  
message returns you to the previous entry for correction.  
Making Numbers Negative  
The key changes the sign of a number.  
To key in a negative number, type the number, then press ,  
In ALG mode, you may press key before or after typing the number.  
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.)  
Exponents of Ten  
Exponents in the Display  
-5  
Numbers with explicit powers of ten (such as 4.2x10 ) are displayed with an E  
-5  
preceding the exponent of 10. Thus 4.2x10 is entered and displayed as 4.2E-5.  
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:  
Shows number being entered.  
  
  
Rounds number to fit the display  
format.  
  
Automatically uses scientific notation  
because otherwise no significant digits  
would appear.  
  
  
  
Keying in Powers of Ten  
The key is used to enter powers of ten quickly. For example, instead of entering  
one million as 1000000 you can simply enter . The following example  
illustrates the process as well as how the calculator displays the result.  
Example:  
-34  
Suppose you want to enter Planck’s constant: 6.6261×10  
Keys:  
  
Display:  
Description  
Enter the mantissa  
_  
x
Equivalent to ×10  
_  
  
z  
Enter the exponent  
  
For a power of ten without a multiplier, as in the example of one million above,  
press the key followed by the desired exponent of ten.  
Other Exponent Functions  
To calculate an exponent of ten (the base 10 antilogarithm), use  . To  
calculate the result of any number raised to a power (exponentiation), use (see  
chapter 4).  
Understanding Entry Cursor  
As you key in a number, the cursor (_) appears and blinks 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:  
Entry not terminated: the number is not  
complete.  
If you execute a function to calculate a result, the cursor disappears because the  
number is complete —entry has been terminated.  
  
Entry is terminated.  
Pressing terminates entry. To separate two numbers, key in the first  
number, press to terminate entry, and then key in the second number  
  
  
A completed number.  
  
  
Another completed number.  
If entry is not terminated (if the cursor is present), backspaces to erase the last  
digit. If entry is terminated (no cursor), acts like and clears the entire  
number. Try it!  
Range of Numbers and OVERFLOW  
499  
The smallest number available on the calculator is –9.99999999999 × 10 ,while  
499  
the largest number is 9.99999999999 × 10  
.
If a calculation produces a result that exceeds the largest possible number, –  
499  
499  
is returned, and  
9.99999999999 × 10  
or 9.99999999999 × 10  
the warning message  appears.  
   
Performing Arithmetic Calculations  
The HP 35s can operate in either RPN mode or in Algebraic mode (ALG). These  
modes affect how expressions are entered. The following sections illustrate the entry  
differences for single argument (or unary) and two argument (or binary) operations.  
Single Argument or Unary Operations  
Some of the numerical operations of the HP 35s require a single number for input,  
such as , , &and k. These single argument operations are entered  
differently, depending on whether the calculator is in RPN or ALG mode. In RPN  
mode, the number is entered first and then the operation is applied. If the   
key is pressed after the number is entered, then the number appears in line 1 and  
the result is shown in line 2. Otherwise, just the result is displayed in line 2 and line  
1 is unchanged. In ALG mode, the operator is pressed first and the display shows  
the function, followed by a set of parentheses. The number is entered between the  
parentheses and then the key is pressed. The expression is displayed in line  
1 and the result is shown in line 2. The following examples illustrate the differences.  
   
Example:  
Calculate 3.4 , first in RPN mode and then in ALG mode.  
2
Keys:  
Display:  
Description:  
Enter RPN mode (if necessary)  
Enter the number  
9()  
  
  
  
Press the square operator  
  
Switch to ALG mode  
9()  
  
  
Enter the square operation  
Insert the number between the  
parentheses  
  
  
  
Press the Enter key to see the result  
  
In the example, the square operator is shown on the key as but displays as  
SQ(). There are several single argument operators that display differently in ALG  
mode than they appear on the keyboard (and differently than they appear in RPN  
mode as well). These operations are listed in the table below.  
Key  
In RPN,RPN Program  
In ALG, Equation, ALG Program  
SQ()  
2
X
?
#
SQRT()  
EXP()  
x  
x
e
x
!
ALOG()  
INV()  
10  
1/x  
Two Argument or Binary Operations  
Two argument operations, such as , , ), and x, are also entered  
differently depending on the mode though the differences are similar to the case for  
single argument operators. In RPN mode, the first number is entered, then the  
second number is placed in the x-register and the two argument operation is  
invoked. In ALG mode, there are two cases, one using traditional infix notation and  
another taking a more function-oriented approach. The following examples illustrate  
the differences.  
 
Example  
Calculate 2+3 and C , first in RPN mode and then in ALG mode.  
6
4
Keys:  
Display:  
Description:  
Switch to RPN mode (if necessary)  
Enter 2, then place 3 in the x-register.  
9()  
  
Note the flashing cursor after the 3;  
don’t press Enter!  
Press the addition key to see the result.  
_  
Enter 6, then place 4 in the x-register.  
  
_  
Press the combinations key to see the  
x  
result.  
  
Switch to ALG mode  
9()  
  
Expression and result are both shown.  
  
Enter the combination function.  
Enter the 6, then move the edit cursor  
past the comma and enter the 4.  
Press Enter to see the result.  
  
x  
  
Õ  
  
  
In ALG mode, the infix operators are , ,, , and . The other two  
argument operations use function notation of the form f(x,y), where x and y are the  
first and second operands in order. In RPN mode, the operands for two argument  
operations are entered in the order Y, then X on the stack. That is, y is the value in  
the y-register and x is the value in the x-register.  
th  
3
The x root of y (') is the exception to this rule. For example, to calculate 8 in  
RPN mode, press  '. In ALG mode, the equivalent  
operation is keyed in as 'Õ.  
As with the single argument operations, some of the two argument operations  
display differently in RPN mode than in ALG mode. These differences are  
summarized in the table below.  
Key  
In RPN, RPN Program  
In ALG, Equation, ALG Program  
x
^
y
x y  
INT÷  
XROOT(, )  
IDIV(, )  
For commutative operations such as and , the order of the operands does  
not affect the calculated result. If you mistakenly enter the operand for a  
noncommutative two argument operation in the wrong order in RPN mode, simply  
press the key to exchange the contents in the x- and y-registers. This is  
explained in detail in Chapter 2 (see the section entitled Exchanging the X- and Y-  
Registers in the Stack).  
Controlling the Display Format  
All numbers are stored with 12-digit precision; however, you may control the  
number of digits used in the display of numbers via the options in the Display menu.  
Press 8 to access this menu. The first four options (FIX, SCI, ENG, and  
ALL) control the number of digits in the display of numbers. During some  
complicated internal calculations, the calculator uses 15–digit precision for  
intermediate results. The displayed number is rounded according to the display  
format.  
Fixed–Decimal Format ()  
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 the number of  
decimal places to be displayed. For 10 or 11 places, press or .  
For example, in the number , the "7", "0", "8", and "9" are the  
decimal digits you see when the calculator is set to FIX 4 display mode.  
11  
-11  
Any number that is too large (10 ) or too small (10 ) to display in the current  
decimal–place setting will automatically be displayed in scientific format.  
 
Scientific Format ()  
SCI format displays a number in scientific notation (one digit before the "" or ""  
radix mark) with up to 11 decimal places and up to three digits in the exponent.  
After the prompt, _, type in the number of decimal places to be displayed. For  
10 or 11 places, press or . (The mantissa part of the number will  
always be less than 10.)  
For example, in the number , the "2", "3", "4", and "6" are the  
decimal digits you see when the calculator is set to SCI 4 display mode. The "5"  
5
following the "E" is the exponent of 10: 1.2346 × 10 .  
If you enter or calculate a number that has more than 12 digits, the additional  
precision is not maintained.  
Engineering Format ()  
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  
3
that use units specified in multiples of 10 (such as micro–, milli–, and kilo–units.)  
After the prompt, _, type in the number of digits you want after the first  
significant digit. For 10 or 11 places, press or .  
For example, in the number , the "2", "3", "4", and "6" are the  
significant digits after the first significant digit you see when the calculator is  
set to ENG 4 display mode. The "3" following the "" is the (multiple of 3)  
3
exponent of 10: 123.46 x 10 .,  
Pressing @ or 2 will cause the exponent display for the  
number being displayed to change in multiples of 3, with the mantissa adjusted  
accordingly.  
Example:  
This example illustrates the behavior of the Engineering format using the number  
12.346E4. It also shows the use of the @ and 2 functions.  
This example uses RPN mode.  
Keys:  
Display:  
_  
Description:  
Choose Engineering format  
8(  
)  
Enter 4 (for 4 significant digits after the  
1 )  
  
  
st  
  
  
  
Enter 12.346E4  
}  
@ or  
  
  
  
2  
@  
Increases the exponent by 3  
Decreases the exponent by 3  
  
  
2  
  
ALL Format ()  
The All format is the default format, displaying numbers with up to 12 digit  
precision. If all the digits don't fit in the display, the number is automatically  
displayed in scientific format.  
Periods and Commas in Numbers () ()  
The HP 35s uses both periods and commas to make numbers easier to read. You  
can select either the period or the comma as the decimal point (radix). In addition,  
you can choose whether or not to separate digits into groups of three using  
thousand separators. The following example illustrates the options.  
 
Example  
Enter the number 12,345,678.90 and change the decimal point to the comma.  
Then choose to have no thousand separator. Finally, return to the default settings.  
This example uses RPN mode.  
Keys:  
Display:  
Description:  
Select full floating point precision  
(ALL format)  
8(  
)  
  
The default format uses the comma  
as the thousand separator and the  
period as the radix.  
  
  
  
Change to use the comma for the  
radix. Note that the thousand  
separator automatically changes to  
the period.  
8()  
  
  
Change to having no comma  
separator.  
8(  
  
  
)  
Return to the default format.  
8()  
  
  
8(  
)  
Complex number display format ( ,  , )  
Complex numbers can be displayed in a number of formats:  ,  , and  
, although  is only available in ALG mode. In the example below, the  
complex number 3+4i is displayed in all three ways.  
 
Example  
Display the complex number 3+4i in each of the different formats.  
Keys:  
Display:  
Description:  
Enable ALG mode  
9()  
6  
Enter the complex number. It displays  
as 3i4, the default format.  
Change to x+yi format.  
8  
( )  
8  
() or  
8×  
×Õ  
  
Change to rθ a format. The radius is  
5 and the angle is approximately  
53.13°.  
θ  
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 Î. This shows  
you the mantissa (but no exponent) of the number for as long as you hold down  
Î.  
Keys:  
  
  
Display:  
  
Description:  
Four decimal places displayed.  
Scientific format: two decimal  
places and an exponent.  
8()  
  
  
Engineering format.  
8()  
 
All significant digits; trailing  
zeros dropped.  
  
8()  
Four decimal places, no exponent.  
  
8()  
  
Reciprocal of 58.5.  
Shows full precision until you release  
  
Î(hold)  
Fractions  
The HP 35s allows you to enter and operate on fractions, displaying them as either  
decimals or fractions. The HP 35s displays fractions in the form a b/c, where a is an  
integer and both b and c are counting numbers. In addition, b is such that 0b<c  
and c is such that 1<c4095.  
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 or a function key to 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.  
The following example illustrates the entry and display of fractions.  
   
Example  
Enter the mixed numeral 12 3/8 and display it in fraction and decimal forms. Then  
enter ¾ and add it to 12 3/8. This example uses RPN mode.  
Keys:  
  
Display:  
Description:  
The decimal point is interpreted in the  
normal way.  
  
nd  
  
 _  
  
When is pressed the 2 time, the  
display switches to fraction mode.  
  
  
Upon entry, the number is displayed  
using the current display format.  
   
   
Switch to fraction display mode.  
É  
  
   
Enter ¾. Note you start with   
because there is no integer part (you  
could type in 0 ¾).  
 _  
Add ¾ to 12 3/8.  
   
É  
Switch back to the current display  
mode.  
  
Refer to chapter 5, "Fractions," for more information about using fractions.  
Messages  
The calculator responds to error conditions by displaying the annunciator.  
Usually, a message will accompany the error annunciator as well.  
To clear a message, press or ; in RPN mode, you will return to the  
stack as it was before the error. In ALG mode, you will return to the last  
expression with the edit cursor at the position of the error so that you can  
correct it.  
 
Any other key also clears the message, though the key function is not entered  
If no message is displayed, but the annunciator appears, then you have pressed  
an inactive or invalid key. For example, pressing  will display because  
the second decimal point has no meaning in this context.  
All displayed messages are explained in appendix F, "Messages".  
Calculator Memory  
The HP 35s has 30KB of memory in which you can store any combination of data  
(variables, equations, or program lines).  
Checking Available Memory  
Pressing   displays the following menu:  
    
  
  
Where  
 is the amount of used indirect variables.  
 is the number of bytes of memory available.  
Pressing the () displays the catalog of direct variables (see "Reviewing  
Variables in the VAR Catalog" in chapter 3). Pressing the () displays the  
catalog of programs.  
1. To enter the catalog of variables, press (); to enter the catalog of  
programs, press ().  
2. To review the catalogs, press Øor ×.  
3. To delete a variable or a program, press   
catalog.  
while viewing it in its  
4. To exit the catalog, press .  
   
Clearing All of Memory  
Clearing all of memory erases all numbers, equations, and programs you've  
stored. It does not affect mode and format settings. (To clear settings as well as  
data, see "Clearing Memory" in appendix B.)  
To clear all of memory:  
1. Press (). You will then see the confirmation prompt    ,  
which safeguards against the unintentional clearing of memory.  
2. Press Ö () .  
 
2
RPN: The Automatic  
Memory Stack  
This chapter explains how calculations take place in the automatic memory stack in  
RPN mode. 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 35s 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 Łukasiewicz  
(1878–1956).  
While conventional algebraic notation places the operators between the relevant  
numbers or variables, Łukasiewicz'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.  
2-1  
   
P a r t 3  
P a r t 2  
“ O l d e s t ” n u m b e r  
T
P a r t 1 0 . 0 0 0 0  
P a r t 3  
P a r t 2  
Z
Y
P a r t 1 0 . 0 0 0 0  
P a r t 3  
P a r t 2  
D i s p l a y e d  
D i s p l a y e d  
P a r t 1 0 . 0 0 0 0  
P a r t 3  
P a r t 2  
X
P a r t 1 0 . 0 0 0 0  
The most "recent" number is in the X–register: this is the number you see in the  
second line of the display.  
Every register is separated into three parts:  
A real number or a 1-D vector will occupy part 1; part 2 and part 3 will be  
null in this case.  
A complex number or a 2-D vector will occupy part 1 and part 2; part 3 will  
be null in this case.  
A 3-D vector will occupy part 1, part 2, and part 3.  
In programming, the stack 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.  
2-2  
The X and Y–Registers are in the Display  
The X and Y–Registers are what you see except when a menu, a message, an  
equation line ,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.  
Clearing the X–Register  
Pressing   
() always clears the X–register to zero; it is also used to  
program this instruction. The key, in contrast, is context–sensitive. It either clears  
or cancels the current display, depending on the situation: it acts like  
1() only when the X–register is displayed. also acts like  
() when the X–register is displayed and digit entry is terminated  
(no cursor present).  
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 the numbers  
as they roll through the x- and y-registers.  
Suppose the stack is filled with 1, 2, 3, 4. (press  
) Pressing four times rolls the numbers  
all the way around and back to where they started:  
1
2
3
4
4
1
2
3
3
4
1
2
2
3
4
1
1
2
3
4
T
Z
Y
X
2-3  
     
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 contents of the registers are rolled  
— the registers themselves maintain their positions, and only the X– and Y–register's  
contents are displayed.  
R(Roll Up)  
The  (roll up) key has a similar function to except that it "rolls" the stack  
contents upward, one register at a time.  
The contents of the X–register rotate into the Y–register; what was in the T–register  
rotates into the X–register, and so on.  
T
Z
Y
X
1
2
3
4
2
3
4
1
3
4
1
2
4
1
2
3
1
2
3
4
  
  
  
  
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 twice restores the original order of the X– and Y–register contents.  
The function is used primarily to swap the order of numbers in a calculation.  
For example, one way to calculate 9 ÷ (13 × 8):  
Press  .  
The keystrokes to calculate this expression from left–to–right are:  
.  
Understand 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.  
Note  
2-4  
 
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  
1. The stack "drops" its contents. The T–(top) register replicates its contents.  
2. The stack "lifts" its contents. The T–register's contents are lost.  
3. The stack drops.  
Notice that when the stack lifts, it replaces the contents of the T– (top) register  
with the contents of the Z–register, and that the former contents of the T–  
register are lost. You can see, therefore, that the stack's memory is limited to  
four numbers.  
Because of the automatic movements of the stack, you do not need to clear  
the X–register before doing a new calculation.  
Most functions prepare the stack to lift its contents when the next number  
enters the X–register. See appendix B for lists of functions that disable stack  
lift.  
2-5  
 
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:  
5+6  
1 lost  
2 lsot  
T
Z
Y
X
1
2
3
4
2
3
4
5
3
4
5
5
3
4
5
6
3
3
4
11  
1
2
3
4
1. Lifts the stack.  
2. Lifts the stack and replicates the X–register.  
3. Does not lift the stack.  
4. Drops the stack and replicates 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 to clear the stack quickly: press 0  
. All stack registers now contain zero. Note, however, that  
you don't need to clear the stack 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 .  
2-6  
 
Filling the stack with a constant  
The replicating effect of together with the replicating effect of stack drop  
(from T into Z) allows you to fill the stack with a numeric constant for calculations.  
Example:  
Given bacterial culture with a constant growth rate of 50% per day, how large  
would a population of 100 be at the end of 3 days?  
Replicates T – register  
T
Z
Y
X
1.5  
1.5  
1.5  
1.5  
1.5  
1.5  
1.5  
100  
1.5  
1.5  
1.5  
150  
1.5  
1.5  
1.5  
225  
1.5  
1.5  
  
1.5  
337.5  
1
2
3
4
5
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 to Clear the Stack  
Clearing the X–register puts a zero in the X–register. The next number you key in (or  
recall) writes over this zero.  
There are four ways to clear the contents of the X–register, that is, to clear x:  
1. Press   
2. Press   
3. Press   
() (Mainly used during program entry.)  
4. Press   
() to clear the X-, Y-, Z-, and T-registers to zero.  
For example, if you intended to enter 1 and 3 but mistakenly entered 1 and 2, this  
is what you should do to correct your error:  
2-7  
 
T
Z
Y
X
1
1
1
2
1
0
1
3
1
2
C
3
1
1
2
3
4
5
1. Lifts the stack  
2. Lifts 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 .)  
Pressing   returns this value into the X–register.  
This ability to retrieve the "last x" has two main uses:  
1. Correcting errors.  
2. Reusing a number in a calculation.  
See appendix B for a comprehensive list of the functions that save x in the LAST X  
register.  
2-8  
 
Correcting Mistakes with LAST X  
Wrong Single Argument Function  
If you execute the wrong single argument function, use  to retrieve  
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 recover  
from these functions in the same manner as from single argument functions.  
Example:  
5
Suppose that you had just computed ln 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 .  
Mistakes with Two Argument Functions  
If you make a mistake with a two argument operation (such as , ), or x),  
you can correct it by using and the inverse of the two argument  
operation.  
1. Press  to recover the second number (x just before the 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 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, key in the correct first number, press  
 to recover the second number, and execute the function again.  
(Press first if you want to clear the incorrect result from the stack.)  
2-9  
 
Example:  
Suppose you made a mistake while calculating  
16 × 19 = 304  
There are three kinds of mistakes you could have made:  
Wrong  
Mistake:  
Correction:  
Calculation:  
Wrong function  
  
  
Ù  
  
Wrong first number  
Wrong second number  
  
  
  
  
  
   
Reusing Numbers with LAST X  
You can use  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 .  
Example:  
96.704+ 52.3947  
Calculate  
52.3947  
 
Keys:  
  
Display:  
  
Description:  
Enters first number.  
Intermediate result.  
  
  
  
  
   
Brings back display from before  
   
.  
Final result.  
Example:  
Two close stellar neighbors of Earth are Rigel Centaurus (4.3 light–years away) and  
15  
Sirius (8.7 light–years away). Use c, the speed of light (9.5 × 10 meters per year)  
to convert the distances from the Earth to these stars into meters:  
15  
To Rigel Centaurus: 4.3 yr × (9.5 × 10 m/yr).  
15  
To Sirius: 8.7 yr × (9.5 × 10 m/yr).  
Keys:  
  
Display:  
  
Description:  
Light–years to Rigel Centaurus.  
Speed of light, c.  
  
_  
  
Meters to R. Centaurus.  
Retrieves c.  
  
   
Meters to Sirius.  
  
Chain Calculations in RPN Mode  
In RPN mode, 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, evaluate (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  
Evaluate the expression in the same way on the HP 35s, starting inside the  
parentheses.  
Keys:  
  
Display:  
  
Description:  
Calculates the intermediate result first.  
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  
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 an  
expression. 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.  
  
  
  
Calculate 4 ÷ [14 + (7 × 3) – 2]:  
Keys:  
Display:  
Description:  
Calculates (7 × 3).  
Calculates denominator.  
Puts 4 before 33 in preparation for  
division.  
  
  
  
  
  
  
  
Calculates 4 ÷ 33, the answer.  
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.  
Work through the problem the same way with the HP 35s, except that you don't  
have to write down intermediate answers—the calculator remembers them for you.  
Keys:  
  
Display:  
  
  
Description:  
First adds (3+4)  
Then adds (5+6)  
  
  
Then multiplies the intermediate  
answers together for the final  
answer.  
Exercises  
Calculate:  
(16.3805x5)  
= 181.0000  
0.05  
Solution:  
   
Calculate:  
[(2+ 3)×(4 + 5)] + [(6 + 7)×(8+ 9)] = 21.5743  
Solution:  
  
  
Calculate:  
(10 – 5) ÷ [(17 – 12) × 4] = 0.2500  
Solution:  
    
or  
   
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   
 .  
If you work the problem from left–to–right, press  
.  
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 noncommutative  
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.  
When using the left–to–right method, be sure that no more  
than four intermediate numbers (or results) will be needed at  
Note  
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:  
Saves 4 and 14 as intermediate  
numbers in the stack.  
_  
At this point the stack is full with  
numbers for this calculation.  
Intermediate result.  
  
  
  
Intermediate result.  
  
  
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:  
   
Calculate:  
2
23 – (13 × 9) + 1/7 = 412.1429  
A Solution:  
   
Calculate:  
(5.4× 0.8) ÷ (12.50.73) = 0.5961  
Solution:  
  
  
or  
   
   
Calculate:  
8.33×(45.2)÷[(8.337.46)×0.32]  
= 4.5728  
4.3×(3.152.75)(1.71×2.01)  
 
A Solution:  
  
  
     
3
Storing Data into Variables  
The HP 35s has 30 KB of memory, in which you can store numbers, equations, and  
programs. 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.)  
Example:  
This example shows you how to store the value 3 in the variable A, first in RPN  
mode and then in ALG mode.  
Keys:  
Display:  
  
Description:  
Switch to RPN mode (if necessary)  
9( )  
Enter the value (3)  
_  
The Store command prompts for a  
letter; note the A…Z annunciator.  
The value 3 is stored in A and  
returned to the stack.  
  
_  
  
A
  
Switch to ALG mode (if necessary)  
9( )  
  
_  
Again, the Store command prompts  
for a letter and the A…Z annunciator  
appears.  
A  
The value 3 is stored in A and the  
result is placed in line 2.  
  
  
3-1  
 
In ALG mode, you can store an expression into a variable; in this case, the value of  
the expression is stored in the variable rather than the expression itself.  
Example:  
Keys:  
  
  
Display:  
  
Description:  
Enter the expression, then  
proceed as in the previous  
example.  
  
Each pink letter is associated with a key and a unique variable. (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 and vectors are stored into, and recalled from, lettered variables by  
means of the Store () and Recall () commands. Numbers may be  
real or complex, decimal or fraction, base 10 or other as supported by the HP 35s.  
To store a copy of a displayed number (X–register) to a direct variable:  
Press  letter–key .  
To recall a copy of a number from a direct variable to the display:  
Press letter–key .  
Example: Storing Numbers.  
23  
Store Avogadro's number (approximately 6.0221 × 10 ) in A.  
3-2  
 
Keys:  
  
  
Display:  
_  
Description:  
Avogadro's number.  
_ “” prompts for variable.  
A  
  
  
_
Stores a copy of Avogadro's number  
in A. This also terminates digit entry .  
Clears the number in the display.  
A..Z The A..Z annunciator Turns on  
Copies Avogadro's number from A  
the display.  
  
A  
  
To recall the value stored in a variable, use the Recall command. The display of this  
command differs slightly from RPN to ALG mode, as the following example  
illustrates.  
Example:  
In this example, we recall the value of 1.75 that we stored in the variable G in the  
last example. This example assumes the HP 35s is still in ALG mode at the start.  
Keys:  
G  
Display:  
  
Description:  
Pressing simply activates A…Z  
mode; no command is pasted into  
line 1.  
In ALG mode, Recall can be used to paste a variable into an expression in the  
command line. Suppose we wish to evaluate 15-2×G, with G=1.75 from above.  
Keys:  
  
Display:  
  
Description:  
  
G  
We now proceed to switch to RPN mode and recall the value of G.  
3-3  
Keys:  
9()  
Display:  
Description:  
Switch to RPN mode  
In RPN mode, pastes the  
command into the edit line.  
 _  
  
G
No need to press .  
  
Viewing a Variable  
The VIEW command () displays the value of a variable without recalling  
that value to the x-register. The display takes the form Variable=Value. If the number  
has too many digits to fit into the display, use Õor Öto view the  
missing digits. To cancel the VIEW display, press or . The VIEW command  
is most often used in programming but it is useful anytime you want to view a  
variable’s value without affecting the stack.  
Using the MEM Catalog  
The MEMORY catalog (u) provides information about the amount of  
available memory. The catalog display has the following format:  
  
  
   
  
where mm,mmm is the number of bytes of available memory and nnn is the amount  
of used indirect variables.  
For more information on indirect variables, see Chapter 14.  
The VAR catalog  
By default, all direct variables from A to Z contain the value zero. If you store a non-  
zero value in any direct variable, that variable’s value can be viewed in the VAR  
Catalog (u()).  
3-4  
     
Example:  
In this example, we store 3 in C, 4 in D, and 5 in E. Then we view these variables  
via the VAR Catalog and clear them as well. This example uses RPN mode.  
Keys:  
(  
Display:  
Description:  
Clear all direct variables  
)  
Store 3 in C, 4 in D, and 5 in E.  
Enter the VAR catalog.  
C  
D  
E  
u()  
  
Note the and annunciators indicating that the Øand ×keys are active  
to help you scroll through the catalog; however, if Fraction Display mode is active,  
the and annunciators will not be active to indicate accuracy unless there is  
only one variable in the catalog. We return to our example, illustrating how to  
navigate the VAR catalog.  
Scroll down to the next direct  
  
Ø
variable with non-zero value: D=4.  
Scroll down once more to see E=5.  
  
Ø
While we are in the VAR catalog, let’s extend this example to show you how to  
clear the value of a variable to zero, effectively deleting the current value. We’ll  
delete E.  
E is no longer in the VAR catalog,  
  
as its value is zero. The next  
variable is C as shown.  
Suppose now that you wish to copy the value of C to the stack.  
The value of C=3 is copied to the  
x-register and 5 (from defining E=5  
previously) moves to the y-register.  
3-5  
To leave the VAR catalog at any time, press either or . An alternate  
method to clearing a variable is simply to store the value zero in it. Finally, you can  
clear all direct variables by pressing   
(). If all direct  
variables have the value zero, then attempting to enter the VAR catalog will display  
the error message “   .  
If the value of a variable has too many digits to display completely, you can use  
Õand Öto view the missing digits.  
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 arithmetic in the variable itself and to store the result there. It uses the  
value in the X–register and does not 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 A. Now A = 12, while 3 is still in  
the display.  
3-6  
   
15  
12  
A
A
Result: 15 – 3  
that is, A – x  
t
t
T
Z
Y
X
T
Z
Y
X
z
y
3
z
y
3
A  
Recall Arithmetic  
Recall arithmetic uses , , , or  to do arithmetic  
in the X–register using a recalled number and to leave the result in the display. Only  
the X–register is affected. The value in the variable remains the same and the result  
replaces the value in the x-register.  
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   A. Now x = 0.25, while 12 is still in A.  
Recall arithmetic saves memory in programs: using   A (one instruction)  
uses half as much memory as A, (two instructions).  
12  
12  
A
A
t
t
z
T
Z
Y
X
T
Z
Y
X
z
y
3
y
Result: 3 ÷ 12  
that is, x ÷ 12  
A  
0.25  
3-7  
 
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.  
D  
E  
F  
  
  
Adds1 to D, E, and F.  
  
D  
E  
F  
  
Displays the current value of D.  
  
D  
E  
F  
  
  
  
  
  
  
Clears the VIEW display; displays X-  
register again.  
Suppose the variables D, E, and F contain the values 2, 3, and 4 from the last  
example. Divide 3 by D, multiply it by E, and add F to the result.  
Keys:  
D  
E  
F  
Display:  
  
Description:  
Calculates 3 ÷ D.  
  
  
3 ÷ D × E.  
3 ÷ D × E + F.  
Exchanging x with Any Variable  
The   key allows you to exchange the contents of x (the displayed X –  
register) with the contents of any variable. Executing this function does not affect the  
Y–, Z–, or T–registers.  
3-8  
 
Example:  
Keys:  
Display:  
  
Description:  
Stores 12 in variable A.  
  
A  
_  
Displays x.  
  
Exchanges contents of the X–register  
and variable A.  
A  
  
Exchanges contents of the X–register  
and variable A.  
A  
12  
3
A
A
t
t
T
Z
Y
X
T
z
y
3
z
Z
y
Y
A  
12  
X
The Variables "I" and "J"  
There are two variables that you can access directly: the variables I and J. Although  
they store values as other variables do, I and J are special in that they can be used  
to refer to other variables, including the statistical registers, using the (I) and (J)  
commands. (I) is found on the key, while (J) is on the key. This is a  
programming technique called indirect addressing that is covered under “Indirectly  
Addressing Variables and Labels” in chapter 14.  
3-9  
 
4
Real–Number Functions  
This chapter covers most of the calculator's functions that perform computations on  
real numbers, including some numeric functions used in programs (such as ABS, the  
absolute–value function). These functions are addressed in groups, as follows:  
Exponential and logarithmic functions.  
Quotient and Remainder of Divisions.  
Power functions. (and )  
Trigonometric functions.  
Hyperbolic functions.  
Percentage functions.  
Physics constants  
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. The examples in this chapter all  
assume the HP 35s is in RPN mode.  
Exponential and Logarithmic Functions  
Put the number in the display, then execute the function- there is no need to press  
.  
4-1  
   
To Calculate:  
Press:  
Natural logarithm (base e)  
Common logarithm (base 10)  
Natural exponential  
  
  
  
  
Common exponential (antilogarithm)  
Quotient and Remainder of Division  
You can use ()and () to produce the  
integer quotient and integer remainder, respectively, from the division of two  
integers.  
1. Key in the first integer.  
2. Press to separate the first number from the second.  
3. Key in the second number. (Do not press .)  
4. Press the function key.  
Example:  
To display the quotient and remainder produced by 58 ÷ 9  
Keys:  
Display:  
  
Description:  
Displays the quotient.  
  
()  
  
()  
Displays the remainder.  
  
Power Functions  
In RPN mode, to calculate a number y raised to a power x, key in y x, then  
press . (For y>0, x can be any number; for y<0, x must be positive.)  
4-2  
   
To Calculate:  
2
15  
Press:  
  
Result:  
  
  
  
  
6
   
10  
  
4
5
2
–1.4  
   
  
  
3
(–1.4)  
th  
In RPN mode, 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.  
To Calculate:  
Press:  
  
Result:  
  
196  
  
3 125  
  
  
  
4 625  
  
1.4 .37893  
  
  
Trigonometry  
Entering π  
Press   to place the first 12 digits of π into the X–register.  
(The number displayed depends on the display format.) Because  is a  
function that returns an approximation of π to the stack, it is not necessary to press  
.  
Note that the calculator cannot exactly represent π, since π is a transcendental  
number.  
4-3  
   
Setting the Angular Mode  
The angular mode specifies which unit of measure to 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 9. A menu will be displayed from which you  
can select an option.  
Option  
  
Description  
Annunciator  
Sets degree mode, which uses decimal  
degrees rather than hexagesimal degrees  
(degrees, minutes, seconds)  
none  
Sets radian mode  
  
RAD  
Sets gradient mode  
  
GRAD  
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.  
   
   
   
Calculations with the irrational number π cannot be expressed  
exactly by the 15–digit internal precision of the calculator. This is  
particularly noticeable in trigonometry. For example, the  
Note  
–13  
calculated sin π (radians) is not zero but –2.0676 × 10  
, a very  
small number close to zero.  
4-4  
   
Example:  
Show that cosine (5/7)π radians and cosine 128.57° are equal (to four significant  
digits).  
Keys:  
Display:  
Description:  
Sets Radians mode; RAD  
annunciator on.  
9()  
5/7 in decimal format.  
Cos (5/7)π.  
  
  
     
  
  
Switches to Degrees mode (no  
annunciator).  
9()  
Calculates cos 128.57°, which is  
the same as cos (5/7)π.  
  
  
Programming Note:  
Equations using inverse trigonometric functions to determine an angle θ, often look  
something like this:  
θ = arctan (y/x).  
If x = 0, then y/x is undefined, resulting in the error:   .  
4-5  
Hyperbolic Functions  
With x in the display:  
To Calculate:  
Hyperbolic sine of x (SINH).  
Press:  
    
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).  
    
    
    
    
    
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.  
To Calculate:  
Press:  
x% of y  
y x   
y x    
Percentage change from y to x. (y0)  
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.  
4-6  
   
Keys:  
Display:  
Description:  
Rounds display to two decimal  
places.  
8()  
  
  
  
  
  
Calculates 6% tax.  
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:  
This year's price dropped about  
2.2% from last year's price.  
Restores FIX 4 format.  
  
  
  
8()  
The order of the two numbers is important for the %CHG function.  
The order affects whether the percentage change is considered  
positive or negative.  
Note  
4-7  
Physics Constants  
There are 41 physics constants in the CONST menu. You can press    
to view the following items.  
CONST Menu  
Items  
Description  
Value  
–1  
Speed of light in vacuum  
299792458 m s  
–2  
Standard acceleration of gravity  
Newtonian constant of  
gravitation  
9.80665 m s  
–11  
3
– 1 –2  
6.673×10  
m kg  
s
3
–1  
Molar volume of ideal gas  
Avogadro constant  
Rydberg constant  
Elementary charge  
Electron mass  
0.022413996 m mol  
  
23  
–1  
mol  
6.02214199×10  
–1  
10973731.5685 m  
–19  
1.602176462×10  
C
kg  
kg  
kg  
  
  
  
–31  
9.10938188×10  
–27  
Proton mass  
Neutron mass  
1.67262158×10  
1.67492716×10  
–27  
–28  
Muon mass  
Boltzmann constant  
Planck constant  
1.88353109×10  
1.3806503×10  
6.62606876×10  
kg  
–1  
  
–23  
J K  
–34  
J s  
J s  
–34  
Planck constant over 2 pi  
1.054571596×10  
–15  
Magnetic flux quantum  
Bohr radius  
2.067833636×10  
Wb  
m
  
–11  
5.291772083×10  
–1  
–12  
Electric constant  
8.854187817×10  
F m  
ε  
–1 –1  
k
Molar gas constant  
Faraday constant  
8.314472 J mol  
–1  
96485.3415 C mol  
–27  
Atomic mass constant  
Magnetic constant  
1.66053873×10  
kg  
–2  
–6  
1.2566370614×10 NA  
–1  
–1  
–1  
–1  
–1  
–24  
Bohr magneton  
  
  
  
  
  
9.27400899×10  
5.05078317×10  
1.410606633×10  
9.28476362×10  
9.662364×10  
J T  
J T  
J T  
J T  
J T  
–27  
–26  
–24  
–27  
Nuclear magneton  
Proton magnetic moment  
Electron magnetic moment  
Neutron magnetic moment  
4-8  
 
Items  
Description  
Value  
–1  
–26  
Muon magnetic moment  
Classical electron radius  
  
  
–4.49044813×10  
J T  
–15  
2.817940285×10  
m
Characteristic impendence of  
vacuum  
376.730313461 Ω  
–12  
Compton wavelength  
2.426310215×10  
m
m
λ  
–15  
Neutron Compton wavelength  
Proton Compton wavelength  
Fine structure constant  
1.319590898×10  
λ
  
  
α
–15  
1.321409847×10  
m
–3  
λ
7.297352533×10  
–2 –4  
–8  
Stefan–Boltzmann constant  
Celsius temperature  
5.6704×10 W m  
K
σ
273.15  
Standard atmosphere  
101325 Pa  
a  
γ   
–1 –1  
Proton gyromagnetic ratio  
267522212 s T  
2
–16  
First radiation constant  
Second radiation constant  
Conductance quantum  
  
  
374177107×10  
W m  
0.014387752 m K  
–5  
7.748091696×10  
S
The base number of natural  
logarithm(natural constant)  
2.71828182846  
Reference: Peter J.Mohr and Barry N.Taylor, CODATA Recommended Values of  
the Fundamental Physical Constants: 1998, Journal of Physical and Chemical  
Reference Data,Vol.28, No.6,1999 and Reviews of Modern Physics,Vol.72,  
No.2, 2000.  
To insert a constant:  
1. Position your cursor where you want the constant inserted.  
2. Press   to display the physics constants menu.  
3. Press ÕÖ×Ø(or, you can press  to access the next  
page, one page at a time) to scroll through the menu until the constant you  
want is underlined, then press to insert the constant.  
Note that constants should be referred to by their names rather than their values,  
when used in expressions, equations, and programs.  
4-9  
Conversion Functions  
The HP 35s supports four types of conversions. You can convert between:  
rectangular and polar formats for complex numbers  
degrees, radians, and gradients for angle measures  
decimal and hexagesimal formats for time (and degree angles)  
various supported units (cm/in, kg/lb, etc)  
With the exception of the rectangular and polar conversions, each of the  
conversions is associated with a particular key. The left (yellow) shift of the key  
converts one way while the right (blue) shift of the same key converts the other way.  
For each conversion of this type, the number you entered is assumed to be  
measured using the other unit. For example, when using ¾to convert a number  
to Fahrenheit degrees, the number you enter is assumed to be a temperature  
measured in Celsius degrees. The examples in this chapter utilize RPN mode. In  
ALG mode, enter the function first, then the number to convert.  
Rectangular/Polar Conversions  
Polar coordinates (r,θ) and rectangular coordinates (x,y) are measured as shown in  
the illustration. The angle θ 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:  
The format for representing complex numbers is a mode setting. You may enter a  
complex number in any format; upon entry, the complex number is converted to the  
format determined by the mode setting. Here are the steps required to set a  
complex number format:  
1. Press 8 and then choose either ( ) or  () in  
RPN mode (in ALG mode, you may also choose  ( )  
2. Input your desired coordinate value (x 6 y, x y 6 or r ?a)  
3. press   
Example: Polar to Rectangular Conversion.  
In the following right triangles, find sides x and y in the triangle on the left, and  
hypotenuse r and angle θ in the triangle on the right.  
10  
r
y
4
30o  
θ
x
3
Keys:  
9()  
8( )  
Display:  
Description:  
Sets Degrees and complex  
coordinate mode.  
Convert rθa (polar) to xiy  
(rectangular).  
?    
Sets complex coordinate  
mode.  
θ  
8  
()  
Convert xiy (rectangular) to  
θ  
6  
rθ a (polar).  
Example: Conversion with Vectors.  
Engineer P.C. Bord has determined that in the RC circuit shown, the total impedance  
is 77.8 ohms and voltage lags current by 36.5º. What are the values of resistance R  
and capacitive reactance X in the circuit?  
C
Use a vector diagram as shown, with impedance equal to the polar magnitude, r,  
and voltage lag equal to the angle, θ, in degrees. When the values are converted  
to rectangular coordinates, the x–value yields R, in ohms; the y–value yields X , in  
C
ohms.  
R
θ
_
36.5o  
R
X
c
77.8 ohms  
C
Keys:  
Display:  
Description:  
Sets Degrees and complex  
coordinate mode.  
9()  
¹8( )  
?  
   
θ  
Enters θ, degrees of voltage lag.  
Enters r, ohms of total  
impedance.  
Calculates x, ohms  
   
resistance, R.  
Calculates y, ohms  
reactance, X  
C
Time Conversions  
The HP 35s can convert between decimal and hexagesimal formats for numbers.  
This is especially useful for time and angles measured in degrees. For example, in  
decimal format an angle measured in degrees is expressed as D.ddd…, while in  
hexagesimal the same angle is represented as D.MMSSss, where D is the integer  
pat of the degree measure, ddd… is the fractional part of the degree measure, MM  
is the integer number of minutes, SS is the integer part of the number of seconds,  
and ss is the fractional part of the number of seconds.  
To convert between decimal format and hours minutes, and seconds:  
1. Enter the number you wish to convert  
2. Press to convert to hours/degrees, minutes, and seconds or press  
5 to convert back to decimal format.  
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:  
Sets FIX 6 display format.  
8()  
  
  
1/7 hour as a decimal fraction.  
Equals 8 minutes and 34.29  
   
  
  
   
seconds.  
  
  
Restores FIX 4 format.  
8()  
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:  
Example  
In this example, we convert an angle measure of 30° to π/6 radians.  
Keys:  
Display:  
  
Description:  
Enter the angle in degrees.  
  
_  
Convert to radians. Read the result  
as 0.5236, a decimal  
µ  
  
  
approximation of π/6.  
Unit Conversions  
The HP 35s has ten unit–conversion functions on the keyboard: kg, lb, ºC,  
ºF, cm, in, l, gal, MILE,KM  
To Convert:  
1 lb  
To:  
kg  
Press:  
  
Displayed Results:  
 (kilograms)  
 (pounds)  
 (°C)  
  
1 kg  
lb  
  
  
  
  
  
32 ºF  
ºC  
ºF  
100 ºC  
1 in  
 (°F)  
cm  
in  
 (centimeters)  
 (inches)  
 (liters)  
100 cm  
1 gal  
l
  
1 l  
gal  
KM  
MILE  
 (gallons)  
(KMS)  
1 MILE  
1 KM  
<  
(MILES)  
;  
 
Probability Functions  
Factorial  
To calculate the factorial of a displayed non-negative integer x (0 x 253), press  
*(the right–shifted key).  
Gamma  
To calculate the gamma function of a noninteger x, Γ(x), key in (x – 1) and press  
*. The x! function calculates Γ(x + 1). The value for x cannot be a negative  
integer.  
Probability  
Combinations  
To calculate the number of possible sets of n items taken r at a time, enter n first,  
x, then r (nonnegative integers only). No item occurs more than once in a  
set, and different orders of the same r items are not counted separately.  
Permutations  
To calculate the number of possible arrangements of n items taken r at a time, enter  
n first, {, then r (nonnegative integers only). No item occurs more than  
once in an arrangement, and different orders of the same r items are counted  
separately.  
Seed  
To store the number in x as a new seed for the random number generator, press  
 ..  
Random number generator  
To generate a random number in the range 0 < x < 1, press  . (The  
number is part of a uniformly–distributed pseudo–random number sequence. It  
passes the spectral test of D. Knuth, The Art of Computer Programming, vol. 2,  
Seminumerical Algorithms, London: Addison Wesley, 1981.)  
       
The RANDOM function 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. If memory is cleared, the seed is reset  
to zero. A seed of zero will result in the calculator generating its own seed.  
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.  
Total number of combinations  
possible.  
  
  
_  
  
x  
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 women grouped six at a  
time.  
  
  
_  
  
Number of combinations of six  
women on the committee.  
Brings total number of  
combinations back into the X–  
register.  
x  
  
Divides combinations of women  
by total combinations to find  
probability that any one  
combination would have all  
women.  
  
Parts of Numbers  
These functions are primarily used in programming.  
Integer part  
To remove the fractional part of x and replace it with zeros, press   
(). (For example, the integer part of 14.2300 is 14.0000.)  
Fractional part  
To remove the integer part of x and replace it with zeros, press   
(). (For example, the fractional part of 14.2300 is 0.2300)  
Absolute value  
To replace a number in the x-register with its absolute value, press  . For  
complex numbers and vectors, the absolute value of:  
1. a complex number in rθa format is r  
x2 + y2  
2. a complex number in xiy format is  
2
2
A 2 + A2 + ⋅⋅ + An  
3. a vector [A1,A2,A3, …An] is  
=
A
1
Argument value  
To extract the argument of a complex number, use =. The argument of a  
complex number:  
1. in rθa format is a  
2. in xiy format is Atan(y/x)  
Sign value  
To indicate the sign of x, press (). If the x value is negative, –  
1.0000 is displayed; if zero, 0.0000 is displayed; if positive, 1.0000 is displayed.  
 
Greatest integer  
To obtain the greatest integer equal to or less than given number, press  
 ().  
Example:  
This example summarizes many of the operations that extract parts of numbers.  
To calculate:  
The integer part of 2.47  
The fractional part of 2.47  
The absolute value of –7  
Press:  
Display:  
 ()  
 ()  
   
  
  
  
The sign value of 9  
The greatest integer equal to  
  
 ()  
  
  
or less than –5.3  
()  
The RND function (  ) 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.  
5
Fractions  
In Chapter 1, the section Fractions introduced the basics of entering, displaying,  
and calculating with fractions. This chapter gives more information on these topics.  
Here is a short review of entering and displaying fractions:  
To enter a fraction, press twice: once after the integer part of a mixed  
number and again between the numerator and denominator of the fractional  
part of the number. To enter 2 3/8, press . To enter 5/8,  
press either  or .  
To toggle Fraction-display mode on and off, press . When  
Fraction-display mode is turned off, the display reverts to the previous display  
format set via the Display menu. Choosing another format via this menu also  
turns off Fraction-display mode, if active.  
Functions work the same with fractions as they do with decimal numbers –  
except for RND, which is discussed later in this chapter.  
The examples in this chapter all utilize RPN mode unless otherwise noted.  
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).  
Example:  
Keys:  
Display:  
   
Description:  
Turns on Fraction–display mode.  
   
  
Enters 1.5; shown as a fraction.  
3
Enters 1 / .  
4
     
Displays x as a decimal number.  
Displays x as a fraction.  
   
  
   
   
5-1  
   
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.  
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).  
The denominator is no greater than 4095.  
The fraction is reduced as far as possible.  
Examples:  
These are examples of entered values and the displayed results. For comparison, the  
internal 12–digit values are also shown. The and annunciators in the last  
column are explained below.  
5-2  
   
Entered Value  
Internal Value  
2.37500000000  
14.4687500000  
4.50000000000  
9.60000000000  
2.83333333333  
0.00183105469  
12349793.0000  
16.0001831055  
Displayed Fraction  
   
3
2 /  
8
15  
   
   
14  
54  
/
32  
/
12  
18  
   
6
/
5
34  
   
/
/
12  
15  
   
  
8192  
12345  
12345678  
3
/
3
  
16 /  
16384  
Accuracy Indicators  
The accuracy of a displayed fraction is indicated by the and annunciators at  
the right 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.  
If is lit, 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.  
If is lit, the fractional part of the internal 12–digit value is slightly greater  
than the displayed fraction — the exact numerator is no more than 0.5 above  
the displayed numerator.  
This diagram shows how the displayed fraction relates to nearby values — ꢄ  
means the exact numerator is "a little above" the displayed numerator, and ꢅ  
means the exact numerator is "a little below".  
0 7/16  
0 7/16  
0 7/16  
6
6.5  
7
7.5  
8
/
/
/
/
/
16  
16  
16  
16  
16  
(0.40625)  
(0.43750)  
(0.46875)  
5-3  
 
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  
2
fractions to have 5 as the denominator, then / is displayed as  because  
3
3.3333  
3
2
the exact fraction is approximately  
/ , "a little above" / . Similarly, – / is  
5
5
3
displayed as  because the true numerator is "a little above" 3.  
Sometimes an annunciator is lit when you wouldn't expect it to be. For example, if  
2
you enter 2 / , you see  , even though that's the exact number you  
3
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.  
Changing the Fraction Display  
In its default condition, the calculator displays a fractional number according to  
certain rules. 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).  
5-4  
   
To set the maximum denominator value, enter the value and then press  
. Fraction-display mode will be automatically enabled. The value you  
enter cannot exceed 4095.  
To recall the /c value to the X–register, press .  
To restore the default value to 4095, press  or enter any value  
greater than 4095 as the maximum denominator. Again, Fraction-display  
mode will be automatically enabled.  
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.  
If the displayed fraction is too long to fit in the display, the annunciator will  
appear; you can then use Öand Õto scroll page by page to see the  
rest of the fraction. To see the number’s decimal representation, press and then  
hold .  
Example:  
This example illustrates the steps required to set the maximum denominator to 3125  
and then show a fraction that is too long for the display.  
Keys:  
  
Display:  
Description:  
Set the maximum denominator to  
3125.  
Note the missing digits in the  
denominator.  
#  
   
Õ
Scroll right to see the rest of the  
denominator.  
  
Notes:  
1. In ALG mode, you can enter an expression in line 1 and then press . In  
this case, the expression is evaluated and the result is used to determine the  
maximum denominator.  
5-5  
2. In ALG mode, you can use the result of a calculation as the argument for the /c  
function. With the value in line 2, simply press . The value in line 2 is  
displayed in Fraction format and the integer part is used to determine the  
maximum denominator.  
3. You may not use either a complex number or a vector as the argument for the /  
c command. The error message “ ” will be displayed.  
Choosing a Fraction Format  
The calculator has three fraction formats. The displayed fractions are always the  
most accurate fractions within the rules for the selected 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    
and   ( /c value is 8 ). Or if the /c value is 12, possible 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 ).  
There are three flags that control the fraction format. These flags are numbered 7, 8,  
and 9. Each flag is either clear or set. Their purposes are as follows:  
Flag 7 toggles fraction-display mode on or off; clear=off and set=on.  
Flag 8 toggles between using any value less than or equal to the /c value  
or using only factors of the /c value; clear = use any value and set = use  
only factors of the /c value.  
Flag 9 operates only if Flag 8 is set and toggles between reducing or not  
reducing the fractions; clear = reduce and set = do not reduce.  
With Flags 8 and 9 appropriately cleared or set, you can get the three fraction  
formats as shown in the table below:  
5-6  
 
To Get This Fraction Format:  
Most precise  
Factors of denominator  
Fixed denominator  
Change These Flags:  
8
Clear  
Set  
9
Clear  
Set  
Set  
You can change flags 8 and 9 to set the fraction format using the steps listed here.  
(Because flags are especially useful in programs, their use is covered in detail in  
chapter 14.)  
1. Press to get the flag menu.  
2. To set a flag, press () and type the flag number, such as 8.  
To clear a flag, press () and type the flag number.  
To see if a flag is set, press () and type the flag number. Press or  
to clear the  or  response.)  
Example:  
This example illustrates the display of fractions in the three formats using the number  
π. This example assumes fraction-display format is active and that Flag 8 is in its  
default state (cleared).  
Keys:  
  
Display:  
Description:  
Sets the maximum /c value back  
to the default.  
Most precise format  
j  
Flag 8 = clear.  
Flag 8 = set;  
   
()  
Factors of denominator format;  
819*5=4095  
   
Flag 9 = set;  
()    
Fixed denominator format  
Return to default format (most  
precise)  
   
()  
(  
   
)  
5-7  
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)  
(2.7699)  
(2.7699)  
(2.7692)  
(2.7500)  
(2.7500)  
Most Precise  
2 77/100  
2 10/13ꢉ  
2 3/4ꢉ  
Factors of Denominator  
Fixed Denominator  
2 1051/1365ꢉ  
2 3153/4095ꢉ  
2 12/16ꢉ  
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  
2
/
16  
Format  
2
2
/
25  
2
2.5  
2.9999  
3
Most precise  
Factors of  
denominator  
Fixed denominator  
For a /c value of 16.  
2
2
2 1/2  
2 2/3  
3
2 9/14  
2 1/2  
2 11/16  
3  
2 5/8ꢉ  
2 0/16 2 8/16 2 11/16  
3 0/16  
2 10/16  
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 and 9. The accuracy  
indicatior turns off if the fraction matches the decimal representation exactly.  
Otherwise, the accuracy indicatior 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.  
5-8  
   
Example:  
Suppose you have a 56 / –inch space that you want to divide into six equal  
3
4
1
sections. How wide is each section, assuming you can conveniently measure /  
16  
inch increments? What's the cumulative roundoff error?  
Keys:  
Display:  
Description:  
Sets Flag 8  
Sets up fraction format for /  
   
  
1
16  
inch increments. (Flags 8 and 9  
should be the same as for the  
previous example.)  
  
D  
  
   
Stores the distance in D.  
   
The sections are a bit wider than 9  
7
/
16  
inches.  
   
   
   
   
   
  
Rounds the width to this value.  
Width of six sections.  
  
D  
 ()  
   
The cumulative round off error.  
Clears flag 8.  
Turns off Fraction–display mode.  
Fractions in Equations  
You can use a fraction in an equation. When an equation is displayed, all  
numerical values in the equation are shown in their entered form. Also, fraction-  
display mode is available for operations involving equations.  
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.  
5-9  
 
Fractions in Programs  
You can use a fraction in a program just as you can in an equation; numerical  
values are shown in their entered form.  
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. The program’s result is displayed using the current display  
format.  
A program can control the fraction display using the /c function and by setting and  
clearing flags 7, 8, and 9. See "Flags" in chapter 14.  
See chapters 13 and 14 for information about working with programs.  
 
6
Entering and Evaluating Equations  
How You Can Use Equations  
You can use equations on the HP 35s in several ways:  
For specifying an equation to evaluate (this chapter).  
For specifying an equation to solve for unknown values (chapter 7).  
For specifying a function to integrate (chapter 8).  
Example: Calculating with an Equation.  
Suppose you frequently need to determine the volume of a straight section of pipe.  
The equation is  
2
V = .25 π d l  
where d is the inside diameter of the pipe, and l is its length.  
You could key in the calculation over and over; for example,  
 calculates the  
1
volume of 16 inches of 2 / –inch diameter pipe (78.5398 cubic inches). However,  
2
by storing the equation, you get the HP 35s 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:  
6-1  
   
Keys:  
Display:  
Description:  
Selects Equation mode, shown by  
the EQN annunciator.  
    
or the current equation in  
line 2  
Begins a new equation,   
turns on the A..Z annunciator so  
you can enter a variable name.  
types   
Digit entry uses the "_" entry  
cursor.  
  
  
_  
_  
  
D  
  
ends the number.  
π_  
types .  
π _  
π_  
π  
Terminates and displays the  
equation.  
Shows the checksum and length  
for 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 right–  
hand side of the equation. Prompts  
for D first; value is the current value of  
D.  
  
value  
1
  
Enters 2 / inches as a fraction.  
2
  
 _  
  
Stores D, prompts for L; value is  
current value of L.  
Stores L; calculates V in cubic inches  
and stores the result in V.  
value  
  
  
  
6-2  
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.  
When displaying equations in the equation list, two equations are displayed at a  
time. The currently active equation is shown on line 2.  
Key  
Operation  
Enters and leaves Equation mode.  
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 . (See "Types of Equations" later in this  
chapter.)  
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.)  
Integrates the displayed equation with respect to the  
variable you specify. (See chapter 8.)  
Deletes the current equation or deletes the element to  
the left of the cursor.  
   
Begins editing the displayed equation, only moving  
the cursor and not deleting any content.  
Öor Õ  
Scroll the current equation display screen.  
Öor Õ  
Steps up or down through the equation list.  
Jumps to the top or bottom of the equation list.  
×or Ø  
×or Ø  
   
Shows the displayed equation's checksum (verification  
value) and length (bytes of memory).  
Recovers the most recently deleted element or  
equation.  
:  
Leaves Equation mode.  
You can also use equations in programs — this is discussed in chapter 13.  
6-3  
 
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.  
To enter an equation:  
You can make an equation as long as you want – it is limited only by the amount of  
available memory.  
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 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 or : 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 instead, the equation is  
saved, but Equation mode is turned off.)  
Equations can contain variables, numbers, vectors, 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 variables in an equation: A through Z,(I) and  
(J). You can use each variable as many times as you want.(For information about (I)  
and (J), see "Indirectly Addressing Variables and Labels" in chapter 14.)  
To enter a variable in an equation, press variable. When you press , the  
A..Z annunciator shows that you can press a variable key to enter its name in the  
equation.  
6-4  
   
Numbers in Equations  
You can enter any valid number in an equation, including base 2, 8 and 16, real,  
complex, and fractional numbers. Numbers are always shown using ALL display  
format, which displays up to 12 characters.  
To enter a number in an equation, you can use the standard number–entry keys,  
including , , and . Do not use for subtraction.  
Functions in Equations  
You can enter many HP 35s functions in an equation. A complete list is given under  
“Equation Functions” later in this chapter. Appendix G, "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 their  
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  to separate them.  
6-5  
   
Parentheses in Equations  
You can include parentheses in equations to control the order in which operations  
are performed. Press 4 to insert parentheses. (For more information, see  
"Operator Precedence" later in this chapter.)  
Example: Entering an Equation.  
Enter the equation r = 2 × c ×(t – a)+25  
Keys:  
Display:  
Description:  
π  
Shows the last equation used in  
the equation list.  
  
Starts a new equation with  
_  
variable R.  
Enters a number  
 _  
  
Enters infix operators.  
_  
  
Enters a prefix function with a left  
parenthesis.  
4
Enters the argument and right  
parenthesis.  
  
Õ  
  _  
Terminates the equation and  
displays it.  
   
  
Shows its checksum and length.  
  
  
Leaves Equation mode.  
Displaying and Selecting Equations  
The equation list contains two built-in equations, 2*2 lin. solve and 3*3 lin. Solve,  
and the equations you've entered. You can display the equations and select one to  
work with.  
6-6  
   
To display equations:  
1. Press . This activates Equation mode and turns on the EQN annunciator.  
The display shows an entry from the equation list:  
   if the equation pointer is at the top of the list.  
The current equation (the last equation you viewed).  
2. Press ×or Øto step through the equation list and view each equation.  
The list "wraps around" at the top and bottom.    marks the  
"top" of the list.  
To view a long equation:  
1. Display the equation in the equation list, as described above. If it's more than  
14 characters long, only 14 characters are shown. The annunciator  
indicates more characters to the right.  
2. Press Õto begin editing the equation at the beginning, or press Öto  
begin editing the equation at the end. Then press Öor Õrepeatedly to  
move the cursor through the equation one character at a time. and ꢆ  
display when there are more characters to the left or right.  
3. Press Öor Õto scroll the long equations in line 2 by a screen.  
To select an equation:  
Display the equation in the equation list, as described above. The displayed  
equation in line 2 is the one that's used for all equation operations.  
Example: Viewing an Equation.  
View the last equation you entered.  
Keys:  
Display:  
  
Description:  
Displays the current equation in the  
equation list.  
Activates cursor to the left of the  
  
Õ
equation  
Activates cursor to the right of the  
equation  
Leaves Equation mode.  
Ö  
_  
6-7  
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. However, you cannot edit or clear the two built-  
in equations 2*2 lin. solve and 3*3 lin. solve. If you attempt to insert a equation  
between the two built-in equations, the new equation will be inserted after 3*3 lin.  
solve.  
To edit an equation you're typing:  
1. Press Öor Õ to move the cursor allowing you to insert characters before  
the cursor.  
2. Move the cursor and press repeatedly to delete the unwanted number or  
function. Pressing when the equation editing line is empty has no effect,  
but pressing on an empty equation line causes the empty equation line  
to be deleted. The display then shows the previous entry in the equation list.  
3. Press (or ) to save the equation in the equation list.  
To edit a saved equation:  
1. Display the desired equation, press Õto activate the cursor at the beginning  
of the equation or press Öto activate the cursor at the end of the  
equation.(See "Displaying and Selecting Equations" above.)  
2. When the cursor is active in the equation, you can edit the equation just like  
you would when entering a new equation.  
3. Press (or ) to save the edited equation in the equation list,  
replacing the previous version.  
Using menus while editing an equation:  
1. When editing an equation, selecting a setting menu (such as 9,  
8, or   
), will end the equation edit status.  
2. When editing an equation, selecting an insert or view menu (such as ,  
, , , , >,,   
and ), the equation will still be in edit mode after inserting the  
item.  
3. The menus , , are disabled in equation mode.  
6-8  
 
To clear a saved equation:  
Scroll the equation list up or down until the desired equation is in line 2 of the  
display, and then press .  
To clear all saved equations:  
In EQN mode, press   
. Select (). The     menu is  
displayed. Select Ö(Y) .  
Example: Editing an Equation.  
Remove 25 in the equation from the previous example.  
Keys:  
Display:  
  
Description:  
Shows the current equation in the  
equation list.  
Activates cursor at the end of the  
equation  
Deletes the number 25.  
Ö
_  
  
_  
  
Shows the end of edited equation  
in the equation list.  
Leaves Equation mode.  
Types of Equations  
The HP 35s works with three types of equations:  
Equalities. The equation contains an "=", and the left side contains more  
than just a single variable. For example, x + y = r is an equality.  
2
2
2
Assignments. The equation contains an "=", and the left side contains just  
a single variable. For example, A = 0.5 × b × h is an assignment.  
6-9  
 
3
Expressions. The equation does not contain an "=". For example, x + 1  
is an expression.  
When you're calculating with an equation, you might use any type of equation —  
although the type can affect how it's evaluated. When you're solving a problem for  
an unknown variable, you'll probably use an equality or assignment. When you're  
integrating a function, you'll probably use an expression.  
Evaluating Equations  
One of the most useful characteristics of equations is their ability to be evaluated —  
to generate numeric values. This is what enables you to calculate a 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 is essentially treated as "–". The value is a measure of how well  
the equation balances.  
The HP 35s 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 of equation.  
returns the value of the equation — unless it's an assignment–type  
equation. For an assignment equation, returns the value of the right  
side only, and also "enters" that value into the variable on the left side — it  
stores the value in the variable.  
The following table shows the two ways to evaluate equations.  
 
Type of Equation  
Equality: g(x) = f(x)  
Result for  
Result for   
g(x) f(x)  
2
2
2
Example: x + y = r  
2
2
2
x + y r  
y f(x)  
A – 0.5 × b × h  
Assignment: y = f(x)  
Example: A = 0.5 × b x h  
f(x)  
0.5 × b × h  
Expression: f(x)  
f(x)  
3
3
Example: x + 1  
x + 1  
Also stores the result in the left–hand variable, A for example.  
To evaluate an equation:  
1. Display the desired equation. (See "Displaying and Selecting Equations"  
above.)  
2. Press or . The equation prompts for a value for each variable  
needed. (If the base of a number in the equation is different from the current  
base, the calculator automatically changes the result to the current base.)  
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.)  
To halt a calculation, press or . The message  is shown in  
line 2.  
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.  
Using ENTER for Evaluation  
If an equation is displayed in the equation list, you can press to evaluate  
the equation. (If you're in the process of typing the equation, pressing only  
ends the equation — it doesn't evaluate it.)  
 
If the 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.  
π  
( ×as required)  
Starts evaluating the  
  
  
assignment equation so the  
value will be stored in V.  
Prompts for variables on the  
right–hand side of the  
equation. The current value  
for D is 2.5.  
  
  
  
Stores D, prompts for L,  
whose current value is 16.  
  
  
Stores L in millimeters;  
  
calculates V in cubic  
  
  
millimeters, stores the result  
in V, and displays V.  
Changes cubic millimelers to  
liters (but doesn't change V.  
  
  
Using XEQ for Evaluation  
If an equation is displayed in the equation list, you can press 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 changed to 35.5 millimeters.  
Keys:  
Display:  
  
Description:  
Displays the desired equation.  
Starts evaluating the equation to  
  
 find its value. Prompts for all  
variables.  
  
Keeps the same V, prompts for D.  
  
  
Stores new D, Prompts for L.  
  
  
Keeps the same L; calculates the  
value of the equation — the  
imbalance between the left and  
right sides.  
  
  
Changes cubic millimeters to liters.  
  
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  
. If the unnamed indirect variable (I) or (J) is in an equation, you will not  
be prompted to for its value, as the current value stored in the unnamed indirect  
variable will be used automatically. (See chapter 14)  
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  
5  in RPN mode, or press 25 in ALG  
mode. Before pressing , the expression will display in line 2, and  
after pressing , the result of the expression will display in line 2.  
To cancel the prompt, press . The current value for the variable remains in  
the X–register and displays in right-side of the line two. 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  .  
In RPN mode,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 lift is enabled, so the value is saved on the stack.  
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:  
   
Order  
Operation  
Parentheses  
Functions  
Power ( )  
Unary Minus ()  
Multiply and Divide  
Add and Subtract  
Equality  
Example  
1
2
3
4
5
6
7
  
  
  
  
,   
,   
  
So, for example, all operations inside parentheses are performed before operations  
outside the parentheses.  
Examples:  
Equations  
Meaning  
3
a × (b ) = c  
  
3
  
(a × b) = c  
a + (b/c) = 12  
(a + b) / c = 12  
2
  
[%CHG ((t + 12), (a – 6)) ]  
Equation Functions  
The following table lists the functions that are valid in equations. Appendix G,  
"Operation Index" also gives this information.  
LN  
LOG  
IP  
INTG  
COS  
COSH  
EXP  
FP  
IDIV  
TAN  
TANH  
ALOG  
RND  
RMDR  
ASIN  
SQ  
ABS  
SQRT  
!
INV  
SGN  
SIN  
ACOS  
ACOSH  
ATAN  
ATANH  
SINH  
ASINH  
%CHG  
nCr  
XROOT  
nPr  
DEG  
L  
RAD  
GAL  
HMSꢇ  
MILE  
HMS  
KM  
KG  
SEED  
+
LB  
ARG  
°C  
RAND  
×
°F  
CM  
IN  
π
÷
σ y  
^
x
sx  
sy  
σ x  
y
r
m
b
x w  
ˆ
x
ˆ
y
2
2
n
Σx  
Σy  
Σxy  
Σx  
Σy  
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, IDIV, RMDR,  
nCr and nPr. Separate the two arguments with a comma.  
In an equation, the XROOT function takes its arguments in the opposite order from  
RPN usage. For example, –83 to is equivalent to .  
All other two argument functions take their arguments in the Y, X order used for  
RPN. For example, 284 xis equivalent to .  
For two argument functions, be careful if the second argument is negative. These are  
valid equations:  
 
  
  
Eight of the equation functions have names that differ from their equivalent  
operations:  
RPN Operation  
Equation function  
2
x
SQ  
SQRT  
x
e
x
EXP  
ALOG  
INV  
x
10  
1/x  
X
y
XROOT  
x
^
y
INT÷  
IDIV  
Example: Perimeter of a Trapezoid.  
The following equation calculates the perimeter of a trapezoid. This is how the  
equation might appear in a book:  
1
1
)
Perimeter = a + b + h ( sinθ sinφ  
+
a
h
φ
θ
b
The following equation obeys the syntax rules for HP 35s equations:  
Parentheses used to group items  
Single letter  
name  
Optional explicit  
multiplication  
Division is done before  
addition  
The next equation also obeys the syntax rules. This equation uses the inverse  
function, , instead of the fractional form, . Notice that  
the SIN function is "nested" inside the INV function. (INV is typed by .)  
  
Example: Area of a Polygon.  
The equation for area of a regular polygon with n sides of length d is:  
1
4
cos(π /n)  
sin(π/n)  
n d 2  
Area =  
d
2
π/n  
You can specify this equation as  
ππ  
Notice how the operators and functions combine to give the desired equation.  
You can enter the equation into the equation list using the following keystrokes:  
  
Õ  
Syntax Errors  
The calculator doesn't check the syntax of an equation until you evaluate the  
equation. If an error is detected,   is displayed and the cursor is  
displayed at the first error location. You have to 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 35s lets you create  
"equations" that might actually be messages. This is especially useful in programs,  
as described in chapter 13.  
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 key to keep the values in the display.  
The checksum is a four–digit hexadecimal value that uniquely identifies this  
equation. 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.  
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:  
π  
Displays the desired equation.  
( ×as required)  
  
  
 (hold)  
Display equation's checksum  
and length.  
π  
(release)  
Redisplays the equation.  
Leaves Equation mode.  
7
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  
2
x – 3y = 10  
If you know the value of y in this equation, then SOLVE can solve for the unknown x.  
If you know the value of x, then SOLVE can solve for the unknown y. This works for  
"word problems" just as well:  
Markup × Cost = Price  
If you know any two of these variables, then SOLVE can calculate the value of the  
third.  
When the equation has only one variable, or when known values are supplied for  
all variables except one, then to solve for x is to find a root of the equation. A root  
of an equation occurs where an equality or assignment equation balances exactly,  
or where an expression equation equals zero.  
Solving an Equation  
To solve an equation (excluding built-in equations) for an unknown variable:  
1. Press and display the desired equation. If necessary, type the equation  
as explained in chapter 6 under "Entering Equations into the Equation List."  
7-1  
   
2. Press  then press the key for the unknown variable. For example,  
press  X to solve for x. The equation then prompts for a value for  
every other variable in the equation.  
3. For each prompt, enter the desired value:  
If the displayed value is the one you want, press .  
If you want a different value, type or calculate the value and press .  
(For details, see "Responding to Equation Prompts" in chapter 6.)  
You can halt a running calculation by pressing or .  
When the root is found, it's stored in the relation 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 value or Zero, and the Z–register contains the value  
of the root D-value(which should be zero).  
For some complicated mathematical conditions, a definitive solution cannot be  
found — and the calculator displays   . See "Verifying the Result"  
later in this chapter, and "Interpreting Results" and "When SOLVE Cannot Find a  
Root" in appendix D.  
For certain equations it helps to provide one or two initial guesses for the unknown  
variable before solving the equation. This can speed up the calculation, direct the  
answer toward a realistic solution, and find more than one solution, if appropriate.  
See "Choosing Initial Guesses for Solve" later in this chapter.  
Example: Solving the Equation of Linear Motion.  
The equation of motion for a free–falling object is:  
1
2
d = v t + / g t  
0
2
where d is the distance, v is the initial velocity, t is the time, and g is the  
0
acceleration due to gravity.  
Type in the equation:  
7-2  
Keys:  
()  
Display:  
Description:  
Clears memory.  
Ö()  
    
    
Selects Equation mode.  
Starts the equation.  
  
    
 G  
  
_  
 _  
  
Terminates the equation  
and displays the left end.  
  
  
  
Checksum and length.  
g (acceleration due to gravity) is included as a variable so you can change it for  
2
2
different units (9.8 m/s or 32.2 ft/s ).  
Calculate how many meters an object falls in 5 seconds, starting from rest. Since  
Equation mode is turned on and the desired equation is already in the display, you  
can start solving for D:  
Keys:  
Display:  
_  
Description:  
  
Prompts for unknown  
variable.  
  
Selects D; prompts for V.  
value  
  
  
  
  
Stores 0 in V; prompts for T.  
value  
  
Stores 5 in T; prompts for  
G.  
value  
  
Stores 9.8 in G; solves for  
  
D.  
  
Try another calculation using the same equation: how long does it take an object to  
fall 500 meters from rest?  
7-3  
Keys:  
Display:  
Description:  
Displays the equation.  
  
  
  
  
Solves for T; prompts for  
D.  
  
Stores 500 in D; prompts  
for V.  
  
  
Retains 0 in V; prompts  
  
for G.  
  
  
Retains 9.8 in G; solves  
for T.  
  
Example: Solving the Ideal Gas Law Equation.  
The Ideal Gas Law describes the relationship between pressure, volume,  
temperature, and the amount (moles) of an ideal gas:  
P × V = N × R × T  
2
where P is pressure (in atmospheres or N/m ), V is volume (in liters), N is the  
number of moles of gas, R is the universal gas constant (0.0821 liter–atm/mole–K  
or 8.314 J/mole–K), and T is temperature (Kelvins: K=°C + 273.1).  
Enter the equation:  
Keys:  
Display:  
Description:  
P  
_  
Selects Equation mode  
and starts the equation.  
  
  
  
_  
  
Terminates and displays  
the equation.  
  
  
  
Checksum and length.  
7-4  
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:  
P  
Display:  
Description:  
  
Solves for P; prompts for V.  
value  
  
Stores 2 in V; prompts for  
N.  
  
value  
  
  
  
Stores .005 in N; prompts  
for R.  
Stores .0821 in R; prompts  
for T.  
value  
  
value  
  
  
Calculates T (Kelvins).  
  
  
Stores 297.1 in T; solves for  
P in 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.  
  
Solves for N; prompts for P.  
  
  
  
Stores .05 in P; prompts for  
  
  
V.  
Stores 5 in V; prompts for  
R.  
Retains previous R; prompts  
for T.  
  
  
  
  
  
Calculates T (Kelvins).  
  
  
  
  
7-5  
Stores 291.1 in T; solves for  
N.  
  
  
  
  
  
Calculates mass in grams,  
N × 28.  
Calculates density in grams  
per liter.  
  
  
Solving built-in Equation  
The built-in equations are: “2*2 lin. solve” (Ax+By=C, Dx+Ey=F) and “3*3 lin.  
Solve”(Ax+By+Cz=D, Ex+Fy+Gz=H, Ix+Jy+Kz=L). If you select one of them, the  
, and key will have no effect. Pressing the will  
request 6 variables (A to F) for the 2*2 case or 12 variables (A to L) for the 3*3  
case, and use them to find x, y for a 2*2 linear equation system or x, y and z for a  
3*3 linear equation system. The result will be saved in variables x, y, and z. The  
calculator can detect cases with infinitely many solutions or no solutions.  
Example: solve the x,y in simultaneous equations  
Keys:  
Display:  
Description:  
    
Enters equation mode.  
    
Ø
    
Displays the built-in  
equation  
    
  
  
  
  
  
  
Prompts for A.  
value  
  
Stores 1 in A; prompts for  
B.  
Stores 2 in B; prompts for  
C.  
Stores 5 in C; prompts for  
D.  
Stores 3 in D; prompts for  
value  
  
value  
  
value  
  
E.  
value  
7-6  
 
  
  
Stores 4 in E ;prompts for F.  
value  
  
  
Stores 11 in F and  
calculates x and y.  
  
  
Ø
value of y  
  
Understanding and Controlling SOLVE  
SOLVE first attempts to solve the equation directly for the unknown variable. If the  
attempt fails, SOLVE changes to an iterative (repetitive) procedure. 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.  
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  
  .  
See appendix D 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.  
7-7  
   
The Y–register (press ) contains the previous estimate for the root or equals  
to zero. 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 D-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 converge on  
a root. (You can see the value in the X–register — the final estimate of the root — by  
pressing or 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 root.  
If the X– and Y–register values are close together, and the Z–register value is  
close to zero, the estimate from the X–register may be an approximation to a  
root.  
Interrupting a SOLVE Calculation  
To halt a calculation, press or , and the message “” will  
be shown. The current best estimate of the root is in the unknown variable; use   
to view it without disturbing the stack, but solving cannot be resumed.  
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).  
7-8  
   
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 the SOLVE  
procedure to the desired answer or range of answers. For example, the  
equation of linear motion  
1
2
d = v t + / gt  
0
2
can have two solutions for t. You can direct the answer to the required  
solution by entering appropriate guesses.  
The example using this equation earlier in this chapter didn't require you 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 (message   ).  
In the following example, the equation has more than one root, but guesses help  
find the desired root.  
7-9  
Example: Using Guesses to Find a Root.  
Using a rectangular piece of sheet metal 40 cm by 80 cm, form an open–top box  
3
having a volume of 7500 cm . You need to find the height of the box (that is, the  
amount to be folded up along each of the four sides) that gives the specified  
volume. A taller box is preferred to a shorter one.  
H
_
40 2H  
40  
H
_
H
80 2H  
H
80  
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:  
Selects Equation mode and starts  
the equation  
_  
  
4  
HÕ  
_  
4  
HÕ  
H  
_  
_  
  
Terminates and displays the  
equation.  
  
Checksum and length.  
  
  
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.  
Displays current equation.  
Solves for H; prompts for V.  
H  
  
_  
  
H  
  
value  
  
Stores 7500 in V; solves for H.  
  
  
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.  
  
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 understand 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 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.  
_
_
_
7500 (40 H) (20 H) 4H  
20,000  
H
_
10  
50  
_
10,000  
For More Information  
This chapter gives you instructions for solving for unknowns or roots over a wide  
range of applications. Appendix D 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.  
 
8
Integrating Equations  
Many problems in mathematics, science, and engineering require calculating the  
definite integral of a function. If the function is denoted by f(x) and the interval of  
integration is a to b, then the integral can be expressed mathematically as  
I = b f (x)dx  
a
f (x)  
I
x
a
b
The quantity I can be interpreted geometrically as the area of a region bounded by  
the graph of the function f(x), the x–axis, and the limits x = a and x = b (provided  
that f(x) is nonnegative throughout the interval of integration).  
The operation (FN) integrates the current equation with respect to a specified  
variable ( d_). The function may have more than one variable.  
8-1  
 
Integrating Equations ( FN)  
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 the  
equation list (press ×or Ø) 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.  
You can halt a running integration calculation by pressing or ,and the  
message “” will be shown in line 2, but the integration cannot be  
resumed. 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  
 and higher , , and  settings. The 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.  
To integrate the same equation with different information:  
If you use the same limits of integration, press   move them into the X– 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 integrand.  
8-2  
 
Example: Bessel Function.  
The Bessel function of the first kind of order 0 can be expressed as  
π cos(x sint)dt  
1
π
J0(x) =  
0
Find the Bessel function for x–values of 2 and 3.  
Enter the expression that defines the integrand's function:  
cos (x sin t )  
Keys:  
Display:  
Description:  
Clears memory.  
()Ö()  
    
    
Selects Equation mode.  
Types the equation.  
X  
  
  
  
  
  
ÕÕ  
_  
  
Terminates the expression and  
displays its left end.  
Checksum and length.  
  
  
  
Leaves Equation mode.  
Now integrate this function with respect to t from zero to π ; x = 2.  
Keys:  
Display:  
Description:  
Selects Radians mode.  
9()  
   
Enters the limits of integration  
(lower limit first).  
  
Displays the function.  
Prompts for the variable of  
integration.  
  
  
 _  
8-3  
Prompts for value of X.  
  
value  
  
x = 2. Starts integrating;  
  
calculates result for  
  
  
π f(t)  
0
The final result for J (2).  
    
  
0
Now calculate J (3) with the same limits of integration. You must re-specify the limits  
0
of integration (0, π) since they were pushed off the stack by the subsequent division  
by π.  
Keys:  
  
Display:  
  
Description:  
Enters the limits of integration  
(lower limit first).  
Displays the current equation.  
Prompts for the variable of  
integration.  
  
  
 _  
  
Prompts for value of X.  
  
x = 3. Starts integrating and  
calculates the result for  
  
  
  
  
π
.
f(t)  
0
The final result for  
    
  
J (3).  
0
Example: Sine Integral.  
Certain problems in communications theory (for example, pulse transmission  
through idealized networks) require calculating an integral (sometimes called the  
sine integral) of the form  
t
sinx  
x
Si (t) =  
(
)dx  
0
Find Si (2).  
8-4  
Enter the expression that defines the integrand's function:  
sinx  
x
If the calculator attempted to evaluate this function at x = 0, the lower limit of  
integration, an error (  ) would result. However, the integration  
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.  
Starts the equation.  
X  
  
The closing right parenthesis is  
required in this case.  
Õ
_  
X  
_  
  
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:  
Display:  
Description:  
Selects Radians mode.  
9()  
X  
Enters limits of integration (lower  
first).  
_  
Displays the current equation.  
  
Calculates the result for Si(2).  
 X  
  
  
  
8-5  
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 E.  
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 the only number of digits  
specified.  
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.  
Interpreting Accuracy  
After calculating the integral, the calculator places the estimated uncertainty of that  
integral's result in the Y–register. Press to view the value of the uncertainty.  
For example, if the integral Si(2) is 1.6054 0.0002, then 0.0002 is its  
uncertainty.  
8-6  
     
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:  
8  
Sets scientific notation with two  
decimal places, specifying that the  
function is accurate to two decimal  
places.  
  
()  
Rolls down the limits of integration  
from the Z–and T–registers into the  
X–and Y–registers.  
  
  
  
Displays the current Equation.  
  
 X  
The integral approximated to two  
decimal places.  
  
  
  
The uncertainty of the  
  
approximation of the integral.  
The integral is 1.61 0.0161. 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 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.  
8-7  
Keys:  
Display:  
Description:  
8  
Specifies accuracy to four decimal  
places. The uncertainty from the  
last example is still in the display.  
  
()  
Rolls down the limits of integration  
from the Z– and T–registers into  
the X– and Y–registers.  
  
  
  
Displays the current equation.  
Calculates the result.  
  
 X  
  
  
  
Note that the uncertainty is about  
1/100 as large as the uncertainty  
of the SCI 2 result calculated  
previously.  
  
8()  
Restores FIX 4 format.  
  
  
9()  
Restores Degrees mode.  
This uncertainty indicates that the result might be correct to only three 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.  
For More Information  
This chapter gives you instructions for using integration in the HP 35s over a wide  
range of applications. Appendix E contains more detailed information about how  
the algorithm for integration works, conditions that could cause incorrect results and  
conditions that prolong calculation time, and obtaining the current approximation to  
an integral.  
8-8  
 
9
Operations with Complex Numbers  
The HP 35s can use complex numbers in the form  
    
It has operations for complex arithmetic (+, –, ×, ÷), complex trigonometry (sin, cos,  
z
z
2
tan), and the mathematics functions –z, 1/z,  
, ln z, and e . (where z and z  
z1  
1
2
are complex numbers).  
The form, x+yi, is only available in ALG mode.  
To enter a complex number:  
Form:   
1. Type the real part.  
2. Press6.  
3. Type the imaginary part.  
Form:   
1. Type the real part.  
2. Press  
3. Type the imaginary part.  
4. Press6.  
Form:   
1. Type the value of r.  
2. Press?.  
3. Type the value of θ.  
The examples in this chapter all utilize RPN mode unless otherwise noted.  
9-1  
 
The Complex Stack  
A complex number occupies part 1 and part 2 of a stack level. In RPN mode, the  
complex number occupying part 1 and part 2 of the X-register is displayed in line 2,  
while the complex number occupying part 1 and part 2 of the Y-register is  
displayed in line 1.  
Part3  
T
Z
Y
X
Part2  
Part1  
Part3  
Part2  
Part1  
Part3  
X1iY1  
or  
Y 1 o r a 1  
X 1 o r r 1  
P a r t 3  
(Display in line 1)  
(Display in line 2)  
r1θ a1  
X 2 i Y 2  
or  
Y 2 o r a 2  
X 2 o r r 2  
Complex Stack  
r2θ a2  
Complex Result,Z  
Complex Operations  
Use the complex operations as you do real operations in ALG and RPN mode.  
To do an operation with one complex number:  
1. Enter the complex number z as described before.  
2. Select the complex function.  
9-2  
   
Functions for One Complex Number, z  
To Calculate: Press:  
Change sign, –z  
Inverse, 1/z  
Natural log, ln z  
Natural antilog, e  
Sin z  
  
  
z
Cos z  
Tan z  
Absolute value, ABS(z)  
Argument value, ARG(z)  
  
=  
To do an arithmetic operation with two complex numbers:  
1. Enter the first complex number, z as described before.  
1
2. Enter the second complex number z as described before.  
2
3. Select the arithmetic operation:  
Arithmetic With Two Complex Numbers, z and z  
1
2
To Calculate:  
Addition, z + z  
Press:  
1
2
Subtraction, z – z  
1
2
Multiplication, z × z  
1
2
Division, z ÷ z  
1
2
z1z2  
Power function,  
9-3  
Examples:  
Here are some examples of trigonometry and arithmetic with complex numbers:  
Evaluate sin (2i3)  
Keys:  
Display:  
Description:  
Sets display format.  
8( )  
6  
Result is 9.1545 i –4.1689.  
   
Evaluate the expression  
z
÷ (z + z ),  
1
2
3
where z = 23 i 13, z = –2i1 z = 4 i– 3  
1
2
3
Perform the calculation as  
Keys:  
Display:  
Description:  
Sets display format  
ENTER z1  
8( )  
6  
   
   
   
   
   
   
   
ENTER z2  
6  
(z + z ). Result is 2 i -2.  
2
3
6  
z
÷(z + z ). Result is 2.5  
1
2
3
i 9.  
Evaluate (4 i –2/5) × (3 i –2/3).  
Keys:  
Display:  
Description:  
Sets display format  
Enters 4i-2/5  
8( )  
6  
   
   
9-4  
Enters 3i-2/3  
6  
   
   
Result is 11.7333i-3.8667  
   
2 , where z = (1i 1).  
Evaluate ez  
Keys:  
Display:  
Description:  
ENTER 1i1Intermediate  
result of  
6  
   
   
   
  
  
–2  
Z ,result is 0i-5  
Final result is  
   
0.8776 i– 0.4794.  
Using Complex Numbers 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.  
i
imaginary  
(a, b)  
r
θ
real  
Example: Vector Addition.  
Add the following three loads.  
9-5  
 
y
62o  
L
185 lb  
2
170 lb  
143o  
L
1
x
L
3
100 lb  
261 o  
Keys:  
Display:  
Description:  
9()  
Sets Degrees mode.  
Sets complex mode  
8  
()  
?  
Enters L  
θ  
1
  
?  
  
?  
  
θ  
Enters L .  
θ  
θꢆ  
θ  
θꢆ  
θꢆ  
2
Enters L and adds L + L  
2
3
3
Adds L + L + L .  
1
2
3
Õ  
Scrolls the screen to see the  
rest of the answer  
  
You can do a complex operation with numbers whose complex forms are different;  
however, the result form is dependent on the setting in 8menu.  
9-6  
Evaluate 1i1+3θ 10+5θ 30  
Keys:  
Display:  
Description:  
Sets Degrees mode.  
Sets complex mode  
9()  
8  
()  
Enters 1i1  
6  
θ  
θ  
θ  
θ  
θ  
θ  
Enters 3θ 10  
?  
Enters 5θ 30 and adds 3θ  
10  
?  
Adds 1i1,result is 9.2088θ  
25.8898  
θ  
Complex Numbers in Equations  
You can type complex numbers in equations. When an equation is displayed, all  
numeric forms are shown as they were entered, like xiy, or rθ a  
When you evaluate an equation and are prompted for variable values, you may  
enter complex numbers. The values and format of the result are controlled by the  
display setting. This is the same as calculating in ALG mode.  
Equations that contain complex numbers can be solved and integrated.  
9-7  
 
Complex Number in Program  
In a program, you can type a complex number. For example, 1i2+3θ 10+5  
θ 30 in program is:  
Program lines: (ALG mode)  
    
Description  
Begins the program  
    
   
When you are running a program and are prompted for values by INPUT  
instructions, you can enter complex numbers. The values and format of the result  
are controlled by the display setting.  
The program that contains the complex number can also be solved and integrated.  
9-8  
 
10  
Vector Arithmetic  
From a mathematical point of view, a vector is an array of 2 or more elements  
arranged into a row or a column.  
Physical vectors that have two or three components and can be used to represent  
physical quantities such as position, velocity, acceleration, forces, moments, linear  
and angular momentum, angular velocity and acceleration, etc.  
To enter a vector:  
1. Press 3  
2. Enter the first number for the vector.  
3. Pressand enter a second number for a 2-D or 3-D vector.  
4. Pressand enter a third number for a 3-D vector.  
The HP 35s cannot handle vectors with more than 3 dimensions.  
Vector operations  
Addition and subtraction:  
The addition and subtraction of vectors require that two vector operands have the  
same length. Attempting to add or subtract vectors of different length produces the  
error message “ .  
1. Enter the first vector  
2. Enter the second vector  
3. Press or   
   
Calculate [1.5,-2.2]+[-1.5,2.2]  
Keys:  
Display:  
Description:  
9()  
Switches to RPN mode(if  
necessary)  
3  
  
Enters [1.5,-2.2]  
  
  
3  
  
Enters [-1.5,2.2]  
Adds two vectors  
  
  
  
  
Calculate [-3.4,4.5]-[2.3,1.4]  
Keys:  
Display:  
Description:  
Switches to ALG mode  
Enters [-3.4,4.5]  
9()  
3 _  
Õ  
Enters [2.3,1.4]  
3  
  
  
  
  
Subtracts two vectors  
Multiplication and divisions by a scalar:  
1. Enter a vector  
2. Enter a scalar  
3. Press for multiplication or for division  
Calculate [3,4]x5  
Keys:  
Display:  
Description:  
Switches to RPN mode  
Enters [3,4]  
9()  
3  
  
  
  
_  
Enters 5 as a scalar  
  
Performs multiplication  
  
Calculate [-2,4]÷2  
Keys:  
9()  
3  
Õ  
Display:  
Description:  
Switches to ALG mode  
Enters [-2,4]  
_  
  
Enters 5 as a scalar  
Performs division  
  
  
  
Absolute value of the vector  
The absolute value function ABS, when applied to a vector, produces the  
magnitude of the vector. For a vector A=(A1, A2, …An), the magnitude is defined  
2
2
A 2 + A2 + ⋅⋅ + An  
as  
=
.
A
1
1. Press  
2. Enter a vector  
3. Press  
For example: Absolute value of vector [5,12]:  
3.The answer is 13. In RPN mode:  
9()3.  
 
Dot product  
Function DOT is used to calculate the dot product of two vectors with the same  
length. Attempting to calculate the dot product of two vectors of different length  
causes an error message “ .  
For 2-D vectors: [A, B], [C, D], dot product is defined as [A, B][C, D]= A x C +B x  
D.  
For 3-D vectors: [A, B, X], [C, D,Y], dot product is defined as [A, B, X][C, D, Y]= A  
x C +B x D+X x Y  
1. Enter the first Vector  
2. Press   
3. Enter the second vector  
4. Press   
Note: The sign,,here means ”dot product” instead of “cross product. For cross  
product, see chapter 17.  
Calculate the dot product of two vectors, [1,2] and [3,4]  
Keys:  
Display:  
Description:  
Switches to ALG mode  
9()  
Enters the first vector [1,2]  
3_  
Õ
  
Executes for dot product,  
and enters the second vector  
The dot product of two  
vectors is 11  
3ꢀ  
  
Calculate the dot product of two vectors, [9,5] and [2.2]  
Keys:  
Display:  
Description:  
Switches to RPN mode  
Enters the first vector [9,5]  
9()  
3  
  
  
and enters the second vector  
[2,2]  
3  
  
  
 
Presses for dot product  
,and the dot product of two  
vectors is 28  
  
Angle between vectors  
The angle between two vectors, A and B, can be found as θ =  
ACOS(AB/ A B )  
Find the angle between two vectors: A=[1,0],B=[0,1]  
Keys:  
Display:  
Description:  
Switches to ALG mode  
Sets Degrees mode  
Arc cosine function  
Enters vector A [1,0]  
9()  
9()  
  
  
  
3  
Õ
 Enters vector B [0,1] for dot  
3ꢀ  
Õ  
product of A and B  
Finds the magnitude of  
vector A [1,0]  
3  
Õ  
ꢆ  
Finds the magnitude of  
vector B [0,1]  
3  
  
ꢆ  
The angle between two  
  
vectors is 90  
  
Find the angle between two vectors: A=[3,4],B=[0,5]  
Keys:  
Display:  
Description:  
Switches to RPN mode  
Sets Degrees mode  
Finds the dot product of  
two vectors  
9()  
9()  
3  
  
3  
  
  
3  
  
  
  
Finds the magnitude of  
vector [3,4]  
 
3  
  
  
Finds the magnitude of  
vector [0,5]  
Multiplies two vectors  
  
  
  
  
Divides two values  
  
  
  
The angle between two  
vectors is 36.8699  
  
Vectors in Equations  
Vectors can be used in equations and in equation variables exactly like real  
numbers. A vector can be entered when prompted for a variable.  
Equations containing vectors can be solved, however the solver has limited ability if  
the unknown is a vector.  
Equations containing vectors can be integrated, however the result of the equation  
nd  
rd  
must be a real or a 1-D vector or a vector with 0 as the 2 and 3 elements.  
 
Vectors in Programs  
Vectors can be used in program in the same way as real and complex numbers  
For example, [5, 6] +2 x [7, 8] x [9, 10] in a program is:  
Program lines:  
   
        
   
Description:  
Begins the program  
[5,6]  
A vector can be entered when prompted for a value for a variable. Programs that  
contain vectors can be used for solving and integrating.  
 
Creating Vectors from Variables or Registers  
It is possible to create vectors containing the contents of memory variables, stack  
registers, or values from the indirect registers, in run or program modes.  
In ALG mode, begin entering the vector by pressing 3. RPN mode works  
similarly to ALG mode, except that the dkey must be pressed first, followed by  
pressing 3 .  
To enter an element containing the value stored in a lettered variable, press h  
and the variable letter.  
To enter an element from a stack register, press the <key and use the Õor Ö  
keys to move the underline symbol so that it is under the stack register to be used  
and press   
To enter an element indirectly indicated by the value in the I or J register, press  
hand either (I) or (J).  
For example, to construct the vector [ C, REGZ, (J) ] in RPN mode, press d  
3, then hC<Õꢀh A.  
 
11  
Base Conversions and Arithmetic and  
Logic  
The BASE menu (  ) allows you to enter numbers and force the display of  
numbers in decimal, binary, octal and hexadecimal base.  
The LOGIC menu(>) provides access to logic functions.  
BASE Menu  
Menu label  
  
Description  
Decimal mode. This is the normal calculator mode  
Hexadecimal mode. The HEX annunciator is displayed  
when this mode is active. Numbers are displayed in  
hexadecimal format. In RPN mode, the keys , ,  
  
, , and act as shortcut to enter the  
digits A to F. In ALG mode, press A, B, C, D, E or F  
to enter the digits A to F.  
Octal mode. The OCT annunciator is displayed when this  
mode is active. Numbers are displayed in Octal format.  
  
  
Binary mode. The BIN annunciator is displayed when  
this mode is active. Numbers are displayed in Binary  
format. If a number has more than 12 digits, the Õ  
and Ökeys allow to view the full number (See  
"Windows for Long Binary Numbers" later in this chapter.)  
placed at the end of a number means that this number is a  
decimal number  
placed at the end of a number means that this number is  
an hexadecimal number. To enter an hexadecimal  
number, type the number followed by “”  
 
placed at the end of a number means that this number is  
an octal number. To enter an octal number, type the  
number followed by “”  
placed at the end of a number means that this number is a  
binary number. To enter a binary number, type the  
number followed by “”  
Examples: Converting the Base of a Number.  
The following keystrokes do various base conversions.  
Convert 125.99 to hexadecimal, octal, and binary numbers.  
10  
Keys:  
  
Display:  
Description:  
Converts the decimal number to  
base 16.  
Base 8.  
Base 2.  
  
()  
  
 ()  
 ()  
 ()  
  
  
Note: When non decimal bases are use, only the integer part of numbers are used  
for display. The fractional parts are kept (unless operations are performed that erase  
them) and will be displayed if the decimal base is selected.  
Convert 24FF to binary base. The binary number will be more than 14 digits (the  
16  
maximum display) long.  
Keys:  
Display:  
  
Description:  
Use the key to type "F".  
 ()  
  
 ()  
The entire binary number does  
 ()  
not fit. The annunciator  
indicates that the number  
continues to the right.  
ꢆ  
Displays the rest of the number.  
The full number is  
Õ  
  
10010011111111 .  
b
Displays the first 14 digits  
again.  
Ö  
ꢆ  
Restores base 10.  
  
()  
you can use menu to enter base-n sign b/o/d/h following the operand to  
represent 2/8/10/16 base number in any base mode. A number without a base  
sign is a decimal number  
Note:  
In ALG mode:  
1. The result’s base mode is determined by the current base mode setting.  
2. If there is no active command line (there is no blinking cursor on line 1),  
changing the base will update line 2 to be in the new base.  
3. After pressing or changing the base mode, calculator will  
automatically add a current base sign b/o/h following the result to represent  
base 2/8/16 number in line 2.  
4. To edit expression again, press Öor Õ  
In RPN mode:  
When you enter a number in line 2, press , and then change the base  
mode, the calculator will convert the base of the numbers in line 1 and line 2, and  
the sign b/o/h will be added following the number to represent base 2/8/16.  
To view the next screen’s content in line 2, press Öor Õto change the  
screen.  
LOGIC Menu  
Description  
Menu label  
  
Logical bit-by-bit "AND" of two arguments.  
For example: AND(1100b,1010b)=1000b  
Logical bit-by-bit "XOR" of two arguments.  
For example: XOR(1101b,1011b)=110b  
Logical bit-by-bit "OR" of two arguments.  
For example: OR(1100b,1010b)=1110b  
Returns the one's complement of the argument. Each bit in  
the result is the complement of the corresponding bit in the  
argument.  
  
  
  
For example: NOT(1011b)=  
111111111111111111111111111111110100b  
Logical bit-by-bit "NAND" of two arguments.  
For example:  
  
NAND(1100b,1010b)=11111111111111111111111  
1111111110111b  
Logical bit-by-bit "NOR" of two arguments.  
  
For example: NOR(1100b,1010b)=  
111111111111111111111111111111110001b  
The AND, OR, XOR, NOT, NAND, NOR” can be used as logic  
functions. Fraction, complex, vector arguments will be seen as an "  
" in logic function.  
Arithmetic in Bases 2, 8, and 16  
You can perform arithmetic operations using , , , and in any base.  
The only function keys that are actually deactivated in HEX mode are , ,  
, , , and . However, you should realize that most operations other  
than arithmetic will not produce meaningful results since the fractional parts of  
numbers are truncated.  
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 display of the number but not the actual  
number in the X–register, arithmetic does alter the number in the X–register.  
If the result of an operation cannot be represented in valid bits, the display shows  
 and then shows the largest positive or negative number possible.  
Example:  
Here are some examples of arithmetic in Hexadecimal, Octal, and Binary modes:  
12F + E9A  
= ?  
16  
16  
Keys:  
Display:  
Description:  
Sets base 16; HEX  
 ()  
annunciator on.  
Result.  
  
()  
 ()   
  
7760 – 4326 =?  
8
8
Sets base 8; OCT  
annunciator on. Converts  
displayed number to octal.  
Result.  
  
 ()  
  
()  
  ()  
  
100 ÷ 5 =?  
8
8
Integer part of result.  
  
()  
()   
  
5A0 + 1001100 =?  
16  
2
Sets base 16; HEX  
annunciator on.  
  
()  
  
()  
Changes to base 2; BIN  
annunciator on. This  
terminates digit entry, so no  
 ()  
  
 ()  
b  
is needed between  
the numbers.  
Result in binary base.  
Result in hexadecimal base.  
Restores decimal base.  
  
  
 ()  
 ()  
  
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 36 bits  
(12 octal digits or 9 hexadecimal digits). Leading zeros are not displayed, but they  
are important because they indicate a positive number. For example, the binary  
representation of 125 is displayed as:  
10  
1111101b  
which is the same as these 36 digits:  
000000000000000000000000000001111101b  
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:  
Enters a positive, decimal  
number; then converts it to  
hexadecimal.  
  
  
()  
   
2's complement (sign  
changed).  
  
  
Binary version; indicates  
 ()  
ꢆ  
more digits. The number is  
negative since the highest bit  
is 1.  
Displays the rest of the  
number by scrolling one  
screen  
Õ  
ꢆ  
Displays the rightmost  
window;  
Õ  
  
Negative decimal number.  
  
 ()  
Range of Numbers  
The 36-bit binary number 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  
Octal  
7FFFFFFFFh  
800000000h  
377777777777o  
400000000000o  
Binary  
0111111111111111111111 10000000000000000000000  
11111111111111b  
34,359,738,367  
0000000000000b  
–34,359,738,368  
Decimal  
Numbers outside of this range can not be entered when a non decimal base is  
selected.  
 
In BIN/OCT/HEX, If a number entered in decimal base is outside the range given  
above, then it produces the message  . Any operation using    
causes an overflow condition, which substitutes the largest positive or negative  
number possible for the too-big number.  
Windows for Long Binary Numbers  
The longest binary number can have 36 digits. Each 14–digit display of a long  
number is called a window.  
36-bit number  
  
Highest Window  
(Displayed)  
Lowest Window  
When a binary number is larger than the 14 digits, the or annunciator (or  
both) appears, indicating in which direction the additional digits lie. Press the  
indicated key (Öor Õ) to view the obscured window.  
Press to display left  
window  
Press to display right  
window  
Ö  
Õ  
    
Using base in program and equations  
Equations and program are affected by the base setting and binary, octal and  
hexadecimal numbers can be entered in equation and in program as well as when  
the calculator prompts for a variable. Results will be displayed according to the  
current base.  
   
12  
Statistical Operations  
The statistics menus in the HP 35s provide functions to statistically analyze a set of  
one– or two–variable data(real numbers):  
Mean, sample and population standard deviations.  
Linear regression and linear estimation (  
and ).  
ˆ
ˆ
x
y
Weighted mean (x weighted by y).  
2
2
Summation statistics: n, Σx, Σy, Σx , Σy , and Σxy.  
Entering Statistical Data  
One– and two–variable statistical data are entered (or deleted) in similar fashion  
using the (or  ) key. Data values are accumulated as summation  
statistics in six statistics registers (-27 through -32), whose names are displayed in  
the SUMS menu. (Press  and see      ).  
Always clear the statistics registers before entering a new set of  
Note  
statistical data (press   
()).  
   
Entering One–Variable Data  
1. Press   
()to clear existing statistical data.  
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  
.  
Entering Two–Variable Data  
nd  
If the data is a pair of variables, enter first the dependent variable (the 2 variable  
of the pair) and press , and then enter the independent variable (the first  
variable of the pair) and press .  
1. Press   
() to clear existing statistical data.  
2. Key in the y–value first and press .  
3. Key in the corresponding x–value and press .  
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  
.  
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 , press  . 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   to retrieve  
them, then press   to delete them. (The incorrect y–value was still in the Y–  
register, and its x–value was saved in the LAST X register.) After deleting the  
incorrect statistical data, calculator will display the value of Y-register in line 1 and  
value of n in line 2.  
Example:  
Key in the x, y–values on the left, then make the corrections shown on the right:  
Initial x, y  
20, 4  
Corrected x, y  
20, 5  
400, 6  
40, 6  
Keys:  
()  
Display:  
Description:  
Clears existing statistical  
data.  
Enters the first new data pair.  
  
  
  
  
Display shows n, the number  
of data pairs you entered.  
Brings back last x–value. Last  
y is still in Y–register.  
  
  
  
  
  
Deletes the last data pair.  
  
  
  
  
Reenters the last data pair.  
Deletes the first data pair.  
  
  
  
  
  
Reenters the first data pair.  
There is still a total of two  
data pairs in the statistics  
registers.  
  
  
  
Statistical Calculations  
Once you have entered your data, you can use the functions in the statistics menus.  
Statistics Menus  
Menu  
L.R.  
Key  
Description  
The linear–regression menu: linear estimation  
   
ˆ ˆ and curve–fitting   . See ''Linear  
   
Regression'' later in this chapter.  
,
y
   
   
The mean menu:   . See "Mean"  
below.  
x
The standard–deviation menu:   σσ.  
See "Sample Standard Deviation" and  
"Population Standard Deviation" later in this  
chapter.  
s,σ  
The summation menu:       
SUMS  
   
. 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 Õ( ) for the mean of the y–values.  
Press ÕÕ( ) for the weighted mean of the x–values using  
the 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:  
Clears the statistics registers.  
Enters the first time.  
  ()  
  
  
Enters the remaining data; six  
data points accumulated.  
  
  
  
  
Calculates the mean time to  
complete the process.  
 ( )  
  
   
  
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)  
Number of Parts (y)  
$4.25  
250  
$4.60  
800  
$4.70  
900  
$4.10  
1000  
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.  
Enters data; displays n.  
  
  
  
  
  
  
  
  
Four data pairs  
accumulated.  
  
  
  
  
Calculates the mean price  
weighted for the quantity  
purchased.  
ÕÕ( )  
  
  
Sample Standard Deviation  
Sample standard deviation is a measure of how dispersed the data values are  
about the mean sample 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.  
Press Õ() for the standard deviation of y–values.  
The (σ) and (σ) items 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:  
x
15.5  
12.5  
9.25  
12.0  
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.  
Enters the remaining data; six  
  ()  
  
  
  
data points entered.  
  
  
  
 Calculates the standard deviation  
  ()  
  
time.  
  
 
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  ÕÕ(σ) for the population standard deviation of the x–  
values.  
Press  ÕÕÕ(σ) for the population standard deviation of  
the y–values.  
Example: Population Standard Deviation.  
Grandma Hinkle 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. Four data points  
accumulated.  
()  
  
  
  
  
Calculates the population  
 ÕÕ(σ)  
  
standard deviation.  
  
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.  
To avoid a   message, enter your data before  
executing any of the functions in the L.R. menu.  
Note  
   
L.R. (Linear Regression) Menu  
Description  
Menu Key  
Estimates (predicts) x for a given hypothetical value of y,  
based on the line calculated to fit the data.  
ˆ
Estimates (predicts) y for a given hypothetical value of x,  
based on the line calculated to fit the data.  
ˆ
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  ( ˆ ) (or Õ(  
).  
ˆ
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)  
Y, Grain Yield  
0.00  
4.63  
20.00  
5.78  
40.00  
6.61  
60.00  
7.21  
80.00  
7.78  
(metric tons per hectare)  
Keys:  
Display:  
Description:  
Clears all previous statistical  
()  
data.  
Enters data; displays n.  
  
  
  
  
  
  
  
  
  
  
Five data pairs entered.  
  
  
  
 ÕÕ()  
Displays linear–regression  
menu.  
ˆ
    
ˆ
  
Correlation coefficient; data  
closely approximate a straight  
line.  
Õ
Õ
ˆ
Slope of the line.  
ˆ     
  
y–intercept.  
ˆ
ˆ
  
  
    
y
8.50  
7.50  
X
(70, y)  
r = 0.9880  
6.50  
5.50  
m = 0.0387  
b = 4.8560  
20  
x
4.50  
0
40  
60  
80  
What if 70 kg of nitrogen fertilizer were applied to the rice field? Predict the grain  
yield based on the above statistics.  
Keys:  
Display:  
  
_  
Description:  
Enters hypothetical x–value.  
  
The predicted yield in tons per  
hectare.  
ˆ
    
ˆ
 Õ(  
)
ˆ
  
Limitations on Precision of Data  
Since the calculator uses finite precision, it follows that there are limitations to  
calculations due to rounding. Here are two examples:  
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  
ˆ
, and y  
must then be added back to the calculation of  
and  
and b must also  
ˆ
x
x
be adjusted. For example, if your x–values were 7776999, 7777000, and  
7777001, you should enter the data as –1, 0, and 1; then add 7777000 back to  
ˆ, be sure to supply an  
y
and ˆ . For b, add back 7777000 × m. To calculate  
x
x
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  
Executing   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 registers:  
() to recall the number of accumulated data sets.  
Press Õ() to recall the sum of the x–values.  
Press ÕÕ() to recall the sum of the y–values.  
Press ÕÕÕ( ), ÕÕÕÕ( ), and  
ÕÕÕÕÕ() to recall the sums of the squares and the sum 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 Øto view 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:  
Clears the statistics registers.  
Stores the first data pair (1,2).  
  
  
Stores the second data pair (3,4).  
  
  
  
Displays VAR catalog and views n  
  
  
register.  
()  
×
views Σxy register.  
  
   
   
2
Views Σy register.  
×
×
  
   
2
Views Σx register.  
  
  
Views Σy register.  
Views Σx register.  
×
×
  
  
  
  
Views n register.  
×
  
  
Leaves VAR catalog.  
  
Access to the Statistics Registers  
The statistics register assignments in the HP 35s are shown in the following table.  
Summation registers should be referred to by names and not by numbers in  
expression, equations and programs.  
Statistics Registers  
Register  
Number  
-27  
Description  
Number of accumulated data pairs.  
Sum of accumulated x–values.  
Sum of accumulated y–values.  
Sum of squares of accumulated x–values.  
Sum of squares of accumulated y–values.  
Sum of products of accumulated x– and y–  
values.  
n
Σx  
Σy  
-28  
-29  
-30  
2
Σx  
2
Σy  
-31  
Σxy  
-32  
 
You can load a statistics register with a summation by storing the number (-27  
through -32) of the register you want in I or J and then storing the summation (value  
7 or A). Similarly, you can press  7 or A (or  
7 or A ) to view (or recall)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 14 for more  
information.  
Part 2  
Programming  
   
13  
Simple Programming  
Part 1 of this manual introduced you to functions and operations that you can use  
manually, that is, by pressing a key for each individual operation. And you saw  
how you can use equations to repeat calculations without doing all of the keystrokes  
each time.  
In part 2, you'll learn how you can use programs for repetitive calculations —  
calculations that may involve more input or output control or more intricate logic. A  
program lets you repeat operations and calculations in the precise manner you  
want.  
In this chapter you will learn how to program a series of operations. In the next  
chapter, "Programming Techniques," you will learn about subroutines and  
conditional instructions.  
Example: A Simple Program.  
To find the area of a circle with a radius of 5, you would use the  
2
formula A = π r and press  
RPN mode: 5     
ALG mode: 5      
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:  
 
RPN mode  
   
ALG mode  
 π  
 π  
   
This very simple program assumes that the value for the radius is in the X– register  
(the display) when the program starts to run. It computes the area and leaves it in  
the X–register.  
In RPN mode, to enter this program into program memory, do the following:  
Keys:  
Display:  
Description:  
(In RPN mode)  
Clears memory.  
()Ö()  
  
Activates Program–entry mode  
(PRGM annunciator on).  
Resets program pointer to PRGM  
   
   
TOP.  
(Radius)  
2
  
   
   
 π  
2
Area = πx  
   
Exits Program–entry mode.  
   
Try running this program to find the area of a circle with a radius of 5:  
Keys:  
Display:  
Description:  
(In RPN mode)  
This sets the program to its  
    
beginning.  
The answer!  
5   
  
In ALG mode, to enter this program into program memory, do the following:  
Keys:  
(In ALG mode)  
Display:  
Description:  
Clears memory.  
()Ö()  
  
Activates Program–entry mode  
(PRGM annunciator on).  
   
Resets program pointer to PRGM  
TOP.  
    
2
 π  
Area = πx  
  
   
Exits Program–entry mode.  
   
Try running this program to find the area of a circle with a radius of 5:  
Keys:  
Display:  
Description:  
(In ALG mode)  
    
This sets the program to its  
beginning.  
Stores 5 into X  
X  
  
  
 The answer!  
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.  
Selecting a Mode  
Programs created and saved in RPN mode should be edited and executed in RPN  
mode, and programs or steps created and saved in ALG mode should be edited  
and executed in ALG mode. If not, the result may be incorrect.  
   
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 return to mark its  
end (such as  ).  
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  
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 ), but a label can use the same  
letter that a variable uses.  
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.  
Programs can not have more than 999 lines.  
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.  
Using RPN, ALG 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 ALG operations (as explained in appendix C).  
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  
Strengths of Equations and  
ALG Operations  
Use less memory.  
Execute faster.  
Easier to write and read.  
Can automatically prompt.  
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 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.  
 
For output, you can display a variable with the VIEW instruction, you can display a  
message derived from an equation, you can display process in line 1, you can  
display the program result in line 2, or you can leave unmarked values on the stack.  
These are covered later in this chapter under "Entering and Displaying Data."  
Entering a Program  
Pressing   toggles the calculator into and out of Program–entry mode —  
turns the PRGM annunciator on and off. Keystrokes in Program–entry mode are  
stored as program lines in memory. Each instruction (command) or expression  
occupies one program line. In ALG mode, you can enter an expression directly in a  
program  
To enter a program into memory:  
1. Press   to activate Program–entry mode.  
2. Press   to display  . This sets the program 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 want all  
programs deleted, press Ö() after the message    .  
3. Give the program a label — a single letter, A through Z. Press  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 can  
clear the existing program instead — press (), use ×or  
Øto find the label, and press   
and .  
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.  
 
5. End the program with a return instruction, which sets the program pointer back  
to   after the program runs. Press .  
6. Press (or  ) to cancel program entry.  
Numbers in program lines are stored 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 turns on.  
2. Enter the equation as you would in the equation list. See chapter 6 for details.  
Use 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 Öand Õto scroll the display.  
Clear functions and backspace key  
Note these special conditions during program entry:  
always cancels program entry. It never clears a number to zero.  
In program line view status, deletes the current program line and Ö/  
Õbegins the edit status. In program line edit status, deletes a  
character before the cursor.  
To program a function to clear the X–register, use   
().  
When you insert or erase a line in a program, GTO and XEQ statements are  
automatically updated if needed.  
For example:  
 
    
   
   
  
   
Now, erase line A002, and line A004 changes to A003 GTO A002”  
Function Names in Programs  
The name of a 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.  
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 to delete the current program line, then reenter the line  
correctly.  
Keys:  
Display:  
Description:  
(In RPN mode)  
  
Activates Program–entry  
mode (PRGM on).  
Clears all of program  
memory.  
() Ö()  
A  
   
Labels this program routine  
A (for "area").  
Enters the three program  
    
   
  
  
lines.  
 π  
   
Ends the program.  
   
  
Displays label A and the  
length of the program in  
   
 (2)  
  
bytes.  
Checksum and length of  
  
  
  
program.  
 
Cancels program entry  
  
(PRGM annunciator off).  
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 operations like the previous program.  
Keys:  
Display:  
Description:  
(In RPN mode)  
Activates Program–entry  
mode; sets pointer to top of  
  
   
memory.  
Labels this program routine  
E (for "equation").  
E  
    
    
Stores radius in variable R  
R  
Selects Equation–entry  
mode; enters the equation;  
returns to Program–entry  
mode.  
   
 R  
   
  
 π  
  
  
Ends the program.  
  
   
Displays label E and the  
length of the program in  
 (2)    
  
bytes.  
Checksum and length of  
program.  
Cancels program entry.  
  
  
  
  
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.  
Executing a Program (XEQ)  
Press label to execute the program labeled with that letter:  
To execute a program from it’s beginning press label . For example,  
press A. The display will show ” ” and execution will  
start at the top of Label A.  
You can also execute a program starting at another position by pressing   
label Line number, for example   .  
If there is only one program in memory, you can also execute it after moving pointer  
to the top of the program line and pressing (run / stop) key. The PRGM  
annunciator displays and the annunciator turns on while the 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 2π. Remember to enter the radius before executing A or E.  
Keys:  
Display:  
Description:  
(In RPN mode)  
A  
Enters the radius, then starts  
program A. The resulting area is  
displayed.  
Calculates area of the second  
circle using program E.  
  
  
  
E  
Calculates area of the third circle.  
    
A  
  
   
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. Set the program pointer to the start of the program (that is, at its LBL  
instruction). The instruction moves the program pointer without starting  
execution.  
3. Press and hold Ø. This displays the current program line. When you release  
Ø, 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 ×. No execution occurs.  
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 Øor ×simply changes the program  
pointer, without executing lines. Holding down a cursor 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:  
(In RPN mode)  
  
Moves program counter to label A.  
  
    
  
Ø(hold) (release)  
Squares input.  
   
Ø(hold) (release)  
  
 
 π  
  
Value of π.  
Ø(hold) (release)  
Ø(hold) (release)  
Ø(hold) (release)  
   
25π.  
  
   
  
End of program. Result is correct.  
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, 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 values in the X and Y registers are visible. (You can use  
PSE for a 1-second look at the X and Y registers.)  
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  
displays a prompt for the given variable. This display includes the existing value for  
the variable, such as  
  
  
where  
"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:  
RPN mode  
ALG mode  
    
    
    
 π  
   
    
   
 π  
   
   
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 variable instruction to bring  
that value back into the stack.  
Since the INPUT instruction also leaves the value you just entered in the  
Xñregister, you don't have to recall the variable at a later time ó 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. Thus the  
program should not assume that the X–, Y–, and Zñregisters' contents will be  
the same before and after the INPUT instruction. If you collect all the data in the  
beginning and then recall them when needed for calculation, then this prevents  
the stack's contents from being altered just before a calculation.  
To respond to a prompt:  
When 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 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   
   in RPN mode, or press     
in ALG mode (Before you press , the expression will be displayed in  
line 2. After you press , the result of expression will replace the  
expression to display in line 2 and be saved in X-register).  
To cancel the INPUT prompt, press . The current value for the variable  
remains in the X–register. If you press to resume the program, the  
canceled INPUT prompt is repeated. If you press during digit entry, it  
clears the number to zero. Press again to cancel the INPUT prompt.  
Using VIEW for Displaying Data  
The programmed VIEW instruction (  variable ) stops a running program  
and displays and identifies the contents of the given variable, such as  
  
  
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 14 characters, such as binary, complex, vector  
numbers, pressing Öand Õdisplays the rest.  
Pressing (or ) erases the VIEW display and shows the X–register.  
Pressing   
clears the contents of the displayed variable.  
Press 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 T015 and T016 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 16 and 17 also ensure that the VIEWed  
variable is in the X–register as well.  
 
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, press to start  
the equation. Press number and math keys to get numbers and symbols. Press   
before each letter. Press to end the equation.  
If flag 10 is set, equations are displayed instead of being evaluated. This means  
you can display any message you enter as an equation. (Flags are discussed in  
detail in chapter 14.)  
When the message is displayed, the program stops — press to resume  
execution. If the displayed message is longer than 14 characters, the ꢆ  
annunciator turns on when the message is displayed. You can then use Õ  
and Öto scroll the display.  
If you don't want the program to stop, see "Displaying Information without  
Stopping" below.  
Example: INPUT, VIEW, and Messages in a Program.  
Write an equation to find the surface area and volume of a cylinder given its radius  
and height. Label the program C (for cylinder), and use the variables S (surface  
area), V (volume), R (radius), and H (height). Use these formulas:  
2
V = πR H  
2
S = 2π R + 2π RH = 2π R ( R + H )  
Keys:  
Display:  
Description:  
(In RPN mode)  
   
Program, entry; clears the  
program memory.  
()  
Ö  
   
   
Labels program.  
    
ÇR  
ÇH  
    
    
Instructions to prompt for  
radius and height.  
 
Keys:  
Display:  
Description:  
(In RPN mode)  
Calculates the volume.  
    
R   
H  
 π  
  
Checksum and length of  
equation.  
   
  
Store the volume in V.  
Calculates the surface area.  
V  
    
   
R  
4 R  
H  
 πꢆ  
  
Checksum and length of  
equation.  
   
  
Stores the surface area in S.  
Sets flag 10 to display  
equations.  
S  
    
   
()   
    
Displays message in  
equations.  
 V  
OL  
  
 A  
RE  
A  
   ꢆ  
Clears flag 10.  
   
()   
 V  
 S  
   
    
Displays volume.  
Displays surface area.  
Ends program.  
    
    
   
   
Displays label C and the  
length of the program in  
   
()  
  
bytes.  
  
  
Checksum and length of  
   
  
program.  
Cancels program entry.  
1
Now find the volume and surface area–of a cylinder with a radius of 2 / cm and  
2
a height of 8 cm.  
Keys:  
Display:  
Description:  
(In RPN mode)  
C  
Starts executing C; prompts for  
R. (It displays whatever value  
happens to be in R.)  
  
value  
1
Enters 2 / as a fraction.  
2
  
  
Prompts for H.  
Message displayed.  
value  
    
  
3
Volume in cm .  
  
  
  
2
Surface area in cm .  
  
Displaying Information without Stopping  
Normally, a program stops when it displays a variable with VIEW or displays an  
equation message. You normally have to press 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.  
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.  
 
Stopping or Interrupting a Program  
Programming a Stop or Pause (STOP, PSE)  
Pressing (run/stop) during program entry inserts a STOP instruction. This  
will display the contents of the X-register and 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.  
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 or . The  
program completes its current instruction before stopping. Press (run/stop) to  
resume the program.  
If you interrupt a program and then press , , or  , you cannot  
resume the program with . Re-execute the program instead (label line  
number).  
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 F.)  
To see the line in the program containing the error–causing instruction, press   
. The program will have stopped at that point. (For instance, it might be a ÷  
instruction, which caused an illegal division by zero.)  
       
Editing a 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.  
To delete a program line:  
1. Select the relevant program or routine and press Øor ×to locate the  
program line that must be changed. Hold the cursor key down to continue  
scrolling.  
2. Delete the line you want to change —press directly (Undo function is  
active). The pointer then moves to the preceding line. (If you are deleting more  
than one consecutive program line, start with the last line in the group.)  
3. Key in the new instruction, if any. This replaces the one you deleted.  
4. Exit program entry ( or   ).  
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 A004 and A005 of a  
program, you would first display line A004, then key in the instruction or  
instructions. Subsequent program lines, starting with the original line A005, are  
moved down and renumbered accordingly.  
To edit operand, expression or equation in a program line:  
1. Locate or display the program line that you want to edit.  
2. Press Õor Öto start editing the program line. These turn on the “_”  
editing cursor, but do not delete anything in the program line  
Õ key actives the cursor to the left of the program line  
Ö key actives the cursor to the end of the program line  
 
3. Moving the cursor”_” and press repeatedly to delete the unwanted  
number or function, then retype the rest of the program line. (After pressing  
, Undo function is active)  
Notice:  
1. When the cursor is active in the program line, Øor ×key will be  
disabled.  
2. When you are editing a program line (cursor active), and the program line is  
empty, using will have no effect. If you want to erase the program line,  
press and the program line will be erased.  
3. You can use Õand Ökey to review long program lines and  
without editing it.  
4. In ALG mode, can not be used as a function, it is used to validate a  
program line.  
5. An equation can be editing in any mode no matter which mode it was  
entered in.  
Program Memory  
Viewing Program Memory  
Pressing   toggles the calculator into and out of program entry (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 you  
can wrap the program pointer from the bottom to the top and reverse. While  
program entry is active, there are four ways to change the program pointer (the  
displayed line):  
×and Øallows you to move from label to label. If no labels  
are defined, It will move to the top or bottom of the program.  
To move more than one line at a time ("scrolling"), continue to hold the Ø  
or ×key.  
   
Press   to move the program pointer to  .  
Press   label nnn to move to a specific line.  
If Program–entry mode is not active (if no program lines are displayed), you can  
also move the program pointer by pressing label line number.  
Canceling Program–entry mode does not change the position of the program  
pointer.  
Memory Usage  
If during program entry you encounter the message  , then 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   
(2) to display the catalog, and press Øor ×to move within the  
list. You can use this catalog to:  
Review the labels in program memory and the memory cost of each labeled  
program or routine.  
Execute a labeled program. (Press or while the label is  
displayed.)  
Display a labeled program. (Press   while the label is displayed.)  
Delete specific programs. (Press   
while the label is displayed.)  
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:  
   
   
  
where 67 is the number of bytes used by the program.  
Clearing One or More Programs  
To clear a specific program from memory  
1. Press   (2)and display (using Øand ×) the  
label of the program.  
2. Press   
.
3. Press to cancel the catalog or to back out.  
To clear all programs from memory:  
1. Press   to display program lines (PRGM annunciator on).  
2. Press   
() to clear program memory.  
3. The message     prompts you for confirmation. Press Ö()  
.  
4. Press   to cancel program entry.  
Clearing all of memory (  
()) also clears all programs.  
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 your calculator are the  
same, then you have correctly entered all the lines of the program. To see your  
checksum:  
1. Press   () for the catalog of program labels.  
2. Display the appropriate label by using the cursor keys, if necessary.  
3. Press and hold   to display checksum and length.  
   
For example, to see the checksum for the current program (the "cylinder" program):  
Keys:  
Display:  
Description:  
(In RPN mode)  
   
Displays label C, which takes  
67 bytes.  
Checksum and length.  
   
()  
  (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 16 and 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 35s are not programmable:  
()  
()  
   
  label line number  
   
   
Ø, ×,Ö, Õ  
   
   
Ø, ×  
:  
()  
Programming with BASE  
You can program instructions to change the base mode using  . 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).  
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 how input is  
interpreted and how output looks during and after program execution, but it does  
not affect the program lines as you enter them.  
Numbers Entered in Program Lines  
Before starting program entry, set the base mode. The current setting for the base  
mode determines the result of program.  
An annunciator tells you which base is the current setting. Compare the program  
lines below in the decimal and non-decimal mode. All decimal and non-decimal  
numbers are left–justified in the calculator's display.  
Decimal mode set:  
Binary mode set:  
:
:
:
:
PRGM  
PRGM BIN  
   
   
Decimal number  
can omit the sign  
“d”  
Binary number  
should add the  
base sign ”b”  
   
   
:
:
:
:
   
Polynomial Expressions and Horner's Method  
Some expressions, such as polynomials, use the same variable several times for their  
solution. For example, the expression  
4
3
2
Ax + Bx + Cx + Dx + E  
uses the variable x four different times. A program to calculate such an expression  
using RPN operations could repeatedly recall a stored copy of x from a variable.  
Example:  
4
3
Write a program using RPN operations for 5x + 2x , then evaluate it for x = 7.  
 
Keys:  
Display:  
Description:  
(In RPN mode)  
    
  
   
    
    
   
 A  
 X  
5
    
   
X  
4
x
   
4
5x  
   
    
   
X  
3
x
   
   
   
   
3
4
2x  
3
5x + 2x  
   
   
   
Displays label A, which  
takes 46 bytes.  
Checksum and length.  
   
  
()  
   
  
  
Cancels program entry.  
  
Now evaluate this polynomial for x = 7.  
Keys:  
(In RPN mode)  
A  
Display:  
Description:  
Prompts for x.  
Result.  
  
value  
  
  
A more general form of this program for any equation  
4
3
2
Ax + Bx + Cx + Dx + E would be:  
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
   
Checksum and length: 9E5E 51  
14  
Programming Techniques  
Chapter 13 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.  
A routine typically starts at a label and ends with an instruction that stops program/  
routing execution such as RTN or STOP.  
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.  
     
If you plan to have only one program in the calculator memory, you can  
separate the routine in various labels. If you plan to have more than one  
program in the calculator memory, it is better to have routines part of the  
main program label, starting at a specific line number.  
A subroutine can itself call other subroutines.  
The flow diagrams in this chapter use this notation:  
   1  
Program execution branches from this line to the  
line number marked 1 ("from 1").  
   1  
Program execution branches from a line number  
marked 1 ("to 1") to this line.  
The example below show you to call a subroutine to change the sign of the number  
you input. Subroutine E that is called from routine D by line     
changes sign of the number. Subroutine E ends with a RTN instruction that sends  
program execution back to routine D (to store and display the result) at line D004.  
See the flow diagrams below.  
    
Starts here.  
    
    
    
1  
2  
Calls subroutine E.  
Returns here.  
    
   
Starts subroutine.  
    
   
1  
Change sign of the number  
Returns to routine D.  
2  
   
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 20 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 20 levels deep causes an   
 error.  
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.  
In RPN mode,  
    
    
    
    
    
    
    
    
    
Starts subroutine here.  
Enters A.  
Enters B.  
Enters C.  
Enters D.  
Recalls the data.  
2
A .  
   
2
2
    
    
    
A + B .  
1  
3  
5  
2
2
2
2
A + B + C  
2 ꢇ  
4 ꢇ  
6 ꢇ  
2
2
2
A + B + C + D  
A2 + B2 + C2 + D2  
  
   
Returns to main routine.  
    
   
Nested subroutine  
135  
   
2
   
Adds x .  
   
Returns to subroutine S.  
246   
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 specific  
program line (label and line number).  
 
A Programmed GTO Instruction  
The GTO label instruction (press label line number) transfers the execution of  
a running program to the specified program line. 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    
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:  
    
Can start here.  
.
.
.
    
Branches to Z001.  
Can start here.  
1  
    
.
.
.
    
Branches to Z001.  
Can start here.  
1  
    
.
.
.
    
Branches to Z001.  
Branch to here.  
1  
    
1  
.
.
.
Using GTO from the Keyboard  
You can use to move the program pointer to a specified label line number  
without starting program execution.  
   
To  :  .  
To a specific line number: label line number (line number < 1000).  
For example, A. For example, press A  
. The display will show ” ” .  
If you want to go to the first line of a label, for example. A001:  
(press and hold), the display will show ” .  
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 A005 is  (that is, is x 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 A007. This rule is commonly known as "Do if true."  
    
.
.
.
Do next if true.  
   
1     
   
2  
Skip next if false.  
2  
    
.
.
.
1     
.
.
.
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.  
Tests of Comparison (x?y, x?0)  
There are 12 comparisons available for programming. Pressing   or   
displays a menu for one of the two categories of tests:  
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. You can use x?y and  
x?0 to compare two numbers, if one of these isn't real number, it will return an error  
message  .  
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 xy?  
< for x<y?  
> for x>y?  
for xy?  
for x=y?  
for x0?  
for x0?  
< for x<0?  
> for x>0?  
for x0?  
for x=0?  
If you execute a conditional test from the keyboard, the calculator will display   
or .  
For example, if x =2 and y =7, test x<y .  
 
Keys:  
Display:  
  
In RPN mode  
In ALG mode  
 ÕÕ(<)  
  
ÕÕ(<)  
Example:  
The "Normal and Inverse–Normal Distributions" program in chapter 16 uses the  
x<y? conditional in routine T:  
Program Lines:  
(In RPN mode)  
Description  
.
.
.
   
Calculates the correction for X  
.
guess  
    
   
Adds the correction to yield a new X  
.
guess  
   
 <  
    
Tests to see if the correction is significant.  
Goes back to start of loop if correction is significant.  
Continues if correction is not significant.  
    
    
Displays the calculated value of X.  
.
.
.
Line T009 calculates the correction for X . Line T013 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 T014; if the value is equal to or greater than  
0.0001, the program skips to line T015.  
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 35s 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   
() Ö() .  
Flags 0, 1, 2, 3, and 4 have no pre-assigned meanings. That is, their states  
will mean whatever you define them 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 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.  
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  
tested. Besides, when using non-decimal bases in programs, flag 6 also gets  
set for   in programs.  
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.  
 
Flag  
Status  
Fraction–Control Flags  
8
7
9
Fraction display  
off; display real  
numbers in the  
current display  
format.  
Fraction  
Reduce fractions to  
smallest form.  
Clear  
denominators not  
greater than the /c  
value.  
(Default)  
Fraction display  
on; display real  
numbers as  
Fraction  
No reduction of  
fractions. (Used  
only if flag 8 is  
set.)  
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 or . Pressing any other key executes that  
key's function.  
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 when 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 is 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, 3 and 4 have annunciators in the display that turn on when the  
corresponding flag is set. The presence or absence of 0, 1, 2, 3 or 4 lets you know  
at any time whether any of these five flags is set or not. However, there is no such  
indication for the status of flags 5 through 11. The states of these flags can be  
determined by executing the FS? instruction from the keyboard. (See "Using Flags"  
below.)  
Using Flags  
Pressing   displays the FLAGS menu:     
After selecting the function you want, you will be prompted for the flag number (0–  
11). For example, press  () to set flag 0; press   
() to set flag 10; press  ()  to  
set flag 11.  
FLAGS Menu  
Menu Key  
 n  
Description  
Set flag. Sets flag n.  
 n  
Clear flag. Clears flag n.  
 n  
Is flag set? Tests the status of flag n.  
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 "" or "".  
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.  
Program Lines:  
(In RPN mode)  
Description:  
    
    
Clears flag 0, the indicator for In X.  
Clears flag 1, the indicator for In Y.  
Prompts for and stores X  
    
    
    
If flag 0 is set…  
   
… takes the natural log of the X-input  
Stores that value in X after flag test  
Prompts for and stores Y  
If flag 1 is set…  
… takes the natural log of the Y-input  
    
    
   
   
    
    
    
   
Stores that value in Y after flag test  
Displays value  
Displays value  
Checksum and length: 16B3 42  
If you write lines S002 CF0 and S003 CF1(as shown above), the flags 0 and 1 are  
cleared so lines S006 and S010 do not take the natural logarithms of the X- and Y-  
inputs.  
If you replace lines S002 and S003 by SF 0 and CF 1, then flag 0 is set so line  
S006 takes the natural log of the X-input.  
If you replace lines S002 and S003 by CF 0 and SF1, then flag 1 is set so line  
S010 takes the natural log of the Y-input.  
If you replace lines S002 and S003 by SF0 and SF1, then flags 0 and 1 are set so  
lines S006 and S010 take the natural logarithms of the X- and Y-inputs.  
Use above program to see how to use flags  
Keys:  
Display:  
Description:  
(In RPN mode)  
Executes label S; prompts for X  
value  
Stores 1 in X; prompts Y value  
  
S  
  
value  
  
value  
  
  
Stores 1 in X ;displays X value  
after flag test  
  
Displays Y value after flag test  
  
  
You can try other three cases. Remember to press  () and  
 () to clear flag 1 and 0 after you try them.  
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 as MESSAGE and are entered as equations:  
1. Set Equation–entry mode by pressing (the EQN annunciator turns on).  
2. Press letter for each alpha character in the message; press   
for each space character.  
3. Press to insert the message in the current program line and end  
Equation–entry mode.  
Program Lines:  
(In RPN mode)  
Description:  
  
  
  
  
  
  
  
  
  
   
Begins the fraction program.  
Clears three fraction flags.  
   
   
   
   
  
Displays messages.  
Selects decimal base.  
Prompts for a number.  
Prompts for denominator (2 – 4095).  
Displays message, then shows the decimal  
   
   
   
number.  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   
  
Sets /c value and sets flag 7.  
   
   
  
Displays message, then shows the fraction.  
  
   
Sets flag 8.  
Displays message, then shows the fraction.  
   
  
  
   
Sets flag 9.  
Displays message, then shows the fraction.  
   
  
  
   
Goes to beginning of program.  
Checksum and length: BE54 123  
Use the above program to see the different forms of fraction display:  
Keys:  
Display:  
Description:  
(In RPN mode)  
Executes label F; prompts for a  
fractional number (V).  
Stores 2.53 in V; prompts for  
denominator (D).  
Stores 16 as the /c value. Displays  
message, then the decimal number.  
  
F  
  
  
value  
  
value  
  
  
  
   
Message indicates the fraction  
format (denominator is no greater  
than 16), then shows the fraction.  
indicates that the numerator is "a  
  ꢅ  
   
little below" 8.  
Message indicates the fraction  
format (denominator is factor of 16),  
then shows the fraction.  
   
  ꢄ  
   
Message indicates the fraction  
format (denominator is 16), then  
shows the fraction.  
Stops the program and clears flag  
   
  ꢄ  
  
  
  
10  
()   
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.  
    
    
    
    
    
 
This routine is an example of an infinite loop. It can be used to collect the initial  
data. After entering the three values, it is up to you to manually interrupt this loop by  
pressing label line number to execute other routines.  
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.  
Program lines:  
(In RPN mode)  
    
Description:  
    
    
    
It is easier to recall A than to remember where it is in the  
stack.  
    
    
    
   
Calculates A B.  
Replaces old A with new result.  
Recalls constant for comparison.  
Is B < new A?  
   Yes: loops to repeat subtraction.  
    
No: displays new A.  
   
Checksum and length: 2737 33  
 
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 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   variable  
For a count–up loop, use   variable  
These functions accomplish the same thing as a FOR–NEXT loop in BASIC:  
 variable = initial–value  final–value  increment  
.
.
.
 variable  
A DSE instruction is like a FOR–NEXT loop with a negative increment.  
After pressing a shifted key for ISG or DSE (   or   ), you 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. An unspecified value for fff is assumed to be 000.  
 
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 ccccccc with fff, and makes program execution skip the next  
program line if this ccccccc > fff.  
    
1  
.
.
.
If current value >  
final value,  
continue loop.  
If current value ≤  
final value, exit  
loop.  
    
2  
    
    
1  
2  
.
.
.
    
1ꢇ  
.
.
.
If current value ≤  
final value,  
continue loop.  
If current value >  
final value, exit  
loop.  
    
2  
    
    
1  
2  
.
.
.
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.  
If the loop-control number is a complex number or vector, it will use the real part or  
first part to control the loop.  
The following program uses ISG to loop 10 times in RPN mode. The loop counter  
(1.010) is stored in the variable Z. Leading and trailing zeros can be left off.  
    
   
    
    
    
   
Press L, then press  Zto see that the loop–control  
number is now 11.0100.  
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 four different keys: 0, 7, 1 , and A.  
These keys are active for many functions that take A through Z as variables or  
labels.  
I and J are variables whose contents can refer to another variable. It holds a  
number just like any other variable (A through Z).  
(I) and (J) are programming functions that directs, "Use the number in I or J to  
determine which variable or label to address."  
This is an indirect address. (A through Z are direct addresses.)  
Both 0 and 7 are used together to create an indirect address and this applies to  
both 1 and A as well.  
By itself, (I) or (J) is either undefined (no number in (I) or (J)) or uncontrolled (using  
whatever number happens to be left over in I or J).  
The Variables "I" and "J"  
You can store, recall, and manipulate the contents of I or J just as you can the  
contents of other variables. You can even solve for I,J and integrate using I or J . The  
functions listed below can use variable "i"(the variable J is the same).  
   
STO I  
RCL I  
STO +,–, × ,÷ I  
RCL +,–, × ,÷ I  
INPUT I  
VIEW I  
FN d I  
SOLVE I  
DSE I  
ISG I  
x < > I  
The Indirect Address, (I) and (J)  
Many functions that use A through Z (as variables or labels) can use (I) or (J) to refer  
to A through Z (variables or labels) or statistics registers indirectly. The function (I) or  
(J) uses the value in variable I to J to determine which variable, label, or register to  
address. The following table shows how.  
 
If I/J contains:  
Then (I)/(J) will address:  
-1  
variable A or label A  
.
.
.
.
.
.
-26  
-27  
variable Z or label Z  
n register  
-28  
-29  
-30  
Σx register  
Σy register  
2
Σx register  
2
-31  
-32  
Σy register  
Σxy register  
0
Unnamed Indirect variables start  
.
.
.
.
.
.
800  
The Max Address is 800  
error:    
I<-32 or I>800 or variables  
undefined  
J<-32 or I>800 or variables  
error:    
undefined  
The INPUT(I) ,INPUT(J) and VIEW(I) ,VIEW(J)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) or (J) as an address. For FN=, (I) or (J) refers  
to a label; for all other functions (I) or (J) refers to a variable or register.  
STO(I)/(J)  
RCL(I)/(J)  
STO +, –,× ,÷, (I)/(J)  
RCL +, –,× ,÷, (I)/(J)  
X<>(I)/(J)  
INPUT(I)/(J)  
VIEW(I)/(J)  
DSE(I)/(J)  
ISG(I)/(J)  
SOLVE(I)/(J)  
FN d(I)/(J)  
FN=(I)/(J)  
You can not solve or integrate for unnamed variables or statistic registers.  
Program Control with (I)/(J)  
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 STO (I) or (J) can store  
value to a different variable at different times. For example, STO(-1) indicates storing  
the value in Variable A. This maintains flexibility by leaving open (until the program  
runs) exactly which variable or program label will be needed.  
Indirect addressing is very useful for counting and controlling loops. The variable I  
or J serves as an index, holding the address of the variable that contains the loop–  
control number for the functions DSE and ISG.  
Equations with (I)/(J)  
You can use (I) or (J) in an equation to specify a variable indirectly. Notice that  
or  means the variable specified by the number in variable I or J (an  
indirect reference), but that I or J and or  (where the user parenthesis are  
used instead of the (I) or (J) key) means variable I or J.  
Unnamed indirect variables  
Placing a positive number into variable I or J allows you to access up to 801 indirect  
variables. The following example indicates how to use them.  
     
Program Lines:  
(In RPN mode)  
    
Description:  
   
    
   
    
Defined the storage address range “0-100” and saved  
“12345” into address 100.  
   
    
   
    
Saves “67890” into address 150. The defined indirect  
storage range is now “0-150.  
   
    
   
    
Stores 0 into indirect register 100. The defined range is  
still “0-150.  
   
    
   
Display “INVALID (I), because address “170” is  
undefined  
   
Note:  
1. If you want to recall the value from an undefined storage address, the error  
message “ ”will be shown. (See A014)  
2. The calculator allocates memory for variable 0 to the last non-zero variable. It is  
important to store 0 in variables after using them in order to release the  
memory. Each allocated indirect register uses 37 bytes of program memory.  
3. There is a maximum of 800 variables.  
15  
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 if  
you're re–solving the same program.)  
3. Solve for the unknown variable: press  variable.  
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 or and the message  will  
appear in line 2. The current best estimate of the root is in the unknown variable;  
use   to view it without disturbing the stack. To resume the calculation,  
press .  
To write a program for SOLVE:  
The program can use equations and ALG or RPN operations — in whatever  
combination is most convenient.  
   
1. Begin the program with a label. This label identifies the function that you want  
SOLVE to evaluate (label).  
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 or ALG 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.  
Example: Program Using ALG.  
Write a program using ALG operations that solves for any unknown in the equation  
for the "Ideal Gas Law." The equation is:  
P x V= N x R x T  
where  
2
P = Pressure (atmospheres or N/m ).  
V = Volume (liters).  
N = Number of moles of gas.  
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:  
(In ALG mode)  
   
Sets Program mode.  
   
   
Type in the program:  
Program Lines:  
(In ALG mode)  
Description:  
    
Identifies the programmed function.  
Stores P for pressure  
Stores V for volume  
Stores N for number of moles of gas  
Stores R for gas constant  
Stores T for temp.  
    
    
    
    
    
 ×××  
Press  
Pressure × volume = Moles × gas constant × temp.  
   
Ends the program.  
Checksum and length: F425 33  
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:  
(In ALG mode)  
Selects "G" — the program.  
G  
SOLVE evaluates to find the value  
of the unknown variable.  
Selects P; prompts for V.  
  
P  
value  
  
Stores 2 in V; prompts for N.  
  
value  
Stores .005 in N; prompts for R.  
  
  
  
value  
  
Stores .0821 in R; prompts for T.  
value  
  
  
Calculates T.  
  
  
  
  
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:  
(In RPN mode)  
Selects Program–entry mode.  
   
Moves program pointer to top of  
   
   
the list of programs.  
Labels the program.  
Enables equation prompting.  
    
    
  H  
   
(1)   
Evaluates the equation, clearing  
flag 11. (Checksum and length:  
EDC8 9).  
P  
V   
N  
R  
T  
   
   
   
Ends the program.  
Cancels Program–entry mode.  
  
Checksum and length of program: DF52 21  
Now calculate the change in pressure of the carbon dioxide if its temperature drops  
by 10 °C from the previous example.  
Keys:  
Display:  
  
Description:  
(In RPN mode)  
Stores previous pressure.  
Selects program “H.”  
L  
 H  
P  
  
  
Selects variable P; prompts for V.  
Retains 2 in V; prompts for N.  
Retains .005 in N; prompts for R.  
Retains .0821 in R; prompts for T.  
Calculates new T.  
  
  
  
  
  
  
  
  
   
  
  
Stores 287.1 in T; solves for new P.  
  
  
Calculates pressure change of the  
gas when temperature drops from  
297.1 K to 287.1 K (negative result  
indicates drop in pressure).  
  
L  
Using SOLVE in a 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  
 variable  
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 want to do further calculations with this number 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 14). 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:  
(In RPN mode)  
Description:  
    
Setup for X.  
   
Index for X.  
    
Checksum and length: 62A0 11  
    
Branches to main routine.  
Setup for Y.  
   
Index for Y.  
    
Checksum and length: 221E 11  
    
Branches to main routine.  
Main routine.  
    
Stores index in I  
    
Defines program to solve.  
Solves for appropriate variable.  
Displays solution.  
   
   
   
Ends program.  
Checksum and length: D45B 18  
    
Calculates f (x,y). Include INPUT or equation  
prompting as required.  
   
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    
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    
variable.  
Notice that FN= is required if you're integrating a programmed function, but not if  
you're integrating an equation from the equation list.  
You can halt a running integration calculation by pressing or and the  
message  will appear line 2. However, the calculation cannot be  
resumed. No information about the integration is available until the calculation  
finishes normally.  
Pressing while an integration calculation is running will cancel the   
operation. In this case, you should start  again from the beginning.  
To write a program for FN:  
The program can use equations, ALG or 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 (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 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 or ALG 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  
x
t
0
Si(t) =  
(
)dx  
This function can be evaluated by integrating a program that defines the integrand:  
    
Defines the function.  
   
The function as an expression. (Checksum and length:  
0EE0 8).  
Ends the subroutine  
   
Checksum and length of program: D57E 17  
Enter this program and integrate the sine integral function with respect to x from 0 to  
2 (t = 2).  
Keys:  
Display:  
Description:  
(In RPN mode)  
Selects Radians mode.  
9(2)  
Selects label S as the integrand.  
 S  
Enters lower and upper limits of  
integration.  
  
_  
Integrates function from 0 to 2;  
 X  
  
displays result.  
  
  
  
Restores Degrees mode.  
9(1)  
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  
  variable  
The programmed FN instruction does not produce a labeled display (= value)  
since this might not 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 () instruction to display  
the result in the X–register after the FN instruction.  
If the PSE instruction immediately follows an equation that is displayed (Flag 10 set)  
during each iteration of integrating or solving, the equation will be displayed for 1  
second and execution will continue until the end of each iteration. During the  
display of the equation, no scrolling or keyboard input is allowed.  
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  
2
DM  
S
)
e(  
/2dD.  
1
S 2π  
D
M
((DM)÷S)2  
The  
÷2 function is calculated by the routine labeled F. Other routines  
e
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:  
 
    
    
    
    
     
Recalls lower limit of integration.  
Recalls upper limit of integration. (X = D.)  
Specifies the function.  
Integrates the normal function using the dummy variable D.  
Restrictions on Solving and Integrating  
The 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  error. Also, SOLVE and FN cannot call a routine that contains  
an label instruction; if attempted, a   or   error  
will be returned. SOLVE cannot call a routine that contains an FN instruction  
(produces a  error), just as FN cannot call a routine that contains a  
SOLVE instruction (produces an  error).  
The SOLVE variable and FN d variable instructions in a program use one of the 20  
pending subroutine returns in the calculator. (Refer to "Nested Subroutines" in  
chapter 14.)  
 
16  
Statistics Programs  
Curve Fitting  
This program can be used to fit one of four models of equations to your data. These  
models are the straight line, the logarithmic curve, the exponential curve and the  
power curve. The program accepts two or more (x, y) data pairs and then calculates  
the correlation coefficient, r, and the two regression coefficients, m and b. The  
ˆ
program includes a routine to calculate the estimates x and ˆ. (For definitions of  
y
these values, see "Linear Regression" in chapter 12.)  
Samples of the curves and the relevant equations are shown below. The internal  
regression functions of the HP 35s are used to compute the regression coefficients.  
   
Exponential Curve Fit  
Straight Line Fit  
E
S
y
y
Mx  
y = Be  
y = B + Mx  
x
x
Logarithmic Curve Fit  
Power Curve Fit  
L
P
y
y
M
y = B + MIn x  
y = Bx  
x
x
To fit logarithmic curves, values of x must be positive. To fit exponential curves,  
values of y must be positive. To fit power curves, both x and y must be positive. A  
 error will occur if a negative number is entered for these 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 12.  
Program Listing:  
Program Lines:  
(In RPN mode)  
Description  
    
This routine sets, the status for the straight–line model.  
    
Clears flag 0, the indicator for ln X.  
    
Clears flag 1, the indicator for In Y.  
    
Branches to common entry point Z.  
Checksum and length: 8E85 12  
    
    
This routine sets the status for the logarithmic model.  
Sets flag 0, the indicator for ln X.  
Clears flag 1, the indicator for ln Y  
Branches to common entry point Z.  
    
    
Checksum and length: AD1B 12  
    
    
This routine sets the status for the exponential model.  
Clears flag 0, the indicator for ln X.  
Sets flag 1, the indicator for ln Y.  
Branches to common entry point Z.  
    
    
Checksum and length: D6F1 12  
    
    
    
This routine sets the status for the power model.  
Sets flag 0, the indicator for ln X.  
Sets flag 1, the indicator for ln Y.  
Checksum and length: 3800 9  
    
   
   
Defines common entry point for all models.  
Clears the statistics registers. Press   
Sets the loop counter to zero for the first input.  
(4Σ)  
Checksum and length: 8611 10  
    
   
Defines the beginning of the input loop.  
Adjusts the loop counter by one to prompt for input.  
   
    
Stores loop counter in X so that it will appear with the  
prompt for X.  
    
Displays counter with prompt and stores X input.  
Program Lines:  
(In RPN mode)  
Description  
    
If flag 0 is set . . .  
   
. . . takes the natural log of the X–input.  
Stores that value for the correction routine.  
Prompts for and stores Y.  
If flag 1 is set . . .  
. . . takes the natural log of the Y–input.  
    
    
    
   
    
    
   
Accumulates B and R as x,y–data pair in statistics registers.  
    
Loops for another X, Y pair.  
Checksum and length: 9560 46  
    
    
    
   
Defines the beginning of the "undo" routine.  
Recalls the most recent data pair.  
Deletes this pair from the statistical accumulation.  
    
Loops for another X, Y pair.  
Checksum and length: A79F 15  
    
   
Defines the start of the output routine  
Calculates the correlation coefficient.  
Stores it in R.  
Displays the correlation coefficient.  
Calculates the coefficient b.  
    
    
   
    
If flag 1 is set takes the natural antilog of b.  
   
    
    
   
Stores b in B.  
Displays value.  
Calculates coefficient m.  
Stores m in M.  
    
    
Displays value.  
Checksum and length: 850C 36  
    
Defines the beginning of the estimation (projection) loop.  
Program Lines:  
(In RPN mode)  
Description  
    
Displays, prompts for, and, if changed, stores x–value in X.  
If flag 0 is set . . .  
   
    
    
    
Branches to K001  
Branches to M001  
Stores ˆ–value in Y.  
y
    
   
Displays, prompts for, and, if changed, stores y–value in Y.  
If flag 0 is set . . .  
    
    
    
Branches to O001  
Branches to N001  
ˆ
Stores x in X for next loop.  
    
Loops for another estimate.  
Checksum and length: C3B7 36  
    
This subroutine calculates ˆ for the straight–line model.  
y
    
    
    
Calculates ˆ = MX + B.  
y
   
Returns to the calling routine.  
Checksum and length: 9688 15  
ˆ
    
    
    
   
   
This subroutine calculates x for the straight–line model.  
ˆ
Calculates x =(Y B) ÷ M.  
Returns to the calling routine.  
Checksum and length: 9C0F 15  
    
This subroutine calculates ˆ for the logarithmic model.  
y
    
   
    
    
Calculates ˆ = M In X + B.  
y
   
Returns to the calling routine.  
Program Lines:  
(In RPN mode)  
Description  
Checksum and length: 889C 18  
ˆ
    
    
    
   
This subroutine calculates x for the logarithmic model.  
(Y – B) ÷ M  
   
ˆ
Calculates x = e  
   
Returns to the calling routine.  
Checksum and length: 0DBE 18  
    
This subroutine calculates ˆ for the exponential model.  
y
    
    
   
MX  
    
Calculates ˆ= Be  
.
y
    
Branches to M005  
Checksum and length: 9327 18  
ˆ
This subroutine calculates x for the exponential model.  
    
    
   
   
   
ˆ
Calculates x = (ln (Y ÷ B)) ÷ M.  
Goes to N005  
    
Checksum and length: 7219 18  
This subroutine calculates ˆ for the power model.  
    
    
    
y
   
    
M
Calculates Y= B (X ).  
    
Goes to K005  
Checksum and length: 11B3 18  
    
This subroutine calculates x for the power model.  
ˆ
Program Lines:  
(In RPN mode)  
Description  
    
   
    
   
1/M  
   
ˆ
Calculates x= (Y/B )  
   
Goes to O005  
Checksum and length: 8524 21  
    
Determines if D001 or B001 should be run  
If flag 1 is set . . .  
   
    
    
Executes D001  
Executes B001  
    
Goes to Y006  
Checksum and length: 4BFA 15  
    
Determines if C001 or A001 should be run  
   
If flag 1 is set . . .  
Executes C001  
Executes A001  
Goes to Y006  
    
    
    
Checksum and length: 1C4D 15  
    
   
Determines if J001 or H001 should be run  
If flag 1 is set . . .  
Executes J001  
Executes H001  
Goes to Y011  
    
    
    
Checksum and length: 0AA5 15  
    
   
Determines if I001 or G001 should be run  
If flag 1 is set . . .  
    
    
    
Executes I001  
Executes G001  
Goes to Y011  
Checksum and length: 666D 15  
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.  
If flag 1 is set in routine N, then I001 is executed. If flag 1 is clear, G001 is  
executed.  
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:  
Sfor a straight line;  
Lfor a logarithmic curve;  
Efor an exponential curve; or  
Pfor 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 made  
an error after you have pressed in step 3 (with the value prompt still  
visible), press again (displaying the value prompt) and press  
Uto undo (remove) the last data pair. If you discover that you  
made an error after step 4, press U. In either case, continue at  
step 3.  
6. After all data are keyed in, press Rto see the correlation  
coefficient, R.  
7. Press to see the regression coefficient B.  
8. Press to see the regression coefficient M.  
ˆ
9. Press to see the value prompt for the x, ˆ–estimation routine.  
y
10. If you wish to estimate ˆ based on x, key in x at the value prompt, then  
y
press to see ˆ ().  
y
ˆ
11. If you wish to estimate x based on y, press until you see the value  
ˆ
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:  
B
Regression coefficient (y–intercept of a straight line); also  
used for scratch.  
M
R
X
Regression coefficient (slope of a straight line).  
Correlation coefficient; also used for scratch.  
The x–value of a data pair when entering data; the  
ˆ
hypothetical x when projecting ˆ; or x (x–estimate)  
when given a hypothetical y.  
y
Y
The y–value of a data pair when entering data; the  
ˆ
hypothetical y when projecting x; or ˆ (y–estimate)  
when given a hypothetical x.  
y
Statistics registers  
Statistical accumulation and computation.  
Example 1:  
Fit a straight line to the data below. Make an intentional error when keying in the  
third data pair and correct it with the undo routine. Also, estimate y for an x value  
of 37. Estimate x for a y value of 101.  
X
Y
40.5  
38.6  
102  
37.9  
100  
36.2  
97.5  
35.1  
95.5  
34.6  
94  
104.5  
Keys:  
Display:  
Description:  
(In RPN mode)  
S  
  
Starts straight–line routine.  
  
  
Enters x–value of data pair.  
  
value  
  
Enters y–value of data pair.  
  
  
  
Enters x–value of data pair.  
  
  
  
Enters y–value of data pair.  
  
  
Now intentionally enter 379 instead of 37.9 so that you can see how to correct  
incorrect entries.  
Keys:  
Display:  
Description:  
(In RPN mode)  
Enters wrong x–value of data pair.  
  
  
  
  
Retrieves  prompt.  
  
  
Deletes the last pair. Now proceed  
with the correct data entry.  
U  
  
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–value of data pair.  
Enters y–value of data pair.  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Calculates the correlation  
coefficient.  
R  
  
Calculates regression coefficient B.  
Calculates regression coefficient M.  
Prompts for hypothetical x–value.  
  
  
  
  
  
  
  
  
  
Stores 37 in X and calculates ˆ.  
y
  
  
ˆ
Stores 101 in Y and calculates x.  
  
Example 2:  
Repeat example 1 (using the same data) for logarithmic, exponential, and power  
curve fits. The table below gives you the starting execution label and the results (the  
correlation and regression coefficients and the x– and y– estimates) for each type of  
curve. You will need to reenter the data values each time you run the program for a  
different curve fit.  
Logarithmic  
Exponential  
Power  
L E P  
To start:  
R
B
M
0.9965  
–139.0088  
65.8446  
0.9945  
51.1312  
0.0177  
0.9959  
8.9730  
0.6640  
98.7508  
38.2857  
98.5870  
38.3628  
98.6845  
38.3151  
Y ( ˆ when X=37)  
y
ˆ
X (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.  
 
y
"Upper tail"  
area  
Q [x]  
x
x
x
2
1
Q(x) = 0.5−  
e
((xx)÷σ ) ÷2dx  
x
σ 2π  
This program uses the built–in integration feature of the HP 35s 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 Listing:  
Program Lines:  
(In RPN mode)  
    
Description  
This routine initializes the normal distribution program.  
Stores default value for mean.  
   
    
    
   
Prompts for and stores mean, M.  
Stores default value for standard deviation.  
    
    
   
Prompts for and stores standard deviation, S.  
Stops displaying value of standard deviation.  
Checksum and length: 70BF 26  
    
This routine calculates Q(X) given X.  
    
    
    
Prompts for and stores X.  
Calculates upper tail area.  
Stores value in Q so VIEW function can display it.  
Displays Q(X).  
Loops to calculate another Q(X).  
    
    
Checksum and length: 042A 18  
    
    
    
    
This routine calculates X given Q(X).  
Prompts for and stores Q(X).  
Recalls the mean.  
Stores the mean as the guess for X, called X  
.
guess  
Checksum and length: A970 12  
    
This label defines the start of the iterative loop.  
Calculates (Q( X )– Q(X)).  
    
guess  
    
    
    
 ꢀ  
    
   
   
Calculates the derivative at X  
.
guess  
Calculates the correction for X  
.
guess  
Program Lines:  
(In RPN mode)  
    
Description  
Adds the correction to yield a new X  
.
guess  
   
   
   
Tests to see if the correction is significant.  
    
Goes back to start of loop if correction is significant.  
Continues if correction is not significant.  
    
    
    
Displays the calculated value of X.  
Loops to calculate another X.  
Checksum and length: EDF4 57  
    
    
    
    
     
   
This subroutine calculates the upper–tail area Q(x).  
Recalls the lower limit of integration.  
Recalls the upper limit of integration.  
Selects the function defined by LBL F for integration.  
Integrates the normal function using the dummy variable D.  
 π  
   
  
    
Calculates S ×  
.
2π  
    
   
Stores result temporarily for inverse routine.  
   
   
   
Adds half the area under the curve since we integrated  
using the mean as the lower limit.  
   
Returns to the calling routine.  
Checksum and length: 8387 52  
    
This subroutine calculates the integrand for the normal  
((X M)÷S)2 ÷2  
function e  
    
    
Program Lines:  
(In RPN mode)  
   
Description  
   
   
   
   
   
   
Returns to the calling routine.  
Checksum and length: B3EB 31  
Flags Used:  
None.  
Remarks:  
The accuracy of this program is dependent on the display setting. For inputs in the  
area between 3 standard deviations, a display of four or more significant figures  
is adequate for most applications.  
At full precision, the input limit becomes 5 standard deviations. Computation time  
is significantly less with a lower number of displayed digits.  
In routine Q, the constant 0.5 may be replaced by 2 and .  
You do not need to key in the inverse routine (in routines I and T) if you are not  
interested in the inverse capability.  
Program Instructions:  
1. Key in the program routines; press when done.  
2. Press S.  
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, 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 I.  
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:  
D
M
Q
S
Dummy variable of integration.  
Population mean, default value zero.  
Probability corresponding to the upper–tail area.  
Population standard deviation, default value of 1.  
Variable used temporarily to pass the value S ×  
T
to the inverse  
2π  
program.  
X
Input value that defines the left side of the upper–tail area.  
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 "3σ" 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:  
(In RPN mode)  
S  
Starts the initialization routine.  
  
  
Accepts the default value of zero for M.  
  
  
  
Accepts the default value of 1 for S.  
Starts the distribution program and  
D  
  
prompts for X.  
value  
  
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.  
Multiplies by the population. Displays  
  
  
the approximate number of blind dates  
in the local population that meet the  
criteria.  
Since your friend has been known to exaggerate from time to time, you decide to  
see how rare a "2σ" date might be. Note that the program may be rerun simply by  
pressing .  
Keys:  
Display:  
Description:  
Resumes program.  
(In RPN mode)  
  
  
  
Enters X–value of 2 and calculates  
Q(X).  
  
  
Multiplies by the population for the  
revised estimate.  
  
  
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 at least 90? What is the score  
that only 10 percent of the students would be expected to have surpassed? What  
would be the score that only 20 percent of the students would have failed to  
achieve?  
Keys:  
Display:  
Description:  
(In RPN mode)  
Starts the initialization routine.  
Stores 55 for the mean.  
  
S  
  
  
  
  
  
Stores 15.3 for the standard deviation.  
Starts the distribution program and  
prompts for X.  
  
D  
  
value  
  
Enters 90 for X and calculates Q(X).  
  
  
Thus, we would expect that only about 1 percent of the students would do better  
than score 90.  
Keys:  
Display:  
Description:  
(In RPN mode)  
Starts the inverse routine.  
  
I  
  
  
Stores 0.1 (10 percent) in Q(X) and  
calculates X.  
  
  
Resumes the inverse routine.  
  
  
  
Stores 0.8 (100 percent minus 20  
  
  
percent) in Q(X) and calculates X.  
Grouped Standard Deviation  
The standard deviation of grouped data, S , is the standard deviation of data  
xy  
points x , x , ... , x , occurring at positive integer frequencies f , f , ... , f .  
1
2
n
1
2
n
(
xifi)2  
f
x 2f −  
i
i
i
Sxg  
=
(
f ) 1  
i
 
This program allows you to input data, correct entries, and calculate the standard  
deviation and weighted mean of the grouped data.  
Program Listing:  
Program Lines:  
(In ALG mode)  
Description  
    
Start grouped standard deviation program.  
Clears statistics registers (-27 through -32).  
   
   
    
Clears the count N.  
Checksum and length: E5BC 13  
    
    
    
   
Input statistical data points.  
Stores data point in X.  
Stores data–point frequency in F.  
Enters increment for N.  
    
    
Recalls data–point frequency f.  
i
Checksum and length: 3387 19  
    
   
Accumulate summations.  
Stores index for register -27.  
    
    
Updates  
in register -27.  
f
i
   
xifi  
   
    
   
    
    
Stores index for register -28.  
Updates  
in register -28.  
xf  
i i  
   
   
    
   
xi2f  
i
Stores index for register -30.  
    
    
Program Lines:  
(In ALG mode)  
Description  
Updates  
xi2f in register -30.  
i
   
    
    
    
    
   
Increments (or decrements) N.  
    
    
    
Displays current number of data pairs.  
Goes to label line numberI for next data input.  
Checksum and length: F6CB 84  
    
   
Calculates statistics for grouped data.  
Grouped standard deviation.  
    
    
Displays grouped standard deviation.  
Weighted mean.  
  
    
    
    
Displays weighted mean.  
Goes back for more points.  
Checksum and length: DAF2 24  
    
   
Undo data–entry error.  
Enters decrement for N.  
    
    
   
Recalls last data frequency input.  
Changes sign of f.  
i
    
    
Adjusts count and summations.  
Checksum and length: 03F4 23  
Flags Used:  
None.  
Program Instructions:  
1. Key in the program routines; press when done.  
2. Press Sto start entering new data.  
3. Key in x –value (data point) and press .  
i
4. Key in f –value (frequency) and press .  
i
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  
i
i
pressed in step 4, press Uand then press again.  
Then go back to step 3 to enter the correct data.  
7. When the last data pair has been input, press Gto 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 and continue at step 3.  
To start a new problem, start at step 2.Variables Used:  
X
F
Data point.  
Data–point frequency.  
N
S
M
i
Data–pair counter.  
Grouped standard deviation.  
Weighted mean.  
Index variable used to indirectly address the correct  
statistics register.  
Register -27  
Register -28  
Register -30  
Summation Σf.  
i
Summation Σx f.  
i i  
2
Summation Σx f.  
i
i
Example:  
Enter the following data and calculate the grouped standard deviation.  
Group  
1
5
2
8
3
13  
4
15  
5
22  
6
37  
x
i
f
17  
26  
37  
43  
73  
115  
i
Keys:  
Display:  
Description:  
(In ALG mode)  
Prompts for the first x .  
  
S  
i
value  
  
Stores 5 in X; prompts for first f.  
i
  
value  
  
Stores 17 in F; displays the counter.  
  
  
  
Prompts for the second x .  
i
  
  
Prompts for second f.  
i
  
  
  
Displays the counter.  
  
  
  
Prompts for the third x .  
i
  
  
Prompts for the third f.  
i
  
  
  
  
Displays the counter.  
  
You erred by entering 14 instead of 13 for x . Undo your error by executing routine  
3
U:  
Removes the erroneous data;  
  
U  
  
displays the revised counter.  
Prompts for new third x.  
  
i
  
  
Prompts for the new third f.  
i
  
  
  
Displays the counter.  
  
  
Prompts for the fourth x.  
i
  
  
Prompts for the fourth f.  
i
  
  
Displays the counter.  
Prompts for the fifth x.  
  
  
  
  
i
  
  
Prompts for the fifth f.  
i
  
  
  
  
Displays the counter.  
  
  
Prompts for the sixth x .  
i
  
  
Prompts for the sixth f.  
i
  
  
G  
  
  
Displays the counter.  
  
  
Calculates and displays the  
grouped standard deviation (sx)  
of the six data points.  
  
Calculates and displays  
  
  
weighted mean ( x ).  
Clears VIEW.  
  
17  
Miscellaneous Programs and Equations  
Time Value of Money  
Given any four of the five values in the "Time–Value–of–Money equation" (TVM),  
you can solve for the fifth value. This equation is useful in a wide variety of financial  
applications such as consumer and home loans and savings accounts.  
The TVM equation is:  
N  
1(1+ I 100)  
P
+ F(1+ (I 100))N + B = 0  
I 100  
Balance, B  
Payments, P  
N
_
N 1  
3
1
2
Future Value, F  
The signs of the cash values (balance, B; payment, P; and future balance, F)  
correspond to the direction of the cash flow. Money that you receive has a positive  
sign while money that you pay has a negative sign. Note that any problem can be  
viewed from two perspectives. The lender and the borrower view the same problem  
with reversed signs.  
   
Equation Entry:  
Key in this equation:  
  
Keys:  
Display:  
Description:  
(In RPN mode)  
    
or current equation  
 _  
Selects Equation  
mode.  
Starts entering  
P  
equation.  
  
4   
4  
I  
  
 ꢆ  
 ꢆ  
_  
_  
Õ  
 NÕ  
 I F  
4  I  
 Õ  
 N  
 B  
  
_  
_  
_  
Terminates the  
equation.  
  
  
  
Checksum and length.  
  (hold)  
Remarks:  
The TVM equation requires that I must be non–zero to avoid a    error.  
If you're solving for I and aren't sure of its current value, press I  
before you begin the SOLVE calculation (I).  
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   
I.  
2. Press . If necessary, press ×or Øto scroll through the equation list  
until you come to the TVM equation.  
3. Do one of the following five operations:  
a. Press Nto calculate the number of compounding periods.  
b. Press Ito 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  
account.  
d. Press Pto calculate periodic payment.  
e. Press Fto calculate future value or balance of a loan.  
4. Key in the values of the four known variables as they are prompted for; press  
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 same variable using different  
data, go back to step 2.  
SOLVE works effectively in this application without initial guesses.  
Variables Used:  
N
I
The number of compounding periods.  
The periodic interest rate as a percentage. (For example, if the  
annual interest rate is 15% and there are 12 payments per year,  
the periodic interest rate, i, is 15÷12=1.25%.)  
The initial balance of loan or savings account.  
The periodic payment.  
B
P
F
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–month) 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:  
(In RPN mode)  
Selects FIX 2 display format.  
8()   
Displays the leftmost part of the  
TVM equation.  
ꢆ  
(Øas needed )  
  
Selects P; prompts for I.  
P  
value  
Converts your annual interest  
rate input to the equivalent  
monthly rate.  
  
   
  
  
  
Stores 0.88 in I; prompts for N.  
value  
  
Stores 36 in N; prompts for F.  
  
value  
  
Stores 0 in F; prompts for B.  
  
value  
  
Calculates B, the beginning  
loan balance.  
  
  
  
  
  
Stores 5750 in B; calculates  
monthly payment, P.  
  
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:  
(In RPN mode)  
Displays the leftmost hart of the TVM  
equation.  
ꢆ  
  
Selects I; prompts for P.  
I  
  
  
Rounds the payment to two decimal  
places.  
Calculates new payment.  
   
  
  
  
  
  
Stores –176.89 in P; prompts for N.  
Retains 36 in N; prompts for F.  
Retains 0 in F; prompts for B.  
  
  
  
  
  
  
Retains 5750 in B; calculates  
monthly interest rate.  
  
  
  
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?  
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.  
Keys:  
Display:  
Description:  
(In RPN mode)  
Displays leftmost part of the TVM  
equation.  
ꢆ  
  
Selects F; prompts for P.  
F  
  
  
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  
negative, indicating that you must,  
pay out this money.  
  
Sets FIX 4 display format.  
8  
()   
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 not 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.  
 
LBL Y  
VIEW Prime  
LBL Z  
Note: x is the  
value in the  
X-register.  
P + 2  
x
LBL P  
Start  
x
3
P
D
LBL X  
yes  
x = 0  
?
no  
yes  
no  
Program Listing:  
Program Lines:  
(In ALG mode)  
    
Description  
This routine displays prime number P.  
    
Checksum and length: 2CC5 6  
    
    
This routine adds 2 to P.  
Checksum and length: EFB2 9  
    
   
   
   
   
This routine stores the input value for P.  
   
   
   
Tests for even input  
Increments P if input an even number.  
Stores 3 in test divisor, D  
Checksum and length: EA89 47  
    
   
   
This routine tests P to see if it is prime.  
Finds the fractional part of P ÷ D.  
Tests for a remainder of zero (not prime).  
If the number is not prime, tries next possibility.  
    
   
   
   
 >  
    
Tests to see whether all possible factors have been tried.  
If all factors have been tried, branches to the display  
routine.  
   
    
Branches to test potential prime with new factor.  
Checksum and length: C6B5 53  
Flags Used:  
None.  
Program Instructions:  
1. Key in the program routines; press when done.  
2. Key in a positive integer greater than 3.  
3. Press Pto run program. Prime number, P will be displayed.  
4. To see the next prime number, press .  
Variables Used:  
P
D
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:  
(In ALG mode)  
  
Calculates next prime number after  
  
789.  
P  
  
  
Calculates next prime number after  
797.  
  
Cross Product in Vectors  
Here is an example showing how to use the program function to calculate the cross  
product.  
Cross product:  
v
× v = (YW ZV )i + (ZU XW)j + (XV YU)k  
1
2
where  
v = X i + Y j + Z k  
1
and  
v =U i + V j + W k  
2
Program Lines:  
(In RPN mode)  
    
Description  
Defines the beginning of the rectangular input/display  
routine.  
    
    
    
Displays or accepts input of X.  
Displays or accepts input of Y.  
Displays or accepts input of Z.  
    
Goes to R001 to input vectors  
Checksum and length: D82E 15  
    
    
Defines the beginning of the vector–enter routine.  
Copies values in X, Y and Z to U, V and W  
respectively.  
    
    
    
    
    
    
Goes to R001 to input vectors  
Checksum and length: B6AF 24  
 
Program Lines:  
(In RPN mode)  
Description  
    
    
    
    
    
   
Defines the beginning of the cross–product routine.  
Calculates (YW ZV), which is the X component.  
    
    
    
    
    
   
Calculates (ZU – WX), which is the Y component.  
    
    
    
    
    
   
    
    
    
    
    
    
Stores (XV YU), which is the Z component.  
Stores X component.  
Stores Y component.  
Goes to R001 to input vectors  
Checksum and length: 838D 72  
Example:  
Calculate the cross product of two vectors, v1=2i+5j+4k and v2=i-2j+3k  
Keys:  
R  
  
Display:  
Description:  
Run R routine to input vector value  
  
  
Input v2 of x-component  
Input v2 of y-component  
Input v2 of z-component  
  
  
  
z  
  
  
  
  
Run E routine to exchange v2 in U,  
V, and W variables  
E  
  
Input v1 of x-component  
Input v1 of y-component  
Input v1 of z-component  
  
  
  
  
  
  
  
  
Run C routine to calculate x-  
component of cross product  
Calculate y-component of cross  
C  
  
product  
  
  
  
Calculate z-component of cross  
product  
Part 3  
Appendixes and Reference  
   
A
Support, Batteries, and Service  
Calculator Support  
You can obtain answers to questions about using your calculator from our  
Calculator Support Department. Our experience shows that many customers have  
similar questions about our products, so we have provided the following section,  
"Answers to Common Questions." If you don't find an answer to your question,  
contact the Calculator Support Department listed on page A–8.  
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 85(5) function (page 1–23).  
Q: How do l change the number of decimal places in the display?  
A: Use the 8 menu (page 1–21).  
Q: How do I clear all or portions of memory?  
A:  
displays the CLEAR menu, which allows you to clear x (the number  
in the X-register), all direct variables, all of memory, all statistical data, all stack  
levels and all indirect variables  
Q: What does an "E" in a number (for example, ) mean?  
A-1  
     
–13  
A: Exponent of ten; that is, 2.51 × 10  
.
Q: The calculator has displayed the message  . What should I do?  
A: You must clear a portion of memory before proceeding. (See appendix B.)  
Q: Why does calculating the sine (or tangent) of π radians display a very small  
number instead of 0?  
A: π 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 (9  
, , or  ).  
Q: What does an annunciator in the display mean?  
A: 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  .  
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  
 
Changing the Batteries  
The calculator is powered by two 3-volt lithium coin batteries, CR2032.  
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.  
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.)  
To install batteries:  
1. Have two 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.  
BK+B  
3. Turn the calculator over and slide off the battery cover.  
4. To prevent memory loss, never remove two old batteries at the same time. Be  
sure to remove and replace the batteries one at a time.  
A-3  
 
Do not mutilate, puncture, or dispose of batteries in fire. The  
batteries can burst or explode, releasing hazardous  
chemicals.  
Warning  
5. Insert a new CR2032 lithium battery, making sure that the positive sign (+) is  
facing outward.  
6. Remove and insert the other battery as in steps 4 through 5. Make sure that the  
positive sign (+) on each battery is facing outward.  
7. Replace the battery compartment cover.  
8. Press .  
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–8.  
The calculator won't turn on (steps 1–4) or doesn't respond when you press  
the keys (steps 1–3):  
1. Reset the calculator. Hold down the key and press . It may be  
necessary to repeat these reset keystrokes several times.  
2. Erase memory. Press and hold down , then press and hold down both  
and 6. Memory is cleared and the   message is  
displayed when you release all three keys.  
A-4  
 
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  .  
4. If the calculator still does not respond to keystrokes, use a thin, pointed  
object to press the RESET hole. Stored data usually remain intact.  
Reset Hole  
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 listed on page A–8.  
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  .  
3. Press the keys in the following sequence:  
A-5  
 
9 × Ö Õ →  
6 Ø →  
4→  
→  
  
If you press the keys in the proper order and they are functioning properly,  
the calculator displays  followed by two–digit numbers. (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 5.  
The calculator displays  followed by a one–digit 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–8). 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 9 starts a continuous self–test that is used at the factory.  
You can halt this factory test by pressing any key.  
A-6  
Warranty  
HP 35s Scientific Calculator; Warranty period: 12 months  
1. HP warrants to you, the end-user customer, that HP hardware, accessories and  
supplies will be free from defects in materials and workmanship after the date  
of purchase, for the period specified above. If HP receives notice of such  
defects during the warranty period, HP will, at its option, either repair or  
replace products which prove to be defective. Replacement products may be  
either new or like-new.  
2. HP warrants to you that HP software will not fail to execute its programming  
instructions after the date of purchase, for the period specified above, due to  
defects in material and workmanship when properly installed and used. If HP  
receives notice of such defects during the warranty period, HP will replace  
software media which does not execute its programming instructions due to  
such defects.  
3. HP does not warrant that the operation of HP products will be uninterrupted or  
error free. If HP is unable, within a reasonable time, to repair or replace any  
product to a condition as warranted, you will be entitled to a refund of the  
purchase price upon prompt return of the product with proof of purchase.  
4. HP products may contain remanufactured parts equivalent to new in  
performance or may have been subject to incidental use.  
5. Warranty does not apply to defects resulting from (a) improper or inadequate  
maintenance or calibration, (b) software, interfacing, parts or supplies not  
supplied by HP, (c) unauthorized modification or misuse, (d) operation outside  
of the published environmental specifications for the product, or (e) improper  
site preparation or maintenance.  
A-7  
 
6. HP MAKES NO OTHER EXPRESS WARRANTY OR CONDITION WHETHER  
WRITTEN OR ORAL. TO THE EXTENT ALLOWED BY LOCAL LAW, ANY  
IMPLIED WARRANTY OR CONDITION OF MERCHANTABILITY,  
SATISFACTORY QUALITY, OR FITNESS FOR A PARTICULAR PURPOSE IS  
LIMITED TO THE DURATION OF THE EXPRESS WARRANTY SET FORTH  
ABOVE. Some countries, states or provinces do not allow limitations on the  
duration of an implied warranty, so the above limitation or exclusion might not  
apply to you. This warranty gives you specific legal rights and you might also  
have other rights that vary from country to country, state to state, or province to  
province.  
7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE REMEDIES IN THIS  
WARRANTY STATEMENT ARE YOUR SOLE AND EXCLUSIVE REMEDIES.  
EXCEPT AS INDICATED ABOVE, IN NO EVENT WILL HP OR ITS SUPPLIERS BE  
LIABLE FOR LOSS OF DATA OR FOR DIRECT, SPECIAL, INCIDENTAL,  
CONSEQUENTIAL (INCLUDING LOST PROFIT OR DATA), OR OTHER  
DAMAGE, WHETHER BASED IN CONTRACT, TORT, OR OTHERWISE. Some  
countries, States or provinces do not allow the exclusion or limitation of  
incidental or consequential damages, so the above limitation or exclusion may  
not apply to you.  
8. The only warranties for HP products and services are set forth in the express  
warranty statements accompanying such products and services. HP shall not be  
liable for technical or editorial errors or omissions contained herein.  
FOR CONSUMER TRANSACTIONS IN AUSTRALIA AND NEW ZEALAND:  
THE WARRANTY TERMS CONTAINED IN THIS STATEMENT, EXCEPT TO  
THE EXTENT LAWFULLY PERMITTED, DO NOT EXCLUDE, RESTRICT OR  
MODIFY AND ARE IN ADDITION TO THE MANDATORY STATUTORY  
RIGHTS APPLICABLE TO THE SALE OF THIS PRODUCT TO YOU.  
Customer Support  
AP  
Country :  
Telephone numbers  
Australia  
1300-551-664 or  
03-9841-5211  
A-8  
China  
010-68002397  
2805-2563  
+65 6100 6682  
+852 2805-2563  
+65 6100 6682  
09-574-2700  
+65 6100 6682  
6100 6682  
2-561-2700  
Hong Kong  
Indonesia  
Japan  
Malaysia  
New Zealand  
Philippines  
Singapore  
South Korea  
Taiwan  
+852 2805-2563  
Thailand  
Vietnam  
+65 6100 6682  
+65 6100 6682  
EMEA  
Country :  
Austria  
Telephone numbers  
01 360 277 1203  
02 620 00 86  
02 620 00 85  
296 335 612  
82 33 28 44  
Belgium  
Belgium  
Czech Republic  
Denmark  
Finland  
09 8171 0281  
01 4993 9006  
069 9530 7103  
210 969 6421  
020 654 5301  
01 605 0356  
02 754 19 782  
2730 2146  
France  
Germany  
Greece  
Netherlands  
Ireland  
Italy  
Luxembourg  
Norway  
23500027  
Portugal  
021 318 0093  
495 228 3050  
0800980410  
913753382  
Russia  
South Africa  
Spain  
Sweden  
08 5199 2065  
022 827 8780  
Switzerland (French)  
A-9  
Switzerland (German)  
Switzerland (Italian)  
United Kingdom  
01 439 5358  
022 567 5308  
0207 458 0161  
LA  
Country :  
Telephone numbers  
Anguila  
1-800-711-2884  
1-800-711-2884  
0-800- 555-5000  
800-8000 800-711-2884  
1-800-711-2884  
1-800-711-2884  
1-800-711-2884  
800-100-193  
Antigua  
Argentina  
Aruba  
Bahamas  
Barbados  
Bermuda  
Bolivia  
Brazil  
0-800-709-7751  
1-800-711-2884  
1-800-711-2884  
001-800-872-2881 +  
800-711-2884  
British Virgin Islands  
Cayman Island  
Curacao  
Chile  
800-360-999  
Colombia  
01-8000-51-4746-8368  
(01-8000-51- HP INVENT)  
Costa Rica  
0-800-011-0524  
1-800-711-2884  
1-800-711-2884  
1-999-119 800-711-2884  
(Andinatel)  
Dominica  
Dominican Republic  
Ecuador  
1-800-225-528 800-711-2884  
(Pacifitel)  
El Salvador  
French Antilles  
French Guiana  
Grenada  
800-6160  
0-800-990-011800-711-2884  
0-800-990-011800-711-2884  
1-800-711-2884  
Guadelupe  
Guatemala  
Guyana  
0-800-990-011800-711-2884  
1-800-999-5105  
159 800-711-2884  
Haiti  
183 800-711-2884  
Honduras  
Jamaica  
Martinica  
Mexico  
800-0-123 800-711-2884  
1-800-711-2884  
0-800-990-011 877-219-8671  
01-800-474-68368 (800 HP  
INVENT)  
Montserrat  
1-800-711-2884  
Netherland Antilles  
001-800-872-2881 ♦  
800-711-2884  
Nicaragua  
Panama  
Paraguay  
1-800-0164 800-711-2884  
001-800-711-2884  
(009) 800-541-0006  
0-800-10111  
Peru  
Puerto Rico  
St. Lucia  
1-877 232 0589  
1-800-478-4602  
St Vincent  
01-800-711-2884  
1-800-711-2884  
1-800-711-2884  
156 800-711-2884  
1-800-711-2884  
01-800-711-2884  
1-800-711-2884  
St. Kitts & Nevis  
St. Marteen  
Suriname  
Trinidad & Tobago  
Turks & Caicos  
US Virgin Islands  
Uruguay  
0004-054-177  
Venezuela  
0-800-474-68368 (0-800 HP  
INVENT)  
NA  
Country :  
Telephone numbers  
Canada  
USA  
800-HP-INVENT  
800-HP INVENT  
Please logon to http://www.hp.com for the latest service and support information.  
Regulatory information  
Federal Communications Commission Notice  
This equipment has been tested and found to comply with the limits for a Class B  
digital device, pursuant to Part 15 of the FCC Rules. These limits are designed to  
provide reasonable protection against harmful interference in a residential  
installation. This equipment generates, uses, and can radiate radio frequency  
energy and, if not installed and used in accordance with the instructions, may cause  
harmful interference to radio communications. However, there is no guarantee that  
interference will not occur in a particular installation. If this equipment does cause  
harmful interference to radio or television reception, which can be determined by  
turning the equipment off and on, the user is encouraged to try to correct the  
interference by one or more of the following measures:  
• Reorient or relocate the receiving antenna.  
• Increase the separation between the equipment and the receiver.  
• Connect the equipment into an outlet on a circuit different from that to which  
the receiver is connected.  
• Consult the dealer or an experienced radio or television technician for help.  
Modifications  
The FCC requires the user to be notified that any changes or modifications made to  
this device that are not expressly approved by Hewlett-Packard Company may void  
the user’s authority to operate the equipment.  
Declaration of Conformity for Products Marked with FCC Logo, United States Only  
This device complies with Part 15 of the FCC Rules. Operation is subject to the  
following two conditions: (1) this device may not cause harmful interference, and (2)  
this device must accept any interference received, including interference that may  
cause undesired operation.  
If you have questions about the product that are not related to this declaration, write  
to  
Hewlett-Packard Company  
P. O. Box 692000, Mail Stop 530113  
Houston, TX 77269-2000  
For questions regarding this FCC declaration, write to  
Hewlett-Packard Company  
P. O. Box 692000, Mail Stop 510101  
   
Houston, TX 77269-2000  
or call HP at 281-514-3333  
To identify your product, refer to the part, series, or model number located on the  
product.  
Canadian Notice  
This Class B digital apparatus meets all requirements of the Canadian Interference-  
Causing Equipment Regulations.  
Avis Canadien  
Cet appareil numérique de la classe B respecte toutes les exigences du Règlement  
sur le matériel brouilleur du Canada.  
European Union Regulatory Notice  
This product complies with the following EU Directives:  
• Low Voltage Directive 2006/95/EC  
• EMC Directive 2004/108/EC  
Compliance with these directives implies conformity to applicable harmonized  
European standards (European Norms) which are listed on the EU Declaration of  
Conformity issued by Hewlett-Packard for this product or product family.  
This compliance is indicated by the following conformity marking placed on the  
product:  
yyyy+  
This marking is valid for non-Telecom products  
This marking is valid for EU non-harmonized  
and EU harmonized Telecom products  
Telecom products.  
(e.g. Bluetooth).  
*Notified body number (used only if  
applicable - refer to the product label)  
Hewlett-Packard GmbH, HQ-TRE, Herrenberger Strasse 140, 71034 Boeblingen,  
Germany  
Japanese Notice  
装置は、 情報処理装置等電波障害自主規制協議会 (VCCI) の基準に基づ く ク  
ラス B 報技術装置です。 この装置は庭環境で使用する こ と を目と し てい  
ますが、 この装置がラ ジオやレビジ ョ ン信機に近し て使用される と、 受信  
障害を引き起こすこ とがあ り ます。  
取扱説明書に従正しい取り扱いを し て く だ さい。  
Disposal of Waste Equipment by Users in Private Household in  
the European Union  
This symbol on the product or on its packaging indicates that this  
product must not be disposed of with your other household waste.  
Instead, it is your responsibility to dispose of your waste equipment  
by handing it over to a designated collection point for the recycling  
of waste electrical and electronic equipment. The separate collection  
and recycling of your waste equipment at the time of disposal will  
help to conserve natural resources and ensure that it is recycled in a  
manner that protects human health and the environment. For more information  
about where you can drop off your waste equipment for recycling, please contact  
your local city office, your household waste disposal service or the shop where you  
purchased the product.  
Perchlorate Material - special handling may apply  
This calculator's Memory Backup battery may contain perchlorate and may require  
special handling when recycled or disposed in California.  
B
User Memory and the Stack  
This appendix covers  
The allocation and requirements of user memory,  
How to reset the calculator without affecting memory,  
How to clear (purge) all of user memory and reset the system defaults, and  
Which operations affect stack lift.  
Managing Calculator Memory  
The HP 35s has 30KB 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 for the  
operation you just attempted. You need to clear some (or all) of user memory. For  
instance, you can:  
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  
13).  
Clear all of user memory (press  
()).  
To see how much memory is available, press . The display shows the  
number of bytes available.  
B-1  
   
To see the memory requirements of specific equations in the equation list:  
1. Press to activate Equation mode. (   or the left end of the  
current equation will be displayed.)  
2. If necessary, scroll through the equation list (press ×or Ø) until you see  
the desired equation.  
3. Press   to see the checksum (hexadecimal) and length (in bytes) of  
the equation. For example,  .  
To see the total memory requirements of specific programs:  
1. Press   () to display the first label in the program list.  
2. Scroll through the program list (press ×or Øuntil you see the desired  
program label and size). For example,   .  
3. Optional: Press   to see the checksum (hexadecimal) and length  
(in bytes) of the program. For example,  .  
To see the memory requirements of an equation in a program:  
1. Display the program line containing the equation.  
2. Press   to see the checksum and length. For example,  
 .  
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.  
If the calculator still does not respond to keystrokes, use a thin, pointed object to  
press the RESET hole.  
The calculator can reset itself if it is dropped or if power is interrupted.  
B-2  
 
Clearing Memory  
The usual way to clear user memory is to press   
(). However,  
there is also a more powerful clearing procedure that resets additional information  
and is useful if the 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):  
1. Press and hold down the key.  
2. Press and hold down ¥.  
3. Press 6. (You will be pressing three keys simultaneously). When you release  
all three keys, the display shows   if the operation is  
successful.  
B-3  
 
Category  
Angular mode  
CLEAR ALL  
Unchanged  
MEMORY CLEAR  
(Default)  
Degrees  
Base mode  
Unchanged  
Unchanged  
Unchanged  
Unchanged  
Unchanged  
Unchanged  
Unchanged  
Unchanged  
Unchanged  
Unchanged  
EQN LIST TOP  
Cleared  
Decimal  
Medium  
“  
“1,000”  
4095  
FIX 4  
Cleared  
xiy  
Off  
Zero  
EQN LIST TOP  
Cleared  
Null  
Contrast setting  
Decimal point  
Thousand separator  
Denominator (/c value)  
Display format  
Flags  
Complex mode  
Fraction–display mode  
Random–number seed  
Equation pointer  
Equation list  
FN = label  
Null  
Program pointer  
Program memory  
Stack lift  
Stack registers  
Variables  
PRGM TOP  
Cleared  
Enabled  
Cleared to zero  
Cleared to zero  
Not defined  
PRGM TOP  
Cleared  
Enabled  
Cleared to zero  
Cleared to zero  
Not defined  
Indirect Variables  
Logic  
Unchanged  
RPN  
Memory may inadvertently be cleared if the calculator is dropped or if power is  
interrupted.  
The Status of Stack Lift  
The four stack registers are always present, and the stack always has a stack–lift  
status. That is to say, the stack lift is always enabled or disabled regarding its  
behavior when the next number is placed in the X–register. (Refer to chapter 2, "The  
Automatic Memory Stack.")  
All functions except those in the following two lists will enable stack lift.  
B-4  
 
Disabling Operations  
The five operations , /, -,   
() and   
() disable stack lift. A number keyed in after one of these disabling  
operations writes over the number currently in the X–register. The Y–, Z– and T–  
registers remain unchanged.  
In addition, when and act like CLx, they also disable stack lift.  
The INPUT function disables stack lift as it halts a program for prompting (so any  
number then entered writes over the X-register), but it enables stack lift when the  
program resumes.  
Neutral Operations  
The following operations do not affect the status of stack lift:  
DEG, RAD,  
GRAD  
PSE  
FIX, SCI,  
ENG, ALL  
SHOW  
DEC, HEX,  
OCT, BIN  
RADIX . RADIX , CLΣ  
* and *  
CLVARS  
h  and STOP ×and Ø  
  
(1)**  
EQN  
  
(2)**  
FDISP  
   
 label nnn  
Errors  
and program  
entry  
Switching binary Digit entry  
windows  
xiy rθ a  
:
Except when used like CLx.  
ꢀꢀ Including all operations performed while the catalog is displayed except  
{} and {} , which enable stack lift.  
B-5  
   
The Status of the LAST X Register  
The following operations save x in the LAST X register in RPN mode:  
x
x
+, –, × , ÷  
e , 10  
2
, x ,  
x
x
y ,  
X
LN, LOG  
I/x, INT÷, Rmdr  
y
SIN, COS, TAN  
ASIN, ACOS, ATAN  
ˆ
ˆ
  
SINH, COSH, TANH  
ASINH, ACOSH, ATANH IP, FP, SGN, INTG,  
RND, ABS  
%, %CHG  
Σ+, Σ–  
HMS, HMS  
!
RCL+, –, ×, ÷  
DEG, RAD  
ARG  
nCr nPr  
x
x
CMPLX +, –, × ,÷  
CMPLX SIN, COS,  
TAN  
CMPLX e , LN, y , 1/x  
kg, lb  
°C, °F  
cm, in  
l, gal  
KM MILE  
Notice that /c does not affect the LAST X register.  
The recall-arithmetic sequence Xhvariable stores x in LASTx and  
Xhvariable stores the recalled number in LASTx.  
In ALG mode, the LAST X register is a companion to the stack: it holds the number  
that is the result of last expression. It supports using the previous expression result in  
ALG mode.  
B-6  
 
Accessing Stack Register Contents  
The values held in the four stack registers, X, Y, Z and T, are accessible in RPN  
mode in an equation or program using the REGX, REGY, REGZ and REGT  
commands.  
To use these instructions, press dfirst. Then, pressing <produces a menu in  
the display showing the X–, Y–, Z–, T–registers. Pressing Õor Öwill move the  
underline symbol, indicating which register is presently selected. Pressing   
will place an instruction into a program or equation that recalls the value of the  
chosen stack register for further use. These are displayed as REGX, REGY, REGZ,  
and REGT.  
For example, a program line entered by first pressing dand then entering the  
instructions REGX x REGY x REGZ x REGT will compute the product of the values in  
the 4 stack registers and place the result into the X-register. It will leave the previous  
values of X, Y, and Z in the stack registers Y, Z, and T.  
Many such efficient uses of values in the stack are possible in this manner that  
would not otherwise be available on the HP35s.  
B-7  
 
C
ALG: Summary  
About ALG  
This appendix summarizes some features unique to ALG mode, including,  
Two argument arithmetic  
Exponential and Logarithmic functions (,,  
,)  
Trigonometric functions  
Parts of numbers  
Reviewing the stack  
Operations with complex numbers  
Integrating an equation  
Arithmetic in bases 2, 8, and 16  
Entering statistical two–variable data  
Press 9() to set the calculator to ALG mode. When the calculator is  
in ALG mode, the ALG annunciator is on.  
In ALG mode, operations are performed in the following order.  
1. Expression in parenthesis.  
2. Factorial ( ! ) function requires inputting values before you press *.  
3. Functions that require inputting values after pressing the function key, for  
example, COS, SIN, TAN, ACOS, ASIN, ATAN, LOG, LN, x2, 1/x,  
, π,  
x
3
, %, RND, RAND, IP, FP, INTG, SGN, nPr, nCr, %CHG, INT÷, Rmdr, ABS,  
x
x
x
e ,10 , unit conversion.  
x y  
x
4.  
and y .  
C-1  
   
5. Unary Minus +/-  
6. ×, ÷  
7. +, –  
8. =  
Doing Two argument Arithmetic in ALG  
This discussion of arithmetic using ALG replaces the following parts that are affected  
by ALG mode. Two argument arithmetic operations are affected by ALG mode:  
Simple arithmetic  
Power functions (, )  
Percentage calculations (or  )  
Permutations and Combinations (x, {)  
Quotient and Remainder of Division ((÷),  
())  
Simple Arithmetic  
Here are some examples of simple arithmetic.  
In ALG mode, you enter the first number, press the operator (, , , ),  
enter the second number, and finally press the key.  
To Calculate:  
12 + 3  
Press:  
  
Display:  
  
  
  
12 – 3  
12 × 3  
12 ÷ 3  
  
  
  
  
  
  
  
  
C-2  
   
Power Functions  
In ALG mode, to calculate a number y raised to a power x, key in y x, then  
press .  
To Calculate:  
Press:  
  
Display:  
  
3
12  
  
  
  
1/3  
Õ64  
64  
(cube root)  
Percentage Calculations  
The Percent Function. The key divides a number by 100.  
To Calculate:  
Press:  
Õ2  
7  
Display:  
  
  
27% of 200  
  
Õ27  
  
  
200 less 27%  
25 plus 12%  
  
Õ12  
  
  
To Calculate:  
Press:  
x% of y  
 y Õx   
Percentage change from y to x. (y0)  
 y Õx   
Example:  
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:  
This year's price dropped  
about 2.2% from last year's  
price.  
  
Õ  
  
  
  
C-3  
   
Permutations and Combinations  
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:  
xÕ  
  
Display:  
  
  
Description:  
Total number of  
combinations possible.  
Quotient and Remainder Of Division  
You can use (÷) and () to produce  
either the quotient or remainder of division operations involving two integers.  
(÷)Integer ÕInteger .  
1
2
()Integer ÕInteger .  
1
2
Example:  
To display the quotient and remainder produced by 58 ÷ 9  
Keys:  
Display:  
  
  
Description:  
Displays the quotient.  
(÷)  
Õ9  
()  
Õ9  
Displays the remainder.  
  
  
Parentheses Calculations  
Use parentheses when you want to postpone calculating an intermediate result until  
you entered more numbers. For example, suppose you want to calculate:  
30  
8512  
× 9  
C-4  
     
If you were to key in , the calculator would  
calculate the result, -107.6471. However, that’s not what you want. To delay the  
division until you’ve subtracted 12 from 85, use parentheses:  
Keys:  
Display:  
  
Description:  
No calculation is done.  
4  
Calculates 85 12.  
Õ  
_  
Calculates 30/73.  
_  
  
Calculates 30/(85 12)  
× 9.  
  
  
You can omit the multiplication sign (×) before a left parenthesis. Implied  
multiplication is not available in Equation mode. For example, the expression 2 × (5  
– 4) can be entered as 4, without the key inserted between 2  
and the left parenthesis.  
Exponential and Logarithmic Functions  
To Calculate:  
Press:  
Display:  
  
  
Natural logarithm (base e)  
  
  
  
  
  
Common logarithm (base 10)  
Natural exponential  
  
  
  
  
  
Common exponential  
(antilogarithm)  
C-5  
 
Trigonometric Functions  
Assume the unit of the angle is 9()  
To Calculate:  
Sine of x.  
Press:  
  
Display:  
  
  
  
Cosine of x.  
Tangent of x.  
Arc sine of x.  
  
  
  
  
  
  
  
  
Arc cosine of x.  
  
  
  
Arc tangent of x.  
  
  
  
Hyperbolic functions  
To Calculate:  
Press:  
Hyperbolic sine of x (SINH).  
  ,key in a number,  
press   
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).  
  , key in a  
number,press   
  , key in a number,  
press   
  , key in a  
number,press   
  , key in a  
number,press   
  , key in a  
number,press   
C-6  
   
Parts of numbers  
To calculate:  
The integer part of 2.47  
Press:  
Display:  
  
()  
  
  
The fractional part of 2.47  
()  
  
  
  
The absolute value of –7  
The sign value of 9  
   
  
  
  
()  
  
The greatest integer equal to  
or less than –5.3  
()  
  
  
  
Reviewing the Stack  
The or key produces a menu in the display— X–, Y–, Z–, T–registers,  
to let you review the entire contents of the stack. The difference between the   
and the   key is the location of the underline in the display. Pressing the  
  displays the underline on the T register; pressing the displays the  
underline on the Y register.  
Pressing displays the following menu:  
     
value  
Pressing   displays the following menu:  
     
value  
You can press and   (along with Õ or Ö) to review the entire  
contents of the stack and recall them. They will appear as , ,   
or  depending on which part of the stacked was recalled and may be  
used in further calculations.  
C-7  
   
The value of X-, Y-, Z-, T-register in ALG mode is the same in RPN mode. After  
normal calculation, solving, programming, or integrating, the value of the four  
registers will be the same as in RPN or ALG mode and retained when you switch  
between ALG and RPN logic modes.  
Integrating an Equation  
1. Key in an equation. (see "Entering Equations into the Equation List" in chapter  
6) and leave Equation mode.  
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 the  
equation list (press ×or Ø) to display the desired equation.  
4. Select the variable of integration: Press   variable. This starts the  
calculation.  
Operations with Complex Numbers  
To enter a complex number:  
Form:   
1. Type the real part.  
2. Press6.  
3. Type the imaginary part.  
Form:   
1. Type the real part.  
2. Press.  
3. Type the imaginary part.  
4. Press 6.  
Form:   
1. Type the value of r.  
2. Press?.  
3. Type the value of θ .  
C-8  
   
To do an operation with one complex number:  
1. Select the function.  
2. Enter the complex number z.  
3. Press to calculate.  
4. The calculated result will be displayed in Line 2 and the displayed form will  
be the one that you have set in 9.  
To do an arithmetic operation with two complex numbers:  
1. Enter the first complex number, z .  
1
2. Select the arithmetic operation.  
3. Enter the second complex number, z .  
2
4. Press to calculate.  
5. The calculated result will be displayed in Line 2 and the displayed form will  
be the one that you have set in 9.  
Here are some examples with complex numbers:  
Examples:  
Evaluate sin (2 + 3i )  
Keys:  
Display:  
  
Description:  
Sets display form  
8( )  
6  
Result is  
9.1545 i–4.1689  
  
   
Examples:  
Evaluate the expression  
z
÷ (z + z ),  
1
2
3
where z = 23 + 13 i, z = –2 + i z = 4 –3 i  
1
2
3
C-9  
Keys:  
Display:  
Description:  
Sets display form  
8Ë  
( )  
4
   
6  
Õ4   
66  
  Result is  
  
2.5000 + 9.0000 i  
Examples:  
Evaluate (4 - 2/5 i) × (3 - 2/3 i)  
Keys:  
Display:  
Description:  
4  
   
6Õ4  
6  
    
   
Result is  
11.7333 i–3.8667  
Arithmetic in Bases 2, 8, and 16  
Here are some examples of arithmetic in Hexadecimal, Octal, and Binary modes:  
Example:  
12F + E9A  
= ?  
16  
16  
Keys:  
Display:  
Description:  
Sets base 16; HEX  
annunciator on.  
()  
 
Result.  
F  
()  
E9A  
()  
  
  
7760 – 4326 =?  
8
8
Sets base 8: OCT  
annunciator on.  
Converts displayed  
number to octal.  
  
()  
  
  
()  
  
  
  
()  
100 ÷ 5 =?  
8
8
  
()   
  
Integer part of result.  
  
()  
5A0 + 10011000 =?  
16  
2
Set base 16; HEX  
annunciator on.  
 ()  
A0  
()  
  
  
  
()  
Result in hexadecimal  
  
base.  
  
Restores decimal base.  
  
  
()  
Entering Statistical Two–Variable Data  
In ALG mode, remember to enter an (x, y) pair in reverse order (y x or y  
x ) so that y ends up in the Y–register and X in the X–register.  
1. Press   
(4Σ) to clear existing statistical data.  
2. Key in the y–value first and press .  
3. Key in the corresponding x–value and press .  
 
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.  
If you wish to delete the incorrect values that were just entered, press z 4.  
After deleting the incorrect statistical data, the calculator will display the last  
statistical data entered in line 1 (top line of the display) and value of n in line 2. If  
there are no statistical data, the calculator will display n=0 in line 2.  
Example:  
After keying in the x, y–values on the left, make the corrections shown on the right:  
Initial x, y  
Corrected x, y  
20, 4  
400, 6  
20, 5  
40, 6  
Keys:  
Display:  
Description:  
Clears existing statistical data.  
{
(4Σ)  
Zꢁꢂ6  
Enters the first new data pair.  
ꢀꢁ Σ-  
ꢂ)ꢁꢁꢁꢁ  
ꢃꢁꢁ Σ-  
ꢀ)ꢁꢁꢁꢁ  
ꢄꢅ !º  
Display shows n, the number of data  
pairs you entered.  
ꢃZ ꢂꢂ  
6
Brings back last x–value. Last y is still  
{ Ž  
ꢃꢁꢁ)ꢁꢁꢁꢁ  
in Y–register.  
ꢃꢁꢁ Σ.  
Deletes the last data pair.  
Reenters the last data pair.  
Deletes the first data pair.  
z 4  
ꢂ)ꢁꢁꢁꢁ  
ꢃꢁ Σ-  
ꢃZ ꢂ6  
ꢀ)ꢁꢁꢁꢁ  
ꢀꢁ Σ.  
Zꢁꢂz  
4
ꢂ)ꢁꢁꢁꢁ  
Reenters the first data pair. There is  
still a total of two data pairs in the  
statistics registers.  
ꢄZꢁꢂ6  
ꢀꢁ Σ-  
ꢀ)ꢁꢁꢁꢁ  
Linear Regression  
Linear regression, or L.R. (also called linear estimation), is a statistical method for  
finding a straight line that best fits a set of x,y–data.  
To find an estimated value for x (or y), key in a given hypothetical value for y  
(or x) ,press , then press   ( ˆ ) (or  Õ( ˆ )).  
To find the values that define the line that best fits your data, press    
followed by (), (), or ().  
D
More about Solving  
This appendix provides information about the SOLVE operation beyond that given in  
chapter 7.  
How SOLVE Finds a Root  
SOLVE first attempts to solve the equation directly for the unknown variable. If the  
attempt fails, SOLVE changes to an iterative(repetitive) procedure. The iterative  
operation is to execute repetitively the specified equation. The value returned by the  
equation is a function f(x) of the unknown variable x. (f(x) is mathematical shorthand  
for a function defined in terms of the unknown variable x.) SOLVE starts with an  
estimate for the unknown variable, x, and refines that estimate with each successive  
execution of the function, f(x).  
If any two successive estimates of the function f(x) have opposite signs, then SOLVE  
presumes that the function f(x) crosses the x–axis in at least one place between the  
two estimates. This interval is systematically narrowed until a root is found.  
For SOLVE to find a root, the root has to exist within the range of numbers of the  
calculator, and the function must be mathematically defined where the iterative  
search occurs. SOLVE always finds a root, provided one exists (within the overflow  
bounds), if one or more of these conditions are met:  
Two estimates yield f(x) values with opposite signs, and the function's graph  
crosses the x–axis in at least one place between those estimates (figure a,  
below).  
f(x) always increases or always decreases as x increases (figure b, below).  
The graph of f(x) is either concave everywhere or convex everywhere (figure  
c, below).  
D-1  
   
If f(x) has one or more local minima or minima, each occurs singly between  
adjacent roots of f(x) (figure d, below).  
f (x)  
f (x)  
x
x
b
a
f (x)  
f (x)  
x
x
d
Function Whose Roots Can Be Found  
In most situations, the calculated root is an accurate estimate of the theoretical,  
c
infinitely precise root of the equation. An "ideal" solution is one for which f(x) = 0.  
However, a very small non–zero value for f(x) is often acceptable because it might  
result from approximating numbers with limited (12–digit) precision.  
D-2  
Interpreting Results  
The SOLVE operation will produce a solution under either of the following  
conditions:  
If it finds an estimate for which f(x) equals zero. (See figure a, below.)  
If it finds an estimate where f(x) is not equal to zero, but the calculated root is  
a 12–digit number adjacent to the place where the function's graph crosses  
the x–axis (see figure b, below). This occurs when the two final estimates are  
neighbors (that is, they differ by 1 in the 12th digit), and the function's value  
is positive for one estimate and negative for the other. Or they are (0, 10  
499  
–499  
). In most cases, f(x) will be relatively close to zero.  
) or (0, –10  
To obtain additional information about the result, press see the previous  
estimate of the root (x), which was left in the Y–register. Press again to see the  
value of f(x), which was left in the Z–register. If f(x) equals zero or is relatively small,  
it is very likely that a solution has been found. However, if f(x) is relatively large, you  
must use caution in interpreting the results.  
Example: An Equation With One Root.  
Find the root of the equation:  
3
2
–2x + 4x – 6x + 8 = 0  
Enter the equation as an expression:  
D-3  
 
Keys:  
Display:  
Description:  
Select Equation mode.  
Enters the equation.  
  
X  
  
X  
X  
  
ꢆ  
  
Checksum and length.  
Cancels Equation mode.  
  
  
Now, solve the equation to find the root:  
Keys:  
Display:  
Description:  
_  
Initial guesses for the root.  
X  
  
Selects Equation mode; displays  
the left end of the equation.  
Solves for X; displays the result.  
ꢆ  
  
  
X  
  
Final two estimates are the same  
to four decimal places.  
f(x) is very small, so the  
  
  
approximation is a good root.  
Example: An Equation with Two Roots.  
Find the two roots of the parabolic equation:  
2
x + x – 6 = 0.  
Enter the equation as an expression:  
D-4  
Keys:  
Display:  
Description:  
Selects Equation mode.  
Enters the equation.  
X  
X  
  
Checksum and length.  
Cancels Equation mode.  
  
  
  
Now, solve the equation to find its positive and negative roots:  
Keys:  
Display:  
Description:  
Your initial guesses for the positive  
root.  
X  
  
_  
Selects Equation mode; displays  
the equation.  
  
Calculates the positive root using  
guesses 0 and 10.  
  
  
X  
  
  
Final two estimates are the same.  
f(x) = 0.  
   
  
Your initial guesses for the  
negative root.  
X  
  
_  
Redisplays the equation.  
  
Calculates negative root using  
guesses 0 and –10.  
  
  
X  
  
f(x) = 0.  
  
    
Certain cases require special consideration:  
If the function's graph has a discontinuity that crosses the x–axis, then the  
SOLVE operation returns a value adjacent to the discontinuity (see figure a,  
below). In this case, f(x) may be relatively large.  
D-5  
Values of f(x) may be approaching infinity at the location where the graph  
changes sign (see figure b, below). This situation is called a pole. Since the  
SOLVE operation determines that there is a sign change between two  
neighboring values of x, it returns the possible root. However, the value for  
f(x) will be relatively large. If the pole occurs at a value of x that is exactly  
represented with 12 digits, then that value would cause the calculation to halt  
with an error message.  
f (x)  
f (x)  
x
x
a
b
Special Case: A Discontinuity and a Pole  
Example: A Discontinuous Function.  
Find the root of the equation:  
IP(x) = 1.5  
Enter the equation:  
Keys:  
Display:  
Description:  
Selects Equation mode.  
Enter the equation.  
()  
  
  
  
  
  
  
Checksum and length.  
Cancels Equation mode.  
D-6  
Now, solve to find the root:  
Keys:  
Display:  
Description:  
Your initial guesses for the root.  
X  
_  
  
Selects Equation mode; displays  
the equation.  
  
Finds a root with guesses 0 and 5.  
  
  
X  
  
Shows root, to 11 decimal places.  
  
   
The previous estimate is slightly  
bigger.  
  
   
f(x) is relatively large.  
  
Note the difference between the last two estimates, as well as the relatively large  
value for f(x). The problem is that there is no value of x for which f(x) equals zero.  
However, at x = 1.99999999999, there is a neighboring value of x that yields an  
opposite sign for f(x).  
Example:  
Find the root of the equation  
x
1= 0  
x2 6  
As x approaches  
, f(x) becomes a very large positive or negative number.  
6
Enter the equation as an expression.  
Keys: Display:  
Description:  
Selects Equation mode.  
Enters the equation.  
X4  
X  
Õ  
  
  
D-7  
Checksum and length.  
Cancels Equation mode.  
  
  
  
Now, solve to find the root.  
Keys:  
  
Display:  
Description:  
Your initial guesses for the root.  
X  
_  
  
Selects Equation mode; displays  
the equation.  
  
No root found for f(x).  
X  
    
When SOLVE Cannot Find a Root  
Sometimes SOLVE fails to find a root. The following conditions cause the message  
  :  
The search terminates near a local minimum or maximum (see figure a,  
below).  
The search halts because SOLVE is working on a horizontal asymptote—an  
area where f(x) is essentially constant for a wide range of x (see figure b,  
below).  
The search is concentrated in a local "flat" region of the function (see figure c,  
below).  
In these cases, the values in the stack will be same as the values before executing  
SOLVE.  
D-8  
 
f (x)  
f (x)  
x
x
b
a
f (x)  
x
c
Case Where No Root Is Found  
Example: A Relative Minimum.  
Calculate the root of this parabolic equation:  
2
x – 6x + 13 = 0.  
It has a minimum at x = 3.  
Enter the equation as an expression:  
Keys:  
Display:  
Description:  
Selects Equation mode.  
Enters the equation.  
X  
X  
  
  
D-9  
Checksum and length.  
Cancels Equation mode.  
   
  
  
Now, solve to find the root:  
Keys:  
Display:  
Description:  
Your initial guesses for the root.  
X  
  
_  
Selects Equation mode; displays  
the equation.  
  
Search fails with guesses 0 and  
X  
    
10  
Example: An Asymptote.  
Find the root of the equation  
1
10 − = 0  
X
Enter the equation as an expression.  
Keys: Display:  
Description:  
Selects Equation mode.  
Enters the equation.  
  
X  
   
  
Checksum and length.  
  
  
Cancels Equation mode.  
Your positive guesses for the root.  
  
X  
_  
  
Selects Equation mode; displays  
the equation.  
Solves for x using guesses 0.005  
and 5.  
  
X  
  
Previous estimate is the same.  
f (x) = 0  
  
  
  
Watch what happens when you use negative values for guesses:  
Keys:  
Display:  
  
Description:  
Your negative guesses for the root.  
X  
  
Selects Equation mode; displays  
the equation.  
  
Solves for X; displays the result.  
  
X  
  
Example: Find the root of the equation.  
[x ÷ (x + 0.3)] 0.5 = 0  
Enter the equation as an expression:  
Keys: Display:  
Description:  
Selects Equation mode.  
Enters the equation.  
X4  
X  
ÕÕ   
ꢆ  
  
  
Checksum and length.  
Cancels Equation mode.  
  
First attempt to find a positive root:  
Keys:  
Display:  
Description:  
Your positive guesses for the  
root.  
X  
  
_  
Selects Equation mode;  
displays the left end of the  
equation.  
ꢆ  
Calculates the root using  
  
X  
  
guesses 0 and 10.  
Now attempt to find a negative root by entering guesses 0 and –10. Notice that the  
function is undefined for values of x between 0 and –0.3 since those values  
produce a positive denominator but a negative numerator, causing a negative  
square root.  
Keys:  
Display:  
Description:  
X  
  
_  
Selects Equation mode; displays  
the left end of the equation.  
No root found for f(x).  
ꢆ  
    
X  
Example: A Local "Flat" Region.  
Find the root of the function  
f(x) = x + 2 if x < –1,  
f(x) = 1 for –1 x 1 (a local flat region),  
f(x) = –x + 2 if x >1.  
In RPN mode, enter the function as the program:  
    
   
   
    
   
   
   
   
   
   
 ꢀ  
   
Checksum and length: 9412 39  
–8  
–8  
.
Solve for X using initial guesses of 10 and –10  
Keys:  
Display:  
Description:  
(In RPN mode)  
Enters guesses.  
  
X_  
    
Selects program "J" as the function.  
Solves for X; displays the result.  
  
  
  
J  
X  
Round–Off Error  
The limited (12–digit) precision of the calculator can cause errors due to rounding  
off, which adversely affect the iterative solutions of SOLVE and integration. For  
example,  
[( x +1) +1015]2 -1030 = 0  
has no roots because f(x) is always greater than zero. However, given initial guesses  
of 1 and 2, SOLVE returns the answer 1.0000 due to round–off error.  
Round–off error can also cause SOLVE to fail to find a root. The equation  
x2 - 7 = 0  
has a root at  
. However, no 12–digit number exactly equals  
, so the  
7
7
calculator can never make the function equal to zero. Furthermore, the function  
never changes sign SOLVE returns the message   .  
 
E
More about Integration  
This appendix provides information about integration beyond that given in chapter  
8.  
How the Integral Is Evaluated  
The algorithm used by the integration operation,  , calculates the integral of  
a function f(x) by computing a weighted average of the function's values at many  
values of x (known as sample points) within the interval of integration. The accuracy  
of the result of any such sampling process depends on the number of sample points  
considered: generally, the more sample points, the greater the accuracy. If f(x) could  
be evaluated at an infinite number of sample points, the algorithm could —  
neglecting the limitation imposed by the inaccuracy in the calculated function f(x) —  
always provide an exact answer.  
Evaluating the function at an infinite number of sample points would take forever.  
However, this is not necessary since the maximum accuracy of the calculated  
integral is limited by the accuracy of the calculated function values. Using only a  
finite number of sample points, the algorithm can calculate an integral that is as  
accurate as is justified considering the inherent uncertainty in f(x).  
The integration algorithm at first considers only a few sample points, yielding  
relatively inaccurate approximations. If these approximations are not yet as accurate  
as the accuracy of f(x) would permit, the algorithm is iterated (repeated) with a  
larger number of sample points. These iterations continue, using about twice as  
many sample points each time, until the resulting approximation is as accurate as is  
justified considering the inherent uncertainty in f(x).  
E-1  
   
As explained in chapter 8, the uncertainty of the final approximation is a number  
derived from the display format, which specifies the uncertainty for the function. At  
the end of each iteration, the algorithm compares the approximation calculated  
during that iteration with the approximations calculated during two previous  
iterations. If the difference between any of these three approximations and the other  
two is less than the uncertainty tolerable in the final approximation, the calculation  
ends, leaving the current approximation in the X–register and its uncertainty in the  
Y–register.  
It is extremely unlikely that the errors in each of three successive approximations —  
that is, the differences between the actual integral and the approximations — would  
all be larger than the disparity among the approximations themselves.  
Consequently, the error in the final approximation will be less than its uncertainty  
(provided that f(x) does not vary rapidly). Although we can't know the error in the  
final approximation, the error is extremely unlikely to exceed the displayed  
uncertainty of the approximation. In other words, the uncertainty estimate in the Y–  
register is an almost certain "upper bound" on the difference between the  
approximation and the actual integral.  
Conditions That Could Cause Incorrect Results  
Although the integration algorithm in the HP 35s is one of the best available, in  
certain situations it — like all other algorithms for numerical integration — might  
give you an incorrect answer. The possibility of this occurring is extremely remote.  
The algorithm has been designed to give accurate results with almost any smooth  
function. Only for functions that exhibit extremely erratic behavior is there any  
substantial risk of obtaining an inaccurate answer. Such functions rarely occur in  
problems related to actual physical situations; when they do, they usually can be  
recognized and dealt with in a straightforward manner.  
Unfortunately, since all that the algorithm knows about f(x) are its values at the  
sample points, it cannot distinguish between f(x) and any other function that agrees  
with f(x) at all the sample points. This situation is depicted below, showing (over a  
portion of the interval of integration) three functions whose graphs include the many  
sample points in common.  
E-2  
 
f (x)  
x
With this number of sample points, the algorithm will calculate the same  
approximation for the integral of any of the functions shown. The actual integrals of  
the functions shown with solid blue and black lines are about the same, so the  
approximation will be fairly accurate if f(x) is one of these functions. However, the  
actual integral of the function shown with a dashed line is quite different from those  
of the others, so the current approximation will be rather inaccurate if f(x) is this  
function.  
The algorithm comes to know the general behavior of the function by sampling the  
function at more and more points. If a fluctuation of the function in one region is not  
unlike the behavior over the rest of the interval of integration, at some iteration the  
algorithm will likely detect the fluctuation. When this happens, the number of  
sample points is increased until successive iterations yield approximations that take  
into account the presence of the most rapid, but characteristic, fluctuations.  
For example, consider the approximation of  
xexdx.  
0
Since you're evaluating this integral numerically, you might think that you should  
499  
represent the upper limit of integration as 10 , which is virtually the largest  
number you can key into the calculator.  
E-3  
x  
Try it and see what happens. Enter the function f(x) = xe .  
Keys: Display:  
Description:  
Select equation mode.  
Enter the equation.  
X   
 X  
   
  
End of the equation.  
Checksum and length.  
  
  
  
Cancels Equation mode.  
Set the display format to SCI 3, specify the lower and upper limits of integration as  
499  
zero and 10 , than start the integration.  
Keys:  
Display:  
Description:  
Specifies accuracy level  
and limits of integration.  
8(2)  
  
  
_  
Selects Equation mode;  
displays the equation.  
Approximation of the  
integral.  
  
  
  
  
X  
The answer returned by the calculator is clearly incorrect, since the actual integral of  
x  
f(x) = xe from zero to is exactly 1. But the problem is not that was  
499  
499  
is  
represented by 10 , since the actual integral of this function from zero to 10  
very close to 1. The reason for the incorrect answer becomes apparent from the  
graph of f(x) over the interval of integration.  
E-4  
f (x)  
x
The graph is a spike very close to the origin. Because no sample point happened to  
discover the spike, the algorithm assumed that f(x) was identically equal to zero  
throughout the interval of integration. Even if you increased the number of sample  
points by calculating the integral in SCI 11 or ALL format, none of the additional  
sample points would discover the spike when this particular function is integrated  
over this particular interval. (For better approaches to problems such as this, see the  
next topic, "Conditions That Prolong Calculation Time.")  
Fortunately, functions exhibiting such aberrations (a fluctuation that is  
uncharacteristic of the behavior of the function elsewhere) are unusual enough that  
you are unlikely to have to integrate one unknowingly. A function that could lead to  
incorrect results can be identified in simple terms by how rapidly it and its low–order  
derivatives vary across the interval of integration. Basically, the more rapid the  
variation in the function or its derivatives, and the lower the order of such rapidly  
varying derivatives, the less quickly will the calculation finish, and the less reliable  
will be the resulting approximation.  
E-5  
Note that the rapidity of variation in the function (or its low–order derivatives) must  
be determined with respect to the width of the interval of integration. With a given  
number of sample points, a function f(x) that has three fluctuations can be better  
characterized by its samples when these variations are spread out over most of the  
interval of integration than if they are confined to only a small fraction of the  
interval. (These two situations are shown in the following two illustrations.)  
Considering the variations or fluctuation as a type of oscillation in the function, the  
criterion of interest is the ratio of the period of the oscillations to the width of the  
interval of integration: the larger this ratio, the more quickly the calculation will  
finish, and the more reliable will be the resulting approximation.  
f (x)  
Calculated integral  
of this function  
will be accurate.  
x
a
b
f (x)  
Calculated integral  
of this function  
may be inaccurate.  
x
a
b
E-6  
In many cases you will be familiar enough with the function you want to integrate  
that you will know whether the function has any quick wiggles relative to the interval  
of integration. If you're not familiar with the function, and you suspect that it may  
cause problems, you can quickly plot a few points by evaluating the function using  
the equation or program you wrote for that purpose.  
If, for any reason, after obtaining an approximation to an integral, you suspect its  
validity, there's a simple procedure to verify it: subdivide the interval of integration  
into two or more adjacent subintervals, integrate the function over each subinterval,  
then add the resulting approximations. This causes the function to be sampled at a  
brand new set of sample points, thereby more likely revealing any previously hidden  
spikes. If the initial approximation was valid, it will equal the sum of the  
approximations over the subintervals.  
Conditions That Prolong Calculation Time  
In the preceding example, the algorithm gave an incorrect answer because it never  
detected the spike in the function. This happened because the variation in the  
function was too quick relative to the width of the interval of integration. If the width  
of the interval were smaller, you would get the correct answer; but it would take a  
very long time if the interval were still too wide.  
Consider an integral where the interval of integration is wide enough to require  
excessive calculation time, but not so wide that it would be calculated incorrectly.  
x  
Note that because f(x) = xe approaches zero very quickly as x approaches , the  
contribution to the integral of the function at large values of x is negligible.  
Therefore, you can evaluate the integral by replacing , the upper limit of  
499  
3
integration, by a number not so large as 10  
— say 10 .  
Rerun the previous integration problem with this new limit of integration:  
Keys:  
Display:  
Description:  
New upper limit.  
_  
  
  
Selects Equation mode; displays the  
equation.  
E-7  
 
  
  
  
Integral. (The calculation takes a  
minute or two.)  
X  
Uncertainty of approximation.  
  
This is the correct answer, but it took a very long time. To understand why, compare  
3
the graph of the function between x = 0 and x = 10 , which looks about the same  
as that shown in the previous example, with the graph of the function between x = 0  
and x = 10:  
f (x)  
x
0
10  
You can see that this function is "interesting" only at small values of x. At greater  
values of x, the function is not interesting, since it decreases smoothly and gradually  
in a predictable manner.  
The algorithm samples the function with higher densities of sample points until the  
disparity between successive approximations becomes sufficiently small. For a  
narrow interval in an area where the function is interesting, it takes less time to  
reach this critical density.  
To achieve the same density of sample points, the total number of sample points  
required over the larger interval is much greater than the number required over the  
smaller interval. Consequently, several more iterations are required over the larger  
interval to achieve an approximation with the same accuracy, and therefore  
calculating the integral requires considerably more time.  
E-8  
Because the calculation time depends on how soon a certain density of sample  
points is achieved in the region where the function is interesting, the calculation of  
the integral of any function will be prolonged if the interval of integration includes  
mostly regions where the function is not interesting. Fortunately, if you must calculate  
such an integral, you can modify the problem so that the calculation time is  
considerably reduced. Two such techniques are subdividing the interval of  
integration and transformation of variables. These methods enable you to change  
the function or the limits of integration so that the integrand is better behaved over  
the interval(s) of integration.  
E-9  
F
Messages  
The calculator responds to certain conditions or keystrokes by displaying a  
message. The symbol comes on to call your attention to the message. For  
significant conditions, the message remains until you clear it. Pressing or   
clears the message and the previous display content will be shown. Pressing any  
other key clears the message but the function of the key will not be executed.  
   
A running program attempted to select a program label  
(label) while an integration calculation was running.  
  
A running program attempted to integrate a program  
(  variable) while another integration calculation  
was running.  
  
A running program attempted to solve a program while  
an integration calculation was running.  
   
The catalog of variables (   () )  
indicates no values stored.  
   
You set a wrong guess number (like a complex number  
or vector) when SOLVING equation for a variable.  
  
The calculator is executing a function that might take a  
while.  
     
     
Allow you to verify clearing everything in memory.  
Allows you to verily clearing the equation you are  
editing. (Occurs only in Equation–entry mode.)  
     
    
Allows you to verify clearing all programs in memory.  
(Occurs only in Program–entry mode.)  
Attempted to divide by zero. (Includes  if Y–  
register contains zero.)  
  
Attempted to enter a program label that already exists  
for another program routine.  
F-1  
 
    
Indicates the "top" of equation memory. The memory  
scheme is circular, so    is also the  
"equation" after the last equation in equation memory.  
  
  
The calculator is calculating the integral of an equation  
or program. This might take a while.  
A running CALCULATE,SOLVE or FN operation was  
interrupted by pressing or in ALG, RPN, EQN,  
or PGM mode.  
   
Data error:  
Attempted to save or calculate error data.  
Attempted to calculate combinations or permutations  
16  
with r >n, with non–integer r or n, or with n 10  
.
Attempted to save a complex number or vector in  
the statistical data.  
Attempted to save a base-n number that contains  
digits greater than the largest base-n number digit  
allowed.  
Attempted to save an invalid data in the statistical  
register using operation.  
Attempt to compare complex numbers or vectors.  
Attempted to use a trigonometric or hyperbolic  
function with an illegal argument:  
°
with x an odd multiple of 90 .  
 or  with x < –1 or x > 1.  
  with x –1; or x 1.  
 with x < 1.  
   
   
Attempted to enter an invalid variable name when  
solving an equation.  
Attempted a factorial or gamma operation with x as a  
negative integer.  
F-2  
Exponentiation error:  
   
th  
Attempted to raise 0 to the 0 power or to a  
negative power.  
Attempted to raise a negative number to a non–  
integer power.  
Attempted to raise complex number (0 + i 0) to a  
number with a negative real part.  
   
   
Attempted an operation with an invalid indirect value ((I)  
is not defined).  
Attempted an operation with an invalid indirect value ((J)  
is not defined).  
  
Attempted to take a logarithm of zero or (0 + i0).  
Attempted to take a logarithm of a negative number.  
All of user memory has been erased (see page ).  
  
   
   
The calculator has insufficient memory available to do  
the operation (See appendix B).  
  
The condition checked by a test instruction is not true.  
(Occurs only when executed from the keyboard.)  
  
Attempted to refer to a nonexistent program label (or line  
number) with ,, or . Note that the error  
 can mean  
you explicitly (from the keyboard) called a program  
label that does not exist; or  
the program that you called referred to another  
label, which does not exist.  
The result of integration does not exist.  
   
The catalog of programs (   () )  
indicates no program labels stored.  
   
No solution could be found for this system of linear  
equations.  
   
Multiple solutions have been found for this system of  
linear equations.  
F-3  
    
  
   
SOLVE (include EQN and PGM mode)cannot find the  
root of the equation using the current initial guesses (see  
page ). These conditions include: bad guess, solution  
not found, point of interest, left unequal to right. A  
SOLVE operation executed in a program does not  
produce this error; the same condition causes it instead  
to skip the next program line (the line following the  
instruction  variable).  
Warning (displayed momentarily); the magnitude of a  
result is too large for the calculator to handle. The  
calculator returns 9.99999999999E499 in the current  
display format. (See "Range of Numbers and Overflow"  
on page .) This condition sets flag 6. If flag 5 is set,  
overflow has the added effect of halting a running  
program and leaving the message in the display until  
you press a key.  
Indicates the "top" of program memory. The memory  
scheme is circular, so   is also the "line" after  
the last line in program memory.  
The calculator is running an equation or program (other  
   
than a SOLVE or FN routine).  
   
Attempted to execute  variable or  d variable  
without a selected program label. This can happen the  
first time that you use SOLVE or FN after the message  
 , or it can happen if the current label  
no longer exists.  
   
  
  
  
A running program attempted to select a program label  
(label) while a SOLVE operation was running.  
A running program attempted to solve a program while  
a SOLVE operation was running.  
A running program attempted to integrate a program  
while a SOLVE operation was running.  
The calculator is solving an equation or program for its  
root. This might take a while.  
  
Attempted to calculate the square root of a negative  
number.  
F-4  
   
Statistics error:  
Attempted to do a statistics calculation with n = 0.  
ˆ
y
Attempted to calculate s s ,  
= 1.  
,
, m, r, or b with n  
ˆ
x
x y  
Attempted to calculate r,  
or  
with x–data only  
ˆ
xw  
x
(all y–values equal to zero).  
ˆ
y
Attempted to calculate  
values equal.  
,
, r, m, or b with all x–  
ˆ
x
  
  
A syntax error was detected during evaluation of an  
expression, equation,, or ". Pressing or  
clears the error message and allows you to correct  
the error.  
   
The magnitude of the number is too large to be  
converted to HEX, OCT, or BIN base; the number must  
be in the range  
–34,359,738,368 n 34,359,738,367.  
st  
   
A running program attempted an 21 nested  label.  
(Up to 20 subroutines can be nested.) Since SOLVE and  
FN each uses a level, they can also generate this error.  
  
The condition checked by a test instruction is true.  
(Occurs only when executed from the keyboard.)  
Self–Test Messages:  
  
The self–test and the keyboard test passed.  
The self–test or the keyboard test failed, and the  
calculator requires service.  
 n  
©       Copyright message displayed after successfully  
completing the self–test.  
F-5  
G
Operation Index  
This section is a quick reference for all functions and operations and their formulas,  
where appropriate. The listing is in alphabetical order by the function's name. This  
name is the one used in program lines. For example, the function named FIX n is  
executed as 8(1) n.  
Nonprogrammable functions have their names in key boxes. For example, .  
Non–letter and Greek characters are alphabetized before all the letters; function  
names preceded by arrows (for example, DEG) are alphabetized as if the arrow  
were not there.  
The last column, marked , refers to notes at the end of the table.  
Name  
Keys and Description  
Page  
1–15  
1
+/–  
Changes the sign of a number.  
Addition. Returns y + x.  
1–19  
1–19  
1–19  
1–19  
6–16  
1
1
1
1
1
+
Subtraction. Returns y x.  
Multiplication. Returns y × x.  
Division. Returns y ÷ x.  
×
÷
Power. Indicates an exponent.  
^
Deletes the last digit keyed in; clears  
x; clears a menu; erases last function  
keyed in an equation; deletes an  
equation; deletes a program step.  
1–4  
1–8  
6–3  
13–7  
1–28  
6–3  
13–11  
13–20  
Displays previous entry in catalog;  
moves to previous equation in  
equation list; moves program pointer  
to previous step.  
×
 
Name  
Keys and Description  
Page  
Displays next entry in catalog; moves  
to next equation in equation list;  
moves program pointer to next line  
(during program entry); executes the  
current program line (not during  
program entry).  
1–28  
6–3  
13–11  
13–20  
Ø
Moves the cursor and does not  
delete any content.  
1–14  
ÖorÕ  
Scrolls the display to show more  
digits to the left and right; displays  
the rest of an equation or binary  
number; goes the next menu page in  
the CONST and SUMS menus.  
1–11  
6–4  
11–8  
Öor Õ  
Goes to the top line of the equation  
or the first line of the last label in  
program mode.  
Goes to the last line of the equation  
or the first line of the next label in  
program mode.  
6–3  
6–3  
6–5  
×  
Ø  
,
1
Separates the two or three  
arguments of a function.  
1/x  
1–18  
4–2  
1
1
Reciprocal.  
x
10  
  Common exponential.  
Returns 10 raised to the × power.  
%
4–6  
4–6  
1
1
1
 Percent.  
Returns (y × x) ÷ 100.  
%CHG  
π
  Percent change.  
Returns (x – y)(100 ÷ y).  
4–3  
  Returns the approximation  
3.14159265359 (12 digits).  
Σ+  
Σ–  
12–2  
12–2  
12–11  
Accumulates (y, x) into statistics  
registers.  
  Removes (y, x) from  
statistics registers.  
Σx  
1
 Õ()  
Returns the sum of x–values.  
G-2  
Name  
Keys and Description  
Page  
2
Σx  
12–11  
1
 ÕÕÕ( )  
Returns the sum of squares of x–  
values.  
Σxy  
12–11  
1
ÕÕÕÕÕ  
()  
Returns the sum of products of x–and  
y–values.  
Σy  
12–11  
12–11  
1
1
 ÕÕ()  
Returns the sum of y–values.  
2
 ÕÕÕÕ( )  
Returns the sum of squares of y–  
values.  
Σy  
σx  
12–7  
12–7  
1
1
 ÕÕ(σ)  
Returns population standard  
deviation of x–values:  
(x x)2 ÷ n  
i
σy  
 ÕÕÕ(σ)  
Returns population standard  
deviation of y–values:  
(y y)2 ÷ n  
i
FN d variable  
8–2  
15–7  
 (   _) variable  
Integrates the displayed equation or  
the program selected by FN=, using  
lower limit of the variable of  
integration in the Y–register and  
upper limit of the variable of  
integration in the X–register.  
( )  
6–6  
1
4parenthesis. press Õto leave  
the parenthesis for further  
calculation.  
[ ]  
10–1  
9–1  
1
1
3: A vector symbol for  
performing vector operations  
θ
?: A complex number symbol  
for performing complex number  
operations  
Name  
A through Z  
Keys and Description  
Page  
6–4  
1
variable Value of named  
variable.  
ABS  
4–17  
1
  Absolute value.  
Returns  
.
x
ACOS  
4–4  
4–6  
1
1
  Arc cosine.  
Returns cos x.  
–1  
ACOSH  
    
Hyperbolic arc cosine.  
–1  
Returns cosh  
x.  
Activates Algebraic mode.  
1–9  
9()  
ALOG  
6–16  
1
  Common exponential.  
Returns 10 raised to the specified  
power (antilogarithm).  
ALL  
1–23  
8()  
Displays all significant digits. May  
have to scroll right  
(Õ) to see all of the digits.  
AND  
ARG  
11–4  
4–17  
1
1
>(1AND)  
Logic operator  
=  
Replaces a complex number with its  
Argument ”θ”  
ASIN  
4–4  
4–6  
1
1
 Arc sine  
–1  
Returns sin  
x.  
ASINH  
    
Hyperbolic arc sine.  
–1  
Returns sinh  
x.  
ATAN  
4–4  
4–6  
1
1
  Arc tangent.  
–1  
Returns tan  
x.  
ATANH  
    
Hyperbolic arc tangent.  
–1  
Returns tanh  
x.  
b
12–11  
1
 ÕÕÕÕ()  
Returns the y–intercept of the  
x
regression line:  
m  
.
y
G-4  
Name  
Keys and Description  
  ()  
Page  
b
11–2  
1
Indicates a binary number  
Displays the base–conversion menu.  
11–1  
11–1  
   
BIN  
 ()  
Selects Binary (base 2) mode.  
Turns on calculator; clears x; clears  
messages and prompts; cancels  
menus; cancels catalogs; cancels  
equation entry; cancels program  
entry; halts execution of an equation;  
halts a running program.  
1–1  
1–4  
1–8  
1–29  
6–3  
13–7  
13–19  
/c  
5–4  
  Denominator.  
Sets denominator limit for displayed  
fractions to x. If x = 1, displays  
current /c value.  
4–14  
1
  Converts ° F to ° C.  
°C  
CF n  
14–12  
  () n  
Clears flag n (n = 0 through 11).  
Displays menu to clear numbers or  
parts of memory; clears indicated  
variable or program from a MEM  
catalog; clears displayed equation;  
1–5  
1–28  
Clears all stored data, equations,  
and programs.  
1–29  
13–23  
13–7  
12–1  
3–6  
()  
Clears all programs (calculator in  
Program mode).  
()  
Clears the displayed equation  
(calculator in Equation mode).  
()  
CLΣ  
(4)  
Clears statistics registers.  
CLVARS  
CLx  
()  
Clears all variables to zero.  
2–3  
2–7  
13–7  
()  
Clears x (the X–register) to zero.  
Name  
Keys and Description  
Page  
CLVARx  
CLSTK  
1–4  
()  
Clears indirect variables whose  
address is greater than the x address  
to zero.  
2–7  
()  
Clears all stack levels to zero.  
4–14  
4–15  
1
1
  Converts inches to  
centimeters.  
CM  
nCr  
xCombinations of n items  
taken r at a time.  
Returns n! ÷ (r! (n – r)!).  
COS  
4–3  
4–6  
1
1
Cosine.  
Returns cos x.  
COSH  
   Hyperbolic  
cosine. Returns cosh x.  
Accesses the 41 physics constants.  
4–8  
  
d
11–1  
1
  ()  
indicates a decimal number  
DEC  
DEG  
11–1  
4–4  
 ()  
Selects Decimal mode.  
9 ()  
Selects Degrees angular mode.  
4–13  
1–21  
1
 Radians to degrees.  
Returns (360/2π) x.  
DEG  
Displays menu to set the display  
format, radix (or ), thousand  
separator, and display format of  
complex number.  
8  
DSE variable  
14–18  
1–15  
  variable  
Decrement, Skip if Equal or less. For  
control number ccccccc.fffii stored in  
a variable, subtracts ii (increment  
value) from ccccccc (counter value)  
and, if the result fff (final value),  
skips the next program line.  
Begins entry of exponents and adds  
"E" to the number being entered.  
Indicates that a power of 10 follows.  
1
G-6  
Name  
Keys and Description  
Page  
ENG n  
1–22  
8()n  
Selects Engineering display with n  
digits following the first digit (n = 0  
through 11).  
Causes the exponent display for the  
number being displayed to change  
in multiple of 3.  
1–22  
@and2  
Separates two numbers keyed in  
sequentially; completes equation  
entry; evaluates the displayed  
equation (and stores result if  
appropriate).  
1–19  
6–4  
6–11  
ENTER  
2–6  
Copies x into the Y–register, lifts y  
into the Z–register, lifts z into the T–  
register, and loses t.  
Activates or cancels (toggles)  
Equation–entry mode.  
6–3  
13–7  
x
4–1  
1
1
e
 Natural exponential.  
Returns e raised to the x power.  
EXP  
6–16  
 Natural exponential.  
Returns e raised to the specified  
power.  
4–14  
5–1  
1
  Converts °C to °F.  
°F  
   
Turns on and off Fraction–display  
mode.  
FIX n  
1–21  
8 () n  
Selects Fixed display with n decimal  
places: 0 n 11.  
Displays the menu to set, clear, and  
test flags.  
14–12  
   
FN = label  
15–1  
15–7  
  label  
Selects labeled program as the  
current function (used by SOLVE and  
FN).  
FP  
4–17  
1
() Fractional part  
of x.  
Name  
Keys and Description  
Page  
FS? n  
14–12  
 () n  
If flag n (n = 0 through 11) is set,  
executes the next program line; if  
flag n is clear, skips the next  
program line.  
4–14  
4–4  
1
  Converts liters to gallons.  
GAL  
GRAD  
9()Sets Grads  
angular mode.  
Sets program pointer to line nnn of  
program label.  
13–21  
 label nnn  
Sets program pointer to PRGM TOP.  
13–21  
11–1  
   
h
1
  ()  
Indicates a hexadecimal number  
HEX  
11–1  
 ()  
Selects Hexadecimal (base 16)  
mode.  
Displays the HYP_ prefix for  
hyperbolic functions.  
4–6  
   
4–13  
1
1
   
HMS  
Hours to hours, minutes, seconds.  
Converts x from a decimal fraction to  
hours–minutes–seconds format.  
4–13  
9–2  
5  
HMSꢇ  
Hours, minutes, seconds to hours.  
Converts x from hours–minutes–  
seconds format to a decimal fraction.  
Used for entering complex numbers  
1
1
6
(I)/(J)  
6–4  
14–21  
7 /A,7  
/A.  
Value of variable whose letter  
corresponds to the numeric value  
stored in variable I/J.  
4–14  
6–16  
1
1
  Converts centimeters to  
inches.  
IN  
IDIV  
(÷) Produces  
the quotient of a division operation  
involving two integers.  
G-8  
Name  
Keys and Description  
Page  
INT÷  
INTG  
4–2  
1
(÷) Produces  
the quotient of a division operation  
involving two integers.  
4–18  
1
() Obtains the  
greatest integer equal to or less than  
given number.  
INPUT variable  
13–13  
  variable  
Recalls the variable to the X–register,  
displays the variable's name and  
value, and halts program execution.  
Pressing (to resume program  
execution) or Ø(to execute the  
current program line) stores your  
input in the variable. (Used only in  
programs.)  
INV  
IP  
6–16  
4–17  
1
1
Reciprocal of argument.  
() Integer part of  
x.  
ISG variable  
14–18  
  variable  
Increment, Skip if Greater.  
For control number ccccccc.fffii  
stored in variable, adds ii (increment  
value) to ccccccc (counter value)  
and, if the result > fff (final value),  
skips the next program line.  
4–14  
4–14  
1
1
1
  Converts pounds to  
kilograms.  
KG  
<Converts miles to.  
kilometers  
KM  
4–14  
2–8  
  Converts gallons to liters.  
L  
LASTx  
   
Returns number stored in the LAST X  
register.  
4–14  
1
   
Converts kilograms to pounds.  
LB  
Name  
LBL label  
Keys and Description  
  label  
Page  
13–3  
Labels a program with a single letter  
for reference by the XEQ, GTO, or  
FN= operations. (Used only in  
programs.)  
LN  
4–1  
4–1  
1
1
 Natural logarithm.  
Returns log x.  
e
LOG  
  Common logarithm.  
Returns log x.  
10  
Displays menu for linear regression.  
12–4  
12–7  
   
m
1
1
 ÕÕÕ()  
Returns the slope of the regression  
2
line: [Σ(x – )(y – )]÷Σ(x –  
)
y
x
x
i
j
i
4–14  
;Converts kilometers to  
miles.  
MILE  
Displays the amount of available  
memory and the catalog menu.  
1–28  
   
Begins catalog of programs.  
13–22  
   
(2)  
Begins catalog of variables.  
3–4  
   
(1)  
Displays menu to set ALG or RPN  
mode or angular modes.  
1–7  
4–4  
9
n
12–11  
1
 ()  
Returns the number of sets of data  
points.  
NAND  
NOR  
NOT  
o
11–4  
11–4  
11–4  
11–2  
11–1  
1
1
1
1
>()  
Logic operator  
>()  
Logic operator  
>()  
Logic operator  
  ()  
Indicates an octal number  
OCT  
  ()  
Selects Octal (base 8) mode.  
Name  
Keys and Description  
>()  
Page  
OR  
11–4  
1
Logic operator  
Turns the calculator off.  
1–1  
   
nPr  
4–15  
1
{Permutations of n items  
taken r at a time. Returns n!÷(n r)!.  
Activates or cancels (toggles)  
Program–entry mode.  
13–6  
   
PSE  
13–18  
13–19  
  Pause.  
Halts program execution briefly to  
display x, variable, or equation, then  
resumes. (Used only in programs.)  
r
12–7  
1
 ÕÕ() Returns the  
correlation coefficient between the x–  
and y–values:  
(x x)(y y)  
i
i
(x x)2 ×(y y)2  
i
i
1–25  
rθ a  
8 ()  
Changes the display of complex  
numbers.  
RAD  
4–4  
4–13  
1–23  
9()  
Selects Radians angular mode.  
1
1
 Degrees to radians.  
Returns (2π/360) x.  
RAD  
RADIX ,  
8(6)  
Selects the comma as the radix mark  
(decimal point).  
RADIX .  
1–23  
4–15  
3–7  
8()  
Selects the period as the radix mark  
(decimal point).  
RANDOM  
RCL variable  
  Executes the RANDOM  
function. Returns a random number  
in the range 0 through 1.  
variable  
Recall.  
Copies variable into the X–register.  
Name  
Keys and Description  
  variable  
Page  
RCL+ variable  
3–7  
Returns x + variable.  
RCL– variable  
RCLx variable  
RCL÷ variable  
RMDR  
3–7  
3–7  
  variable.  
Returns x – variable.  
  variable.  
Returns x × variable.  
3–7  
 variable.  
Returns x ÷ variable.  
6–16  
1
1
() Produces  
the remainder of a division operation  
involving two integers.  
RND  
4–18  
5–8  
 Round.  
Rounds x to n decimal places in FIX n  
display mode; to n + 1 significant  
digits in SCI n or ENG n display  
mode; or to decimal number closest  
to displayed fraction in Fraction–  
display mode.  
1–9  
9()Activates Reverse  
Polish notation.  
RTN  
13–4  
14–1  
 Return.  
Marks the end of a program; the  
program pointer returns to the top or  
to the calling routine.  
Rꢀ  
2–3  
C–7  
Roll down.  
Moves t to the Z–register, z to the Y–  
register, y to the X–register, and x to  
the T–register in RPN mode.  
Displays the X,Y,Z,T menu to review  
the stack in ALG mode.  
Rꢁ  
2–3  
C–7  
  Roll up.  
Moves t to the X–register, z to the T–  
register, y to the Z–register, and x to  
the Y–register in RPN mode.  
Displays the X,Y,Z,T menu to review  
the stack in ALG mode.  
Displays the standard–deviation  
Menu.  
12–4  
   
Name  
Keys and Description  
Page  
SCI n  
SEED  
1–22  
8() n  
Selects Scientific display with n  
decimal places. (n = 0 through 11.)  
4–15  
 . Restarts the random–  
x
number sequence with the seed  
.
SF n  
14–12  
4–17  
 () n  
Sets flag n (n = 0 through 11).  
SGN  
1
() Indicates the  
sign of x.  
Shows the full mantissa (all 12 digits)  
of x (or the number in the current  
program line); displays hex  
6–19  
13–23  
   
checksum and decimal byte length  
for equations and programs.  
SIN  
4–3  
4–6  
1
1
Sine.  
Returns sin x.  
SINH  
   Hyperbolic sine.  
Returns sinh x.  
7–1  
15–1  
SOLVE variable  
 variable  
Solves the displayed equation or the  
program selected by FN=, using  
initial estimates in variable and x.  
Inserts a blank space character  
during equation entry.  
14–14  
1
  
SQ  
6–16  
6–16  
3–2  
1
1
 Square of argument.  
SQRT  
Square root of x.  
STO variable  
 variable  
Store. Copies x into variable.  
STO + variable  
STO – variable  
STO × variable  
STO ÷ variable  
3–6  
3–6  
3–6  
3–6  
  variable  
Stores variable + x into variable.  
  variable  
Stores variable – x into variable.  
  variable  
Stores variable × x into variable.  
  variable  
Stores variable ÷ x into variable.  
Name  
Keys and Description  
Run/stop.  
Page  
STOP  
13–19  
Begins program execution at the  
current program line; stops a running  
program and displays the X–register.  
Displays the summation menu.  
12–4  
12–6  
   
sx  
1
1
  ()  
Returns sample standard deviation of  
x–values:  
(x x)2 ÷ (n 1)  
i
sy  
12–6  
 Õ()  
Returns sample standard deviation of  
y–values:  
(y y)2 ÷ (n 1)  
i
TAN  
4–3  
4–6  
1
1
Tangent. Returns tan x.  
TANH  
   Hyperbolic  
tangent.  
Returns tanh x.  
VIEW variable  
3–4  
13–15  
  variable  
Displays the labeled contents of  
variable without recalling the value  
to the stack.  
Evaluates the displayed equation.  
6–12  
14–1  
XEQ label  
label  
Executes the program identified by  
label.  
2
x
4–2  
4–2  
1
1
 Square of x.  
Square root of x.  
x
th  
4–2  
1
1
 The x root of y.  
X
y
12–4  
(  
)
x
Returns the mean of x values:  
Σ x ÷ n.  
i
Name  
Keys and Description  
  (  
Given a y–value in the X–register,  
returns the xestimate based on the  
Page  
12–11  
1
)
ˆ
x
ˆ
regression line:  
= (y – b) ÷ m.  
ˆ
x
!
4–15  
1
*Factorial (or gamma).  
Returns (x)(x – 1) ... (2)(1), or Γ (x +  
1).  
XROOT  
6–16  
12–4  
1
1
 The argument root of  
1
argument .  
2
ÕÕ( w )Returns  
w
x
weighted mean of x values: (Σyx ) ÷  
i i  
Σy.  
i
Displays the mean (arithmetic  
average) menu.  
12–4  
3–8  
   
x<> variable  
  x exchange.  
Exchanges x with a variable.  
x<>y  
2–4  
x exchange y.  
Moves x to the Y–register and y to  
the X–register.  
Displays the "x?y" comparison tests  
14–7  
14–7  
   
menu.  
xy  
  ()  
If xy, executes next program line;  
if x=y, skips next program line.  
xy?  
x<y?  
x>y?  
xy?  
14–7  
14–7  
14–7  
14–7  
 Õ()  
If xy, executes next program line;  
if x>y, skips next program line.  
 ÕÕ(<)  
If x<y, executes next program line;  
if xy, skips next program line.  
 ÕÕÕ (>)  
If x>y, executes next program line;  
if xy, skips next program line.  
 ÕÕÕÕ()  
If xy, executes next program line;  
if x<y, skips next program line.  
Name  
Keys and Description  
Page  
x=y?  
14–7  
 ÕÕÕÕÕ()  
If x=y, executes next program line;  
if xy, skips next program line.  
Displays the "x?0" comparison tests  
14–7  
14–7  
   
menu.  
x0?  
  ()  
If x0, executes next program line;  
if x=0, skips the next program line.  
x0?  
x<0?  
x>0?  
x0?  
x=0?  
14–7  
14–7  
14–7  
14–7  
14–7  
 Õ()  
If x0, executes next program line;  
if x>0, skips next program line.  
 ÕÕ(<)  
If x<0, executes next program line;  
if x0, skips next program line.  
 ÕÕÕ(>)  
If x>0, executes next program line;  
if x0, skips next program line.  
 ÕÕÕÕ()  
If x0, executes next program line;  
if x<0, skips next program line.  
 ÕÕÕÕÕ(=)  
If x=0, executes next program line;  
if x0, skips next program line:  
XOR  
xiy  
11–4  
4–11  
1
>()  
Logic operator  
8( )  
Changes display of complex  
numbers.  
x+yi  
1–25  
12–4  
8( )  
Changes display of complex  
numbers. ALG mode only.  
1
 Õ( )  
Returns the mean of y values.  
Σy ÷ n.  
i
y
Name  
Keys and Description  
Page  
12–11  
1
 Õ( ˆ )  
ˆ
y
Given an x–value in the X–register,  
returns the y–estimate based on the  
regression line:  
= m x + b.  
ˆ
y
x
4–2  
1
y
Power.  
Returns y raised to the x power.  
th  
Notes:  
1. Function can be used in equations.  
Index  
flags 14-12  
Special Characters  
list of 1-13  
FN. See integration  
% functions 4-6  
low-power 1-1, A-3  
shift keys 1-2  
1-15  
in fractions 1-26  
π 4-3, A-2  
ꢄ ꢅ annunciator  
in fractions 5-2  
in fractions 5-3  
annunciators  
equations 6-7  
binary numbers 11-8  
equations 13-7  
. See backspace key  
_. See digit-entry cursor  
. See integration  
annunciators 1-3  
annunciator 1-1, A-3  
answers to questions A-1  
arithmetic  
binary 11-4  
general procedure 1-18  
hexadecimal 11-4  
intermediate results 2-12  
long calculations 2-12  
octal 11-4  
order of calculation 2-14  
stack operation 2-5, 9-2  
assignment equations 6-9, 6-11, 6-12,  
7-1  
B
backspace key  
canceling VIEW 3-4  
clearing messages 1-4  
clearing X-register 2-3, 2-7  
deleting program lines 13-20  
equation entry 1-4  
leaving menus 1-4, 1-8  
operation 1-4  
A
A…Z annunciator 1-3, 3-2, 6-4  
absolute value (real number) 4-17  
addressing  
indirect 14-20, 14-21, 14-23  
ALG 1-9  
compared to equations 13-4  
in programs 13-4  
balance (finance) 17-1  
base  
Algebraic mode 1-9  
ALL format. See display format  
in equations 6-5  
affects display 11-6  
arithmetic 11-4  
converting 11-2  
default B-4  
programs 11-8, 13-25  
setting 11-1  
in programs 13-7  
setting 1-23  
alpha characters 1-3  
angles  
base mode  
between vectors 10-5  
converting format 4-13  
converting units 4-13  
implied units 4-4, A-2  
angular mode 4-4, A-2, B-4  
annunciators  
default B-4  
equations 6-5, 6-11, 13-25  
programming 13-25  
setting 13-25  
batteries 1-1, A-3  
Bessel function 8-3  
best-fit regression 12-7, 16-1, C-13  
BIN annunciator 11-1  
alpha 1-3  
battery 1-1, A-3  
Index-1  
 
binary numbers. See numbers  
arithmetic 11-4  
cash flows 17-1  
catalogs  
converting to 11-2  
range of 11-7  
scrolling 11-8  
typing 11-1  
leaving 1-4  
program 1-28, 13-22  
using 1-28  
variable 1-28, 3-4  
chain calculations 2-12  
change-percentage functions 4-6  
changing sign of numbers 1-15, 9-3  
checksums  
viewing all digits 11-8  
borrower (finance) 17-1  
branching 14-2, 14-16, 15-7  
C
equations 6-19, 13-7, 13-24  
programs 13-22  
%CHG arguments 4-6, C-3  
Å
CLEAR menu 1-5  
adjusting contrast 1-1  
canceling prompts 1-4  
canceling VIEW 3-4  
clearing messages 1-4  
clearing X-register 2-3, 2-7  
leaving catalogs 1-4  
leaving menus 1-4, 1-8  
on and off 1-1  
clearing  
equations 6-9  
general information 1-4  
memory 1-29, A-1  
numbers 1-17  
programs 1-29, 13-23  
statistics registers 12-2  
variables 1-28  
operation 1-4  
X-register 2-3, 2-7  
/c value 5-4  
clearing memory A-4, B-3  
combinations 4-15  
commas (in numbers) 1-23, A-1  
comparison tests 14-7  
complex numbers  
leaving Equation mode 6-3  
leaving Equation mode 6-4  
canceling prompts 6-14  
stopping SOLVE 7-8  
stopping integration 8-2  
leaving Program mode 13-7  
leaving Program mode 13-7  
canceling prompts 13-15  
interrupting programs 13-19  
stopping SOLVE 15-1  
stopping integration 15-8  
/c value B-4  
argument value 4-17  
coordinate systems 9-5  
entering 9-1  
on stack 9-2  
operations 9-2  
viewing 9-2  
conditional tests 14-6, 14-7, 14-9, 14-  
12, 14-17  
/c value B-6  
calculator  
constant (filling stack) 2-7  
Continuous Memory 1-1  
contrast adjustment 1-1  
conversion functions 4-10  
conversions  
adjusting contrast 1-1  
default settings B-4  
environmental limits A-2  
questions about A-1  
resetting A-4, B-2  
angle format 4-13  
angle units 4-13  
self-test A-5  
coordinates 4-10  
shorting contacts A-5  
testing operation A-4, A-5  
turning on and off 1-1  
length units 4-14  
mass units 4-14  
number bases 10-1, 11-1  
Index-2  
temperature units 4-14  
time format 4-13  
volume units 4-14  
backspacing 1-4  
meaning 1-17  
EQN annunciator  
coordinates  
in equation list 6-4, 6-7  
in Program mode 13-7  
EQN LIST TOP 6-7, F-2  
equality equations 6-9, 6-11, 7-1  
equation list  
converting 4-10  
correlation coefficient 12-8, 16-1  
cosine (trig) 4-4, 9-3, C-6  
curve fitting 12-8, 16-1  
adding to 6-4  
D
displaying 6-6  
editing 6-8  
EQN annunciator 6-4  
in Equation mode 6-3  
operation summary 6-3  
Equation mode  
Decimal mode. See base mode  
decimal point A-1  
degrees  
angle units 4-4, A-2  
converting to radians 4-14  
denominators  
backspacing 1-4, 6-8  
during program entry 13-7  
leaving 1-4, 6-3  
shows equation list 6-3  
starting 6-3, 6-7  
controlling 5-4, 14-10, 14-14  
range of 1-26, 5-2  
setting maximum 5-4  
discontinuities of functions D-5  
display  
adjusting contrast 1-1  
X-register shown 2-3  
display format  
equations  
and fractions 5-9  
as applications 17-1  
base mode 6-5, 6-11, 13-25  
checksums 6-19, 13-7, 13-24  
compared to ALG 13-4  
compared to RPN 13-4  
controlling evaluation 14-11  
deleting 1-5, 6-9  
affects integration 8-2, 8-6, 8-7  
affects rounding 4-18  
default B-4  
periods and commas in 1-23, A-1  
setting 1-21, A-1  
do if true 14-6, 15-6  
DSE 14-18  
deleting in programs 13-20  
displaying 6-6  
displaying in programs 13-16, 13-  
18, 14-11  
editing 1-4, 6-8  
editing in programs 13-7, 13-20  
entering 6-4, 6-8  
entering in programs 13-7  
evaluating 6-10, 6-11, 6-12, 7-7,  
13-4, 14-11  
E
clearing stack 2-6  
copying viewed variable 13-15  
duplicating numbers 2-6  
ending equations 6-4, 6-8, 13-7  
evaluating equations 6-10, 6-11  
separating numbers 1-17, 2-6  
stack operation 2-6  
functions 6-5, 6-16, G-1  
in programs 13-4, 13-7, 13-24,  
14-11  
(exponent) 1-16  
E in numbers 1-15, 1-22, A-1  
ENG format 1-22 See also display  
format  
integrating 8-2  
lengths 6-19, 13-7, B-2  
list of. See equation list  
long 6-7  
entry cursor  
Index-3  
memory in 13-16  
multiple roots 7-9  
clearing 14-12  
default states 14-9  
equation evaluation 14-11  
equation prompting 14-11  
fraction display 14-10  
meanings 14-9  
no root 7-8  
numbers in 6-5  
numeric value of 6-10, 6-11, 7-1,  
7-7, 13-4  
operation summary 6-3  
parentheses 6-5, 6-6, 6-15  
precedence of operators 6-14  
prompt for values 6-11, 6-13  
prompting in programs 14-11,  
15-1, 15-8  
operations 14-12  
overflow 14-9  
setting 14-12  
testing 14-9, 14-12  
unassigned 14-9  
flow diagrams 14-2  
FN=  
roots 7-1  
scrolling 6-7, 13-7, 13-16  
solving 7-1, D-1  
in programs 15-6, 15-10  
integrating programs 15-8  
solving programs 15-1  
fractional-part function 4-17  
Fraction-display mode  
affects rounding 5-8  
affects VIEW 13-15  
setting 5-1, A-2  
stack usage 6-11  
storing variable value 6-12  
syntax 6-14, 13-16  
TVM equation 17-1  
types of 6-9  
uses 6-1  
variables in 6-3, 7-1  
with (I)/(J) 14-23  
fractions  
accuracy indicator 5-2, 5-3  
and equations 5-9  
and programs 5-10, 13-15, 14-9  
denominators 1-26, 5-4, 14-10,  
14-14  
error messages F-1  
errors  
clearing 1-4  
correcting 2-8, F-1  
estimation (statistical) 12-8, 16-1  
executing programs 13-10  
exponential curve fitting 16-1  
exponential functions 1-16, 4-1, 9-3,  
C-5  
exponents of ten 1-15, 1-16  
expression equations 6-10, 6-11, 7-1  
displaying 5-2, 5-4, A-2  
flags 14-9  
formats 5-6  
not statistics registers 5-2  
reducing 5-2, 5-6  
rounding 5-8  
round-off 5-8  
setting format 5-6, 14-10, 14-14  
typing 1-26  
F
FN. See integration  
functions  
in equations 6-5, 6-16  
list of G-1  
names in display 13-8  
nonprogrammable 13-24  
real-number 4-1  
not programmable 5-10  
toggles display mode 5-1, A-2  
toggles flag 14-9  
factorial function 4-15  
financial calculations 17-1  
FIX format 1-21 See also display format  
flags  
single argument 1-18, 2-9  
two argument 1-19, 2-9, 9-3  
future balance (finance) 17-1  
annunciators 14-12  
Index-4  
difficult functions E-2, E-7  
display format 8-2, 8-6, 8-7  
evaluating programs 15-7  
how it works E-1  
G
finds PRGM TOP 13-6, 13-21, 14-  
6
in programs 15-10  
finds program labels 13-10, 13-  
22, 14-5  
finds program lines 13-22, 14-5  
gamma function 4-15  
limits of 8-2, 15-8, C-8, E-7  
memory usage 8-2  
purpose 8-1  
restrictions 15-11  
results on stack 8-2, 8-6  
stopping 8-2, 15-8  
subintervals E-7  
go to. See GTO  
grads (angle units) 4-4, A-2  
Grandma Hinkle 12-7  
Greatest integer 4-18  
grouped standard deviation 16-18  
GTO 14-4, 14-17  
time required 8-6, E-7  
transforming variables E-9  
uncertainty of result 8-2, 8-6, E-2  
using 8-2, C-8  
guesses (for SOLVE) 7-2, 7-7, 7-8, 7-  
12, 15-6  
H
help about calculator A-1  
HEX annunciator 11-1  
hex numbers. See numbers  
arithmetic 11-4  
variable of 8-2, C-8  
intercept (curve-fit) 12-8, 16-1  
interest (finance) 17-3  
intermediate results 2-12  
inverse function 9-3  
inverse hyperbolic functions 4-6  
inverse trigonometric functions 4-4, C-6  
inverse-normal distribution 16-11  
ISG 14-18  
converting to 11-2  
range of 11-7  
typing 11-1  
hexadecimal numbers. See hex  
numbers  
Horner's method 13-26  
humidity limits for calculator A-2  
hyperbolic functions 4-6, C-6  
J
j 3-9, 14-20, 14-21  
(j) 14-20  
K
I
keys  
i 3-9, 14-20  
alpha 1-3  
letters 1-3  
shifted 1-3  
(i) 14-20, 14-21, 14-23  
imaginary part (complex numbers) 9-1,  
C-8  
indirect addressing 14-20, 14-21, 14-  
23  
L
LAST X register 2-8, B-6  
LASTx function 2-8  
INPUT  
always prompts 14-11  
entering program data 13-12  
in integration programs 15-8  
in SOLVE programs 15-2  
responding to 13-14  
integer-part function 4-17  
integration  
lender (finance) 17-1  
length conversions 4-14  
letter keys 1-3  
limits of integration 8-2, 15-8, C-8  
linear regression (estimation) 12-8, 16-  
1
logarithmic curve fitting 16-1  
accuracy 8-2, 8-6, E-1  
Index-5  
logarithmic functions 4-1, 9-3, C-5  
logic  
usage B-1  
MEMORY CLEAR A-4, B-3, F-3  
MEMORY FULL B-1, F-3  
menu keys 1-6  
AND 11-4  
NAND 11-4  
NOR 11-4  
menus  
NOT 11-4  
example of using 1-8  
general operation 1-6  
leaving 1-4, 1-8  
OR 11-4  
XOR 11-4  
loop counter 14-18, 14-23  
looping 14-16, 14-17  
Łukasiewicz 2-1  
list of 1-6  
messages  
clearing 1-4  
displaying 13-16, 13-18  
in equations 13-16  
responding to 1-27, F-1  
summary of F-1  
M
program catalog 1-28, 13-22  
reviews memory 1-28  
variable catalog 1-28  
mantissa 1-25  
mass conversions 4-14  
math  
minimum of function D-8  
modes. See angular mode, base mode,  
Equation mode, Fraction-display  
mode, Program-entry mode  
MODES menu  
angular mode 4-4  
money (finance) 17-1  
multiplication, dividision 10-2  
complex-number 9-1  
general procedure 1-18  
intermediate results 2-12  
long calculations 2-12  
order of calculation 2-14  
real-number 4-1  
N
negative numbers 1-15, 9-3, 11-6  
nested routines 14-2, 15-11  
normal distribution 16-11  
numbers. See binary numbers, hex  
numbers, octal numbers, variables  
bases 10-1, 13-25  
stack operation 2-5, 9-2  
maximum of function D-8  
mean menu 12-4  
means (statistics)  
calculating 12-4  
normal distribution 16-11  
memory  
changing sign of 1-15, 9-3  
clearing 1-4, 1-5, 1-17  
complex 9-1  
amount available 1-28  
clearing 1-5, 1-29, A-1, A-4, B-1,  
B-3  
clearing equations 6-9  
clearing programs 1-28, 13-6, 13-  
22  
clearing statistics registers 12-2  
clearing variables 1-28  
full A-1  
display format 1-21, 11-6  
E in 1-15, A-1  
editing 1-4, 1-17  
exchanging 2-4  
finding parts of 4-17  
fractions in 1-26, 5-1  
in equations 6-5  
in programs 13-7  
maintained while off 1-1  
programs 13-21, B-2  
size 1-28, B-1  
internal representation 11-6  
large and small 1-15, 1-17  
negative 1-15, 9-3, 11-6  
performing arithmetic calculations  
stack 2-1  
Index-6  
1-18  
power curve fitting 16-1  
power functions 1-17, 4-2, 9-3  
precedence (equation operators) 6-14  
precision (numbers) 1-25, D-13  
present value. See financial  
calculations  
periods and commas in 1-23, A-1  
precision D-13  
prime 17-7  
range of 1-17, 11-7  
real 4-1  
recalling 3-2  
reusing 2-6, 2-10  
rounding 4-18  
showing all digits 1-25  
storing 3-2  
truncating 11-6  
typing 1-15, 1-16, 11-1  
PRGM TOP 13-4, 13-7, 13-21, F-4  
prime number generator 17-7  
probability  
functions 4-15  
normal distribution 16-11  
program catalog 1-28, 13-22  
program labels  
branching to 14-2, 14-4, 14-16  
checksums 13-23  
O
Ä 1-1  
clearing 13-6  
OCT annunciator 11-1, 11-4  
octal numbers. See numbers  
arithmetic 11-4  
duplicate 13-6  
entering 13-4, 13-6  
executing 13-10  
converting to 11-2  
range of 11-7  
indirect addressing 14-20, 14-21,  
14-23  
typing 11-1  
moving to 13-22  
one-variable statistics 12-2  
overflow  
purpose 13-4  
typing name 1-3  
flags 14-9, F-4  
viewing 13-22  
result of calculation 1-17, 11-5  
setting response 14-9, F-4  
testing occurrence 14-9  
program lines. See programs  
program names. See program labels  
program pointer 13-6, 13-11, 13-19,  
13-21, B-4  
P
Program-entry mode 1-4, 13-6  
programs. See program labels  
ALG operations 13-4  
π A-2  
parentheses  
in arithmetic 2-12  
base mode 13-25  
in equations 6-5, 6-6, 6-15  
pause. See PSE  
payment (finance) 17-1  
percentage functions 4-6  
periods (in numbers) 1-23, A-1  
permutations 4-15  
Physics constants 4-8  
polar-to-rectangular coordinate conver-  
sion 4-10, 9-5  
poles of functions D-6  
polynomials 13-26  
branching 14-2, 14-4, 14-6, 14-  
16  
calculations in 13-13  
calling routines 14-1, 14-2  
catalog of 1-28, 13-22  
checksums 13-22, 13-23, B-2  
clearing 13-6, 13-22, 13-23  
clearing all 13-6, 13-23  
comparison tests 14-7  
conditional tests 14-7, 14-9, 14-  
12, 14-17, 15-6  
population standard deviations 12-7  
power annunciator 1-1, A-3  
data input 13-5, 13-13, 13-14  
data output 13-5, 13-14, 13-18  
Index-7  
deleting 1-28  
deleting all 1-5  
variables in 13-12, 15-1, 15-7  
prompts  
deleting equations 13-7, 13-20  
deleting lines 13-20  
designing 13-3, 14-1  
editing 1-4, 13-7, 13-20  
editing equations 13-7, 13-20  
entering 13-6  
equation evaluation 14-11  
equation prompting 14-11  
equations in 13-4, 13-7  
errors in 13-19  
affect stack 6-14, 13-14  
clearing 1-4, 6-14, 13-15  
equations 6-13  
INPUT 13-12, 13-14, 15-2, 15-8  
programmed equations 14-11, 15-  
1, 15-9  
responding to 6-13, 13-14  
showing hidden digits 6-14  
PSE  
pausing programs 13-19, 15-10  
preventing program stops 14-11  
executing 13-10  
flags 14-9, 14-12  
Q
for integration 15-7  
for SOLVE 15-1, D-1  
fractions with 5-8, 13-15, 14-9  
functions not allowed 13-24  
indirect addressing 14-20, 14-21,  
14-23  
questions A-1  
quotient and remainder of division 4-2  
R
inserting lines 13-6, 13-20  
interrupting 13-19  
ending prompts 6-11, 6-14, 7-2,  
13-15  
lengths 13-22, 13-23, B-2  
line numbers 13-22  
loop counter 14-18  
looping 14-16, 14-17  
memory usage 13-22  
messages in 13-16, 13-18  
moving through 13-11  
not stopping 13-18  
numbers in 13-7  
interrupting programs 13-19  
resuming programs 13-16, 13-19  
running programs 13-22  
stopping integration 8-2, 15-8  
stopping SOLVE 7-8, 15-1  
Rand R2-3, C-7  
radians  
angle unit 4-4  
angle units A-2  
converting to degrees 4-14  
radix mark A-1  
pausing 13-19  
prompting for data 13-12  
purpose 13-1  
random numbers 4-15, B-4  
RCL 3-2, 13-14  
resuming 13-16  
return at end 13-4  
RCL arithmetic 3-7  
routines 14-1  
real numbers  
RPN operations 13-4  
running 13-10  
operations 4-1  
real part (complex numbers) 9-1  
recall arithmetic 3-7  
showing long number 13-7  
stepping through 13-11  
stopping 13-14, 13-16, 13-19  
techniques 14-1  
testing 13-11  
using integration 15-10  
using SOLVE 15-6  
rectangular-to-polar coordinate conver-  
sion 4-10, 9-5  
regression (linear) 12-7, 16-1  
resetting the calculator A-4, B-2  
return (program). See programs  
Reverse Polish Notation. See RPN  
Index-8  
rolling the stack 2-3, C-7  
root functions 4-3  
roots. See SOLVE  
checking 7-7, D-3  
in programs 15-6  
multiple 7-9  
none found 7-8, D-8  
of equations 7-1  
of programs 15-1  
rounding  
shift keys 1-3  
sign (of numbers) 1-15, 9-3, 11-6  
sign conventions (finance) 17-1  
Sign value 4-17  
sine (trig) 4-4, 9-3, A-2, C-6  
single-step execution 13-11  
slope (curve-fit) 12-8, 16-1  
SOLVE  
checking results 7-7, D-3  
discontinuity D-5  
fractions 5-8, 13-18  
numbers 4-18  
evaluating equations 7-1, 7-7  
evaluating programs 15-2  
flat regions D-8  
how it works 7-7, D-1  
in programs 15-6  
initial guesses 7-2, 7-7, 7-8, 7-12,  
15-6  
minimum or maximum D-8  
multiple roots 7-9  
round-off  
fractions 5-8  
integration 8-6  
SOLVE D-13  
statistics 12-10  
trig functions 4-4  
routines  
calling 14-1  
no restrictions 15-11  
no root found 7-8, 15-6, D-8  
pole D-6  
nesting 14-2, 15-11  
parts of programs 14-1  
RPN  
purpose 7-1  
compared to equations 13-4  
in programs 13-4  
origins 2-1  
results on stack 7-2, 7-7, D-3  
resuming 15-1  
round-off D-13  
running programs 13-10  
stopping 7-2, 7-8  
using 7-1  
S
stack. See stack lift  
Î
affected by prompts 6-14, 13-14  
complex numbers 9-2  
effect of 2-6  
equation usage 6-11  
exchanging with variables 3-8  
exchanging X and Y 2-4  
filling with constant 2-7  
long calculations 2-12  
operation 2-1, 2-5, 9-2  
program calculations 13-14  
program input 13-12  
program output 13-12  
purpose 2-1, 2-2  
equation checksums 6-19, B-2  
equation lengths 6-19, B-2  
number digits 1-25, 13-7  
program checksums 13-22, B-2  
program lengths 13-22, B-2  
prompt digits 6-14  
14-14  
sample standard deviations 12-6  
SCI format. See display format  
in programs 13-7  
setting 1-22  
scrolling  
binary numbers 11-8  
equations 6-7, 13-7, 13-16  
seed (random number) 4-15  
self-test (calculator) A-5  
registers 2-1  
reviewing 2-3, C-7  
rolling 2-3, C-7  
separate from variables 3-2  
Index-9  
size limit 2-4, 9-2  
unaffected by VIEW 13-15  
stack lift. See stack  
default state B-4  
syntax (equations) 6-14, 6-19, 13-16  
T
tangent (trig) 4-4, 9-3, A-2, C-6  
temperatures  
disabling B-4  
enabling B-4  
converting units 4-14  
limits for calculator A-2  
test menus 14-7  
not affecting B-5  
operation 2-5  
standard deviations  
calculating 12-6, 12-7  
grouped data 16-18  
normal distribution 16-11  
standard-deviation menu 12-6, 12-7  
statistical data. See statistics registers  
clearing 1-5, 12-2  
correcting 12-2  
testing the calculator A-4, A-5  
time formats 4-13  
time value of money 17-1  
T-register 2-5  
trigonometric functions 4-4, 9-3, C-6  
troubleshooting A-4, A-5  
turning on and off 1-1  
TVM 17-1  
twos complement 11-4, 11-6  
two-variable statistics 12-2  
entering 12-1  
initializing 12-2  
one-variable 12-2  
U
precision 12-10  
sums of variables 12-11  
two-variable 12-2  
uncertainty (integration) 8-2, 8-6  
units conversions 4-14  
statistics  
V
calculating 12-4  
curve fitting 12-8, 16-1  
distributions 16-11  
grouped data 16-18  
one-variable data 12-2  
operations 12-1  
variable catalog 1-28, 3-4  
variables  
accessing stack register contents B-  
7
arithmetic inside 3-6  
catalog of 1-28, 3-4  
clearing 1-28  
two-variable data 12-2  
statistics menus 12-1, 12-4  
statistics registers. See statistical data  
accessing 12-12  
clearing all 1-5  
clearing while viewing 13-15  
exchanging with X 3-8  
in equations 6-3, 7-1  
in programs 13-12, 15-1, 15-7  
indirect addressing 14-20, 14-21  
names 3-1  
clearing 1-5, 12-2  
contain summations 12-1, 12-11,  
12-12  
correcting data 12-2  
initializing 12-2  
no fractions 5-2  
viewing 12-11  
number storage 3-1  
of integration 8-2, 15-7, C-8  
polynomials 13-26  
STO 3-2, 13-12  
STO arithmetic 3-6  
STOP 13-19  
storage arithmetic 3-6  
subroutines. See routines  
sums of statistical variables 12-11  
program input 13-14  
program output 13-15, 13-18  
recalling 3-2, 3-4  
separate from stack 3-2  
showing all digits 13-15  
Index-10  
solving for 7-1, 15-1, 15-6, D-1  
storing 3-2  
testing 14-7  
unaffected by VIEW 13-15  
storing from equation 6-12  
typing name 1-3  
viewing 3-4, 13-15, 13-18  
vectors  
absolute value 10-3  
addition, subtraction 10-1  
angle between two vectors  
10-5  
coordinate conversions 4-12, 9-5  
creating vectors from variables or  
registers 10-8  
cross product 17-11  
dot product 10-4  
in equation 10-6  
in program 10-7  
VIEW  
displaying program data 13-15,  
13-18, 15-6  
displaying variables 3-4  
no stack effect 13-15  
stopping programs 13-15  
volume conversions 4-14  
W
weight conversions 4-14  
weighted means 12-4  
windows (binary numbers) 11-8  
X
evaluating equations 6-10, 6-12  
running programs 13-10, 13-22  
X ROOT arguments 6-17  
X-register  
affected by prompts 6-14  
arithmetic with variables 3-6  
clearing 1-5, 2-3, 2-7  
clearing in programs 13-7  
displayed 2-3  
during programs pause 13-19  
exchanging with variables 3-8  
exchanging with Y 2-4  
not clearing 2-5  
part of stack 2-1  
Index-12  

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