HP 35s scientific calculator
user's guide
H
HP part number F2215AA-90001
Edition 1
Contents
1. Getting Started............................................................1-1
Contents
1
Contents
3
7. Solving Equations........................................................7-1
8. Integrating Equations ..................................................8-1
9. Operations with Complex Numbers .............................9-1
10.Vector Arithmetic ......................................................10-1
Contents
5
11.Base Conversions and Arithmetic and Logic................11-1
12.Statistical Operations ................................................12-1
6
Contents
15.Solving and Integrating Programs..............................15-1
17.Miscellaneous Programs and Equations......................17-1
A. Support, Batteries, and Service ................................... A-1
Service...................................................................................A-8
Contents
9
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.
σ σ
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)
Rꢀ R ꢁ
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 ꢂ or ꢃ 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:
_
ꢃ
8
_
or
ꢀ
Pressing another menu key replaces the old menu with the new one.
Keys:
Display:
_
8
ꢃ
ꢃ
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.
1-10 Getting Started
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.
1-12 Getting Started
HP 35s Annunciators
Meaning
Annunciator
Chapter
The " (Busy)" annunciator appears while
an operation, equation, or program is
executing.
When in Fraction–display mode (press
ꢄ
ꢅ
5
), only one of the "ꢄ" or "ꢅ" 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.
ꢂ,ꢃ
A..Z
The alphabetic keys are active.
3
1
Attention! Indicates a special condition or an
error.
Battery power is low.
A
1-14 Getting Started
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.
1-16 Getting Started
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.
1-18 Getting Started
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.
1-20 Getting Started
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.
1-22 Getting Started
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.
1-24 Getting Started
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 0≤b<c
and c is such that 1<c≤4095.
Entering Fractions
Fractions can be entered onto the stack at any time:
1. Key in the integer part of the number and press . (The first
separates the integer part of the number from its fractional part.)
2. Key in the fraction numerator and press again. The second
separates the numerator from the denominator.
3. Key in the denominator, then press 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.
1-26 Getting Started
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 .
1-28 Getting Started
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
Rꢀ (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.
Rꢁ (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) ÷ (12.5− 0.73) = 0.5961
Solution:
or
Calculate:
8.33×(4− 5.2)÷[(8.33− 7.46)×0.32]
= 4.5728
4.3×(3.15− 2.75)− (1.71×2.01)
A Solution:
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 ꢂ and ꢃ annunciators indicating that the Øand ×keys are active
to help you scroll through the catalog; however, if Fraction Display mode is active,
the ꢄ and ꢅ 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. (y≠ 0)
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: ꢇkg, ꢇlb, ꢇºC,
ꢇºF, ꢇcm, ꢇin, ꢇl, ꢇgal, ꢇMILE,ꢇKM
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 ꢄ and ꢅ 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
ꢈ
ꢉ
/
/
12
15
8192
12345
12345678
3
/
3
16 /
16384
Accuracy Indicators
The accuracy of a displayed fraction is indicated by the ꢄ and ꢅ 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 ꢅ 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 ꢄ 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 ꢄ because
3
3.3333
3
2
the exact fraction is approximately
/ , "a little above" / . Similarly, – / is
5
5
3
displayed as ꢄ 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 ꢄ, 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 ꢆ 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
ꢉ
3
ꢈ
2 9/14
ꢈ
2 1/2
2 11/16
ꢈ
ꢈ
3ꢈ
2 5/8ꢉ
2 0/16 2 8/16 2 11/16
3 0/16
ꢈ
2 10/16
ꢉ
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.
ꢉ
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.
5-10 Fractions
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 ꢆ 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 2 5 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
ꢇDEG
ꢇL
ꢇRAD
ꢇGAL
HMSꢇ
ꢇMILE
ꢇHMS
ꢇKM
ꢇKG
SEED
+
ꢇLB
ARG
–
ꢇ°C
RAND
×
ꢇ°F
ꢇCM
ꢇIN
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, 28 4 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
ππ
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."
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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:
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Keys:
()
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