Long results

If the result is too long to fit on the display line, or if you want to see an expression in textbook format, press to highlight it and then press. Type to start a negative number or to insert a negative sign. To raise a negative number to a power, enclose it in parentheses. For example, (5)2 = 25, whereas 52 = 25.

Negative numbers

Scientific notation (powers of 10)
A number like or 3.is written in scientific notation, that is, in terms of powers of ten. This is simpler to work with than 50000 or 0.000000321. To enter numbers like these, use EEX. (This is easier than using 10.) ( ) ( ) Calculate ---------------------------------------------------3

EEX EEX EEX

13 23
Explicit and implicit multiplication
Implied multiplication takes place when two operands appear with no operator in between. If you enter AB, for example, the result is A*B. However, for clarity, it is better to include the multiplication sign where you expect multiplication in an expression. It is clearest to enter AB as A*B.
Implied multiplication will not always work as expected. For example, entering A(B+4) will not give A*(B+4). Instead an error message is displayed: Invalid User Function. This is because the calculator interprets A(B+4) as meaning evaluate function A at the value B+4, and function A does not exist. When in doubt, insert the * sign manually. You need to use parentheses to enclose arguments for functions, such as SIN(45). You can omit the final parenthesis at the end of an edit line. The calculator inserts it automatically. Parentheses are also important in specifying the order of operation. Without parentheses, the hp 39g+ calculates according to the order of algebraic precedence (the next topic). Following are some examples using parentheses. Entering. Calculates. sin (45 + ) sin (45) + 85 9

Parentheses

Algebraic precedence order of evaluation
Functions within an expression are evaluated in the following order of precedence. Functions with the same precedence are evaluated in order from left to right. 1. Expressions within parentheses. Nested parentheses are evaluated from inner to outer. 2. Prefix functions, such as SIN and LOG. 3. Postfix functions, such as ! 4. Power function, ^, NTHROOT. 5. Negation, multiplication, and division. 6. Addition and subtraction. 7. AND and NOT. 8. OR and XOR. 9. Left argument of | (where). 10.Equals, =.

Largest and smallest numbers Clearing numbers
The smallest number the hp 39g+ can represent is 1 10499(1E499). A smaller result is displayed as zero. The largest number is 9.(1E499). A greater result is displayed as this number. clears the character under the cursor. When the cursor is positioned after the last character, deletes the character to the left of the cursor, that is, it performs the same as a backspace key.

CANCEL

) clears the edit line.
CLEAR clears all input and output in the display, including the display history.

Using previous results

The HOME display ( ) shows you four lines of input/output history. An unlimited (except by memory) number of previous lines can be displayed by scrolling. You can retrieve and reuse any of these values or expressions.
Input Last input Edit line

Output Last output

When you highlight a previous input or result (by pressing ), the and menu labels appear.
To copy a previous line To reuse the last result To repeat a previous line
Highlight the line (press ) and press. The number (or expression) is copied into the edit line. Press ANS (last answer) to put the last result from the HOME display into an expression. ANS is a variable that is updated each time you press. To repeat the very last line, just press. Otherwise, highlight the line (press ) first, and then press. The highlighted expression or number is re-entered. If the previous line is an expression containing the ANS, the calculation is repeated iteratively. See how (50), and 50
retrieves and reuses the last result updates ANS (from 50 to 75 to 100).
You can use the last result as the first expression in the edit ANS. Pressing , , , or line without pressing , (or other operators that require a preceding argument) automatically enters ANS before the operator. You can reuse any other expression or value in the HOME display by highlighting the expression (using the arrow keys), then pressing. See Using previous results on page 1-22 for more details. The variable ANS is different from the numbers in HOMEs display history. A value in ANS is stored internally with the full precision of the calculated result, whereas the displayed numbers match the display mode.

Set up the plot

You can change the scales of the x and y axes, graph resolution, and the spacing of the axis ticks. 3. Display plot settings.

SETUP-PLOT

Note: For our example, you can leave the plot settings at their default values since we will be using the Auto Scale feature to choose an appropriate y axis for our x axis settings. If your settings do not match this example, press default values. 4. Specify a grid for the graph.

to restore the

Plot the functions

5. Plot the functions.

Change the scale
6. You can change the scale to see more or less of your graphs. In this example, choose Auto Scale. (See VIEWS menu options on page 2-14 for a description of Auto Scale). Select Auto Scale
7. Trace the linear function. 6 times
Note: By default, the tracer is active. 8. Jump from the linear function to the quadratic function.
Analyse graph with FCN functions
9. Display the Plot view menu.
From the Plot view menu, you can use the functions on the FCN menu to find roots, intersections, slopes, and areas for a function defined in the Function aplet (and any Function-based aplets). The FCN functions act on the currently selected graph. See FCN functions on page 3-10 for further information.
To find a root of the quadratic function
10.Move the cursor to the graph of the quadratic equation by pressing the or key. Then move the cursor so that it is near x = 1 by pressing the or key. Select Root
The root value is displayed at the bottom of the screen. Note: If there is more than one root (as in our example), the coordinates of the root closest to the current cursor position are displayed.
To find the intersection of the two functions
11.Find the intersection of the two functions.
12.Choose the linear function whose intersection with the quadratic function you wish to find.
The coordinates of the intersection point are displayed at the bottom of the screen. Note: If there is more than one intersection (as in our example), the coordinates of the intersection point closest to the current cursor position are displayed.
To find the slope of the quadratic function
13.Find the slope of the quadratic function at the intersection point. Select Slope The slope value is displayed at the bottom of the screen.

The FCN functions are: Function Root Description Select Root to find the root of the current function nearest the cursor. If no root is found, but only an extremum, then the result is labeled EXTR: instead of ROOT:. (The root-finder is also used in the Solve aplet. See also Interpreting results on page 7-6.) The cursor is moved to the root value on the x-axis and the resulting x-value is saved in a variable named ROOT. Select Extremum to find the maximum or minimum of the current function nearest the cursor. This displays the coordinate values and moves the cursor to the extremum. The resulting value is saved in a variable named EXTREMUM. Select Slope to find the numeric derivative at the current position of the cursor. The result is saved in a variable named SLOPE. Select Signed area to find the numeric integral. (If there are two or more expressions checkmarked, then you will be asked to choose the second expression from a list that includes the x-axis.) Select a starting point, then move the cursor to selection ending point. The result is saved in a variable named AREA.

Extremum

Signed area

Function Intersection

Description (Continued) Select Intersection to find the intersection of two graphs nearest the cursor. (You need to have at least two selected expressions in Symbolic view.) Displays the coordinate values and moves the cursor to the intersection. (Uses Solve function.) The resulting xvalue is saved in a variable named ISECT.

Shading area

You can shade a selected area between functions. This process also gives you an approximate measurement of the area shaded. 1. Open the Function aplet. The Function aplet opens in the Symbolic view. 2. Select the expressions whose curves you want to study. 3. Press to plot the functions.
4. Press or to position the cursor at the starting point of the area you want to shade. 5. Press 6. Press. , then select Signed area and press
7. Press , choose the function that will act as the boundary of the shaded area, and press. 8. Press the or key to shade in the area.
9. Press to calculate the area. The area measurement is displayed near the bottom of the screen. To remove the shading, press to re-draw the plot.
Plotting a piecewise-defined function
Suppose you wanted to plot the following piecewisedefined function. x + 2 ;x 1 f ( x ) = x2 ; 1 < x x ;x 1
2. Highlight the line you want to use, and enter the expression. (You can press line, or 2

CHARS CLEAR

to delete an existing

to clear all lines.)

1 > 1

CHARS AND

Note: You can use the menu key to assist in the entry of equations. It has the same effect as pressing.
About the Parametric aplet
The Parametric aplet allows you to explore parametric equations. These are equations in which both x and y are defined as functions of t. They take the forms x = f ( t ) and y = g ( t ).

Note: the T-value is filled in with the position of the cursor from the Plot view. 7. Ensure that the T value is highlighted, and solve the equation.
Use this equation to solve for another variable, such as velocity. How fast must a bodys initial velocity be in order for it to travel 50 m within 3 seconds? Assume the same acceleration, 4 m/s2. Leave the last value of V as the initial guess. 3 50
Using variables in equations
You can use any of the real variable names, A to Z and. Do not use variable names defined for other types, such as M1 (a matrix variable).

Home variables

All home variables (other than those for aplet settings, like Xmin and Ytick) are global, which means they are shared throughout the different aplets of the calculator. A value that is assigned to a home variable anywhere remains with that variable wherever its name is used. Therefore, if you have defined a value for T (as in the above example) in another aplet or even another Solve equation, that value shows up in the Numeric view for this Solve equation. When you then redefine the value for T in this Solve equation, that value is applied to T in all other contexts (until it is changed again). This sharing allows you to work on the same problem in different places (such as HOME and the Solve aplet) without having to update the value whenever it is recalculated.
As the Solve aplet uses existing variable values, be sure to check for existing variable values that may affect the CLEAR to reset all solve process. (You can use values to zero in the Solve aplets Numeric view if you wish.) Functions defined in other aplets can also be referenced in the Solve aplet. For example, if, in the Function aplet, you define F1(X)=X2+10, you can enter F1(X)=50 in the Solve aplet to solve the equation X2+10=50.

Aplet variables

About the Statistics aplet
The Statistics aplet can store up to ten data sets at one time. It can perform one-variable or two-variable statistical analysis of one or more sets of data. The Statistics aplet starts with the Numeric view which is used to enter data. The Symbolic view is used to specify which columns contain data and which column contains frequencies. You can also compute statistics values in HOME and recall the values of specific statistics variables. The values computed in the Statistics aplet are saved in variables, and many of these variables are listed by the function accessible from the Statistics aplets Numeric view screen.

You can type PREDX and PREDY into the edit line, or you can copy these function names from the MATH menu under the Stat-Two category. HINT In cases where more than one fit curve is displayed, the PREDY function uses the most recently calculated curve. In order to avoid errors with this function, uncheck all fits except the one that you want to work with, or use the Plot View method.
About the Inference aplet
The Inference capabilities include calculation of confidence intervals and hypothesis tests based on the Normal Z-distribution or Students t-distribution. Based on the statistics from one or two samples, you can test hypotheses and find confidence intervals for the following quantities: mean proportion difference between two means difference between two proportions

Example data

When you first access an input form for an Inference test, by default, the input form contains example data. This example data is designed to return meaningful results that relate to the test. It is useful for gaining an understanding of what the test does, and for demonstrating the test. The calculators on-line help provides a description of what the example data represents.
Getting started with the Inference aplet
This example describes the Inference aplets options and functionality by stepping you through an example using the example data for the Z-Test on 1 mean.

Open the Inference aplet

1. Open the Inference aplet. Select Inference. The Inference aplet opens in the Symbolic view.

Inference aplet

Inference aplets SYMB view keys
The table below summarizes the options available in Symbolic view. Hypothesis Tests Z: 1 , the Z-Test on 1 mean Z: 1 2, the Z-Test on the difference of two means Z: 1 , the Z-Test on 1 proportion Confidence Intervals Z-Int: 1 , the confidence interval for 1 mean, based on the Normal distribution Z-Int: 1 2, the confidence interval for the difference of two means, based on the Normal distribution Z-Int: 1 , the confidence interval for 1 proportion, based on the Normal distribution Z-Int: 1 2, the confidence interval for the difference of two proportions, based on the Normal distribution T-Int: 1 , the confidence interval for 1 mean, based on the Students t-distribution T-Int: 1 2, the confidence interval for the difference of two means, based on the Students t-distribution
Z: 1 2, the Z-Test on the difference in two proportions T: 1 , the T-Test on 1 mean T: 1 2, the TTest on the difference of two means
If you choose one of the hypothesis tests, you can choose the alternative hypothesis to test against the null hypothesis. For each test, there are three possible choices for an alternative hypothesis based on a quantitative comparison of two quantities. The null hypothesis is always that the two quantities are equal.Thus, the alternative hypotheses cover the various cases for the two quantities being unequal: <, >, and. In this section, we will use the example data for the Z-Test on 1 mean to illustrate how the aplet works and what features the various views present.

The MATH menu

The MATH menu provides access to math functions and programming constants. The MATH menu is organized by category. For each category of functions on the left, there is a list of function names on the right. The highlighted category is the current category.
When you press , you see the menu list of Math categories in the left column and the corresponding functions of the highlighted category in the right column. The menu key indicates that the MATH FUNCTIONS menu list is active. to display the MATH menu. The

To select a function

1. Press
categories appear in alphabetical order. Press or to scroll through the categories. To skip directly to a category, press the first letter of the categorys name. Note: You do not need to press
Using mathematical functions

first.

2. The list of functions (on the right) applies to the currently highlighted category (on the left). Use and to switch between the category list and the function list. 3. Highlight the name of the function you want and press. This copies the function name (and an initial parenthesis, if appropriate) to the edit line.

Function categories

Calculus Complex numbers Constant Hyperbolic trigonometry (Hyperb.) Lists Loop Matrices (Matrices) Polynomial (Polynom.) Probability (Prob.) Real numbers (Real) Two-variable statistics (Stat-Two) Symbolic Tests Trigonometry (Trig)
Math functions by category

Syntax

Each functions definition includes its syntax, that is, the exact order and spelling of a functions name, its delimiters (punctuation), and its arguments. Note that the syntax for a function does not require spaces.
Functions common to keyboard and menus
These functions are common to the keyboard and MATH menu.
For a description, see on page 11-8. For a description, see ARG on page 11-7. For a description, see on page 11-7.
For a description, see AND on page 11-19.
For a description, see COMB(5,2) returns 10. That is, there are ten different ways that five things can be combined two at a time.! on page 11-12. For a description, see on page 11-10. For a description, see Scientific notation (powers of 10) on page 1-20. For a description, see on page 11-7.
The multiplicative inverse function finds the inverse of a square matrix, and the multiplicative inverse of a real or complex number. Also works on a list containing only these object types.

Keyboard functions

The most frequently used functions are available directly from the keyboard. Many of the keyboard functions also accept complex numbers as arguments.
Add, Subtract, Multiply, Divide. Also accepts complex numbers, lists and matrices. value1+ value2, etc.
Natural exponential. Also accepts complex numbers. e^value Example e^5 returns 148.413159103 Natural logarithm. Also accepts complex numbers. LN(value) Example LN(1) returns 0

= (equals)

Sets an equality for an equation. This is not a logical operator and does not store values. (See Test functions on page 11-18.) expression1=expression2

ISOLATE

Isolates the first occurrence of variable in expression=0 and returns a new expression, where variable=newexpression. The result is a general solution that represents multiple solutions by including the (formal) variables S1 to represent any sign and n1 to represent any integer. ISOLATE(expression, variable) Examples ISOLATE(2*X+8,X) returns -4 ISOLATE(A+B*X/C,X) returns -(A*C/B)

LINEAR?

Tests whether expression is linear for the specified variable. Returns 0 (false) or 1 (true). LINEAR?(expression, variable) Example LINEAR?((X^2-1)/(X+1),X) returns 0
Solves quadratic expression=0 for variable and returns a new expression, where variable=newexpression. The result is a general solution that represents both positive and negative solutions by including the formal variable S1 to represent any sign: + or . QUAD(expression, variable) Example QUAD((X-1)2-7,X) returns (2+s1*5.29150262213)/2
Encloses an expression that should not be evaluated numerically. QUOTE(expression) Examples QUOTE(SIN(45)) F1(X) stores the expression SIN(45) rather than the value of SIN(45). Another method is to enclose the expression in single quotes. For example, X^3+2*X F1(X) puts the expression X^3+2*X into F1(X) in the Function aplet.

| (where)

Evaluates expression where each given variable is set to the given value. Defines numeric evaluation of a symbolic expression. expression|(variable1=value1, variable2=value2,.) Example 3*(X+1)|(X=3) returns 12.

Test functions

The test functions are logical operators that always return either a 1 (true) or a 0 (false). < Less than. Returns 1 if true, 0 if false. value1<value2 Less than or equal to. Returns 1 if true, 0 if false. value1value2 == Equals (logical test). Returns 1 if true, 0 if false. value1==value2 Not equal to. Returns 1 if true, 0 if false. value1value2 > Greater than. Returns 1 if true, 0 if false. value1>value2
Greater than or equal to. Returns 1 if true, 0 if false. value1value2
Compares value1 and value2. Returns 1 if they are both non-zero, otherwise returns 0. value1 AND value2

; EXP.ANG;0; The program EXP.ANG is a small routine that is called by other programs that the aplet uses. This entry specifies that the program EXP.ANG is transferred when the aplet is transferred, but the space in the first quotes ensures that no entry appears on the menu. START;EXP.S;7: This specifies the Start menu option. The program that is associated with this entry, EXP.S, runs automatically when you start the aplet. Because this menu option specifies view 7, the VIEWS menu opens when you start the aplet. You only need to run this program once to configure your aplets VIEWS menu. Once the aplets VIEWS menu is configured, it remains that way until you run SETVIEWS again. You do not need to include this program for your aplet to work, but it is useful to specify that the program is attached to the aplet, and transmitted when the aplet is transmitted. 7. Return to the program catalog. The programs that you created should appear as follows: 8. You must now the program EXP.SV to execute the SETVIEWS command and create the modified VIEWS menu. Check that the name of the new aplet is highlighted in the Aplet view. 9. You can now return to the Aplet library and press to run your new aplet.

Programming commands

This section describes the commands for programming with hp 39g+. You can enter these commands in your program by typing them or by accessing them from the Commands menu.

Programming 16-13

Aplet commands
Checks (selects) the corresponding function in the current aplet. For example, Check 3 would check F3 if the current aplet is Function. Then a checkmark would appear next to F3 in Symbolic view, F3 would be plotted in Plot view, and evaluated in Numeric view. CHECK n:

SELECT

Selects the named aplet and makes it the current aplet. Note: Quotes are needed if the name contains spaces or other special characters. SELECT apletname:

SETVIEWS

The SETVIEWS command is used to define entries in the VIEWS menu for aplets that you customize. See Customizing an aplet on page 16-9 for an example of using the SETVIEWS command. When you use the SETVIEWS command, the aplets standard VIEWS menu is deleted and the customized menu is used in its place. You only need to apply the command to an aplet once. The VIEWS menu changes remain unless you apply the command again. Typically, you develop a program that uses the SETVIEWS command only. The command contains a trio of arguments for each menu option to create, or program to attach. Keep the following points in mind when using this command: The SETVIEWS command deletes an aplets standard Views menu options. If you want to use any of the standard options on your reconfigured VIEWS menu, you must include them in the configuration. When you invoke the SETVIEWS command, the changes to an aplets VIEWS menu remain with the aplet. You need to invoke the command on the aplet again to change the VIEWS menu. All the programs that are called from the VIEWS menu are transferred when the aplet is transferred, for example to another calculator or to a PC. As part of the VIEWS menu configuration, you can specify programs that you want transferred with the aplet, but are not called as menu options. For example, these can be sub-programs that menu

FORTOSTEP.END

FOR name=start-expression TO end-expression [STEP increment]; loop-clause END FOR A=1 TO 12 STEP 1; DISP 3;A: END Note that the STEP parameter is optional. If it is omitted, a step value of 1 is assumed.

Terminates loop. BREAK:

Matrix commands
The matrix commands take variables M0M9 as arguments.

ADDCOL

Add Column. Inserts values into a column before column_number in the specified matrix. You enter the values as a vector. The values must be separated by commas and the number of values must be the same as the number of rows in the matrix name. ADDCOL name;[value1,.,valuen];column_number:

ADDROW

Add Row. Inserts values into a row before row_number in the specified matrix. You enter the values as a vector. The values must be separated by commas and the number of values must be the same as the number of columns in the matrix name. ADDROW name;[value1,., valuen];row_number:

DELCOL

Delete Column. Deletes the specified column from the specified matrix. DELCOL name;column_number:

DELROW

Delete Row. Deletes the specified row from the specified matrix. DELROW name;row_number:

EDITMAT

Starts the Matrix Editor and displays the specified matrix. If used in programming, returns to the program when user presses. EDITMAT name:

RANDMAT

Creates random matrix with a specified number of rows and columns and stores the result in name (name must be M0.M9). The entries will be integers ranging from 9 to 9. RANDMAT name;rows;columns:
Redimensions the specified matrix or vector to size. For a matrix, size is a list of two integers {n1,n2}. For a vector, size is a list containing one integer {n}. REDIM name;size:
Replaces portion of a matrix or vector stored in name with an object starting at position start. start for a matrix is a list containing two numbers; for a vector, it is a single number. Replace also works with lists and graphics. REPLACE name;start;object:
Multiplies the specified row_number of the specified matrix by value. SCALE name;value;rownumber:

SCALEADD

Multiplies the row of the matrix name by value, then adds this result to the second specified row. SCALEADD name;value;row1;row2:
Extracts a sub-objecta portion of a list, matrix, or graphic from objectand stores it into name. start and end are each specified using a list with two numbers for a matrix, a number for vector or lists, or an ordered pair, (X,Y), for graphics. SUB name;object;start;end:

SWAPCOL

Swaps Columns. Exchanges column1 and column2 of the specified matrix. SWAPCOL name;column1;column2:

SWAPROW

Enables you to choose between 1-variable and 2-variable statistics in the Statistics aplet. Does not appear in the Plot Setup input form. Corresponds to the and menu keys in Numeric View. In a program, store the constant name (or its number) into the variable StatMode. 1VAR =1, 2VAR=2. Example 1VAR or 1 StatMode StatMode

Note variables

The following aplet variable is available in Note view.

NoteText

Use NoteText to recall text previously entered in Note view.

Sketch variables

The following aplet variables are available in Sketch view.
Sets a page in a sketch set. A sketch set can contain up to 10 graphics. The graphics can be viewed one at a time using the and keys. The Page variable refers to the currently displayed page of a sketch set. In a program, type graphicname Page

PageNum

Sets a number for referring to a particular page of the sketch set (in Sketch view). In a program, type the page that is shown when SKETCH is pressed. n PageNum
Aplets are the application environments where you explore different classes of mathematical operations. You can extend the capability of the hp 39g+ in the following ways: Create new aplets, based on existing aplets, with specific configurations such as angle measure, graphical or tabular settings, and annotations. Transmit aplets between hp 39g+ calculators via an infra red link. Download e-lessons (teaching aplets) from Hewlett-Packards Calculator web site. Program new aplets. See chapter 16, Programming, for further details.
Creating new aplets based on existing aplets
You can create a new aplet based on an existing aplet. To create a new aplet, save an existing aplet under a new name, then modify the aplet to add the configurations and the functionality that you want. Information that defines an aplet is saved automatically as it is entered into the calculator. To keep as much memory available for storage as possible, delete any aplets you no longer need.
This example demonstrates how to create a new aplet by saving a copy of the built-in Solve aplet. The new aplet is saved under the name TRIANGLES contains the formulas commonly used in calculations involving right-angled triangles.

Extending aplets

1. Open the Solve aplet and save it under the new name. Solve

TRIANGLES

2. Enter the four formulas: O H A H O A C 3. Decide whether you want the aplet to operate in Degrees, Radians, or Grads.

Degrees

4. View the Aplet Library. The TRIANGLES aplet is listed in the Aplet Library.
The Solve aplet can now be reset and used for other problems.

Using a customized aplet

To use the Triangles aplet, simply select the appropriate formula, change to the Numeric view and solve for the missing variable. Find the length of a ladder leaning against a vertical wall if it forms an angle of 35o with the horizontal and extends 5 metres up the wall. 1. Select the aplet. TRIANGLES

COT CSC SEC

Program constants
The program constants are: Category Angle Available name Degrees Grads Radians Standard Fixed Cobweb Stairstep Linear LogFit ExpFit Power Stat1Var Stat2Var Hist BoxW QuadFit Cubic Logist User Sci Eng Fraction

SeqPlot S1.5fit

StatMode StatPlot

Program commands

The program commands are: Category Aplet Command CHECK SELECT SETVIEWS UNCHECK IF THEN ELSE END ARC BOX ERASE FREEZE DISPLAY DISPLAY GROB GROBNOT GROBOR GROBXOR FOR = TO STEP END DO ADDCOL ADDROW DELCOL DELROW EDITMAT RANDMAT PRDISPLAY PRHISTORY PRVAR BEEP CHOOSE DISP DISPTIME EDITMAT FREEZE DO1VSTATS RANDSEED DO2VSTATS SETDEPEND SETINDEP

Branch

CASE IFERR RUN STOP LINE PIXOFF PIXON TLINE MAKEGROB PLOT PLOT REPLACE SUB ZEROGROB UNTIL END WHILE REPEAT END BREAK REDIM REPLACE SCALE SCALEADD SUB SWAPCOL SWAPROW

Drawing

Graphic

Matrix

Print Prompt
GETKEY INPUT MSGBOX PROMPT WAIT SETFREQ SETSAMPLE

Stat-One Stat-Two

Status messages
Message Bad Argument Type Bad Argument Value Infinite Result Insufficient Memory Meaning Incorrect input for this operation. The value is out of range for this operation. Math exception, such as 1/0. You must recover some memory to continue operation. Delete one or more matrices, lists, notes, or programs (using catalogs), or custom (not builtin) aplets (using MEMORY). Not enough data points for the calculation. For two-variable statistics there must be two columns of data, and each column must have at least four numbers. Array argument had wrong dimensions. Need two columns with equal numbers of data values. The function or command you entered does not include the proper arguments or order of arguments. The delimiters (parentheses, commas, periods, and semi-colons) must also be correct. Look up the function name in the index to find its proper syntax. The | (where) function attempted to assign a value to the variable of integration or summation index.
Insufficient Statistics Data
Invalid Dimension Invalid Statistics Data Invalid Syntax

glossary R-1 graph analyzing statistical data in 8-19 auto scale 2-14 box-and-whisker 8-16 capture current display 16-20 cobweb 6-1 comparing 2-5 connected points 8-17 defining the independent variable 16-35 drawing axes 2-7 expressions 3-3 grid points 2-7 histogram 8-15 in Solve aplet 7-7 one-variable statistics 8-18 overlaying 2-16 scatter 8-15, 8-16 split-screen view 2-15 splitting into plot and close-up 2-14 splitting into plot and table 2-14 stairsteps 6-1 statistical data 8-15 t values 2-6 tickmarks 2-6 tracing 2-8
i 11-8 implied multiplication 1-21 importing graphics 15-6 notes 15-8 increasing display contrast 1-2 indefinite integral using symbolic variables 11-23 independent values adding to table 2-19 independent variable defined for Tracing mode 16-32 inference confidence intervals 9-15 hypothesis tests 9-8 One-Proportion Z-Interval 9-17 One-Sample Z-Interval 9-15 One-Sample Z-Test 9-8 Two-Proportion Z-Interval 9-17 Two-Proportion Z-Test 9-11 Two-Sample T-Interval 9-19 Two-Sample Z-Interval 9-16 infinite result R-17 infrared transmission of aplets 17-5 initial guess 7-5 input forms resetting default values 1-9 setting Modes 1-12 insufficient memory R-17 insufficient statistics data R-17 integer rank matrix 13-12 integer scaling 2-14, 2-16 integral definite 11-6 indefinite 11-23 integration 11-6 interpreting intermediate guesses 7-7 intersection 3-11 invalid dimension R-17 statistics data R-17 syntax R-17 inverse hyperbolic cosine 11-8 inverse hyperbolic functions 11-9
inverse hyperbolic sine 11-8 inverse hyperbolic tangent 11-8 inverting matrices 13-8 isect variable 16-32
keyboard editing keys 1-5 entry keys 1-5 inactive keys 1-8 list keys 14-2 math functions 1-7 menu keys 1-4 Notepad keys 15-8 shifted keystrokes 1-6
labeling axes 2-7 parts of a sketch 15-5 letters, typing 1-6 library, managing aplets in 17-5 linear fit 8-13 list arithmetic with 14-7 calculate sequence of elements 14-8 calculating product of 14-8 composed from differences 14-7 concatenating 14-7 counting elements in 14-9 creating 14-1, 14-3, 14-4, 14-5 deleting 14-6 deleting list items 14-3 displaying 14-4 displaying list elements 14-4 editing 14-3 finding statistical values in list elements 14-9 generate a series 14-8 list function syntax 14-6 list variables 14-1 returning position of element in 14-8 reversing order in 14-8 sending and receiving 14-6 sorting elements 14-9 storing elements 14-1, 14-4, 14-5 storing one element 14-6 logarithm 11-4

New! HP 39gs Graphing Calculator Classroom Kit
Article Next Announcing the HP 39gs Graphing Calculator Classroom Kit for Middle Grades. Easy to use and powerful, HP Calculators for grades 6-8 are designed to perform above expectations for math and science students on all levels. These reliable calculators are equipped with easy-to-use problem solving tools, flexible connectivity & customizing options, plus award-winning HP support. Our HP 39gs Graphing Calculator Classroom Kit is a unique solution that provides teachers with the tools and training to engage and motivate middle school students in mathematics. We believe that successful integration of technology into the mathematics classroom is enhanced when it is undertaken as a group or team effort, a belief that is reinforced by independent studies, so we designed the Classroom Kit to help middle school teachers establish a mathematics leadership team on campus. The Classroom Kit features a three-day professional development workshop and supporting curricula to help ensure the successful integration of the HP 39gs Graphing Calculator into math classes. The HP 39gs Graphing Calculator is the heart of the HP Graphing Calculator Classroom Kit, and this all-inone kit includes calculators, carrying case, classroom-ready teaching materials, and a 3-day teacher workshop. The HP 39gs Graphing Calculator was designed by teachers for classroom use and to align with NCTM (National Council of Teachers of Mathematics) Standards. Importantly, and uniquely, the HP 39gs has dedicated keys for symbolic, graphic and numeric views. The Classroom Kit allows teachers to save time in their digital classroom. Using our kit and included calculator emulator software, the teacher can project a virtual calculator during the lesson so students can follow along easily. Plus, teachers can use HP Aplets, either that they've written or downloaded from numerous free resources online. Teachers can set up powerful examples beforehand, saved with common names to ease retention, and then send them to students wirelessly! By utilizing this powerful feature, and given the ability to collect or create these aplets beforehand, teachers save precious class time, and are secure in the knowledge that students will quickly get on the same page, ready to learn. See the Graphing Calculator Classroom Kit for more information.
Feature Calculator of the Month: HP 10bII Financial Calculator
Previous Article Next The HP 10bII Financial Calculator is the smart choice for business, finance and accounting needs, for professionals and students alike. With over 100 time-saving business functions you can calculate loan payments, interest rates and conversions, standard deviation, TVM, cash flows and more. Algebraic notation makes it easy to learn and use. With so many features, at a great price, the HP 10bII adds up to a wise investment. The HP 10bII is permitted for use on the CFP Certification Exam [1]. Fun facts about the HP 10bII Financial Calculator: 2
Born: Introduced December 1st, 2001 Replaced the HP 10b, which was introduced January 3rd, 1989. HP also introduced the HP 20s and HP 21s on the same date. The HP 10BII has a memory called the M register that is usable directly from the basic keyboard. There are three keys associated with the M register. The M key will store the number presently displayed into the M register. Note that this will overwrite any value previously stored in the M register. The RM key will recall the number presently in the M register to the display where it can be used. The M+ key will add the number presently in the display to the number already in the M register. If pressed repeatedly, it will add the number in the display to the number in the M register each time. This is often used to keep up with a running total. For example, to solve ( 2 + 3 ) ( ) using the M register, press 5 - 2 = M 2 + 3 RM = To quickly find the total of 12, 44, 17 and 36, press 12 M 44 M+ 17 M+ 36 M+ RM It is HP's lowest cost financial model, well suited for finance students who do not need the advanced features of other HP models, but who want HP reliability. It provides a simple interface, all the important operations needed in common financial calculations but avoids the complications of programming, scientific functions and RPN.
Click here for more information about the HP 10bII Financial Calculator. [1] CFP is a registered trademark of the Certified Financial Planner Board of Standards, Inc.

RPN Tip #4

Previous Article Next Introduction (by Wlodek Mier-Jedrzejowicz, Richard J. Nelson, & Jake Schwartz) Hewlett-Packard has always made calculators with a user interface, the method used to solve problems, most suitable for the general class of problems to be solved. The four basic types of calculator user interface are: Arithmetic, ATH; Algebraic, ALG; Reverse Polish Notation, RPN; and Command Line Interface, CLI. RPN is, over all, the most effective and efficient way to solve a very large class of problems, and HP is well known for its RPN machines. RPN is a Really Productive Notation RPN is different, and to most people it is unfamiliar. It is easy to learn and it is the most efficient way to solve a large class of problems because you treat each part of the problem in exactly the same way. The rules, compared to Algebraic, ALG, are fewer and simpler. The purpose of the Newsletter series of RPN Tips is to describe RPN techniques so new and experienced users may improve their calculator skill. The User Interface The two most popular general purpose user interfaces for midrange calculators are RPN and ALG. The user interface is the means by which the calculator user presses keys to solve problems. The user interface is the "rules" that the user (and the machine) must follow in order to give the machine the numbers (data) and especially the operators +, -, , & so the machine solves the problem correctly. Suppose you had the problem of adding one and two and multiplying the result by three. The problem would be mathematically written as (1 + 2) 3 = 9. In order for the machine to know that the problem is not 1 + (2 3) = 7, both the user and the calculator must know the rules of what the problem is, and how it is to be solved. Using parentheses and inserting the operators between the data is the problem form taught 3
in school and most people are familiar with it. The use of parentheses (and brackets and curly braces) is usually taught in a beginning algebra class. The problem, 1+23=, and pressing the keys as shown, is often used to identify the user interface. If the answer is 7 it is ALG. If the answer is 9 it is ATH. It is interesting to note that without parentheses you cant solve the problem using RPN. This is another example of the rules. The calculator, however, cant think like we can so it must be presented with the data and the operators in a very precise order. This precise order is where the rules are very important. The simple example problem above requires the machine to understand the proper sequence of performing the operators. The parentheses group the various portions of the problem and they show how the portions are related. In addition to knowing the correct order of the data and the operators, the machine must also be able to store intermediate results so that they may be correctly integrated with the current calculation. In this case the addition of one and two is performed first and the result is then multiplied by three. This method of solving the problem, e.g. (1 + 2) 3, and representing and solving problems of this type is called Algebraic, ALG, notation. From a mathematical logic perspective this is called infix notation. In order to present the problem to the calculator the three numbers must be entered, the two operators (+ and ) must be entered, and the order in which calculation is to be made must be conveyed. In addition to this, the machine must know that a previously calculated result is to be used with another number and operator. The parentheses group the various operations and the general rule is to solve the problem from the inside out by performing the operations in the inner most parentheses first. This is true no matter what calculator you are using. There is another notation called postfix notation. Using postfix notation the problem above may be represented as + 3. Solving the problem in postfix notation is what the machine does internally regardless if the user interface is ALG or RPN. Infix is 1 + 2 3; Postfix is + 3. RPN uses postfix notation as the logical interface. RPN does not require an equal key. RPN does not use parentheses. The machine must, however, know when the user presses the 1 and the 2 key that there are two numbers of interest, 1 and 2, and not the single number 12. In other words any number keyed into the calculator must be terminated (completed). RPN machines use the ENTER key to terminate numeric inputs. In terms of the keys pressed to solve our example problem the notation would be: 1 ENTER 2 + 3. The desired answer, 9, appears in the display immediately when the multiply key, , is pressed. The intermediate result, of 1 + 2, also appears when the add operator is pressed and 3 appears in the display. Equal (or a key) is not used for the RPN user interface. The ability to integrate parts of a calculator problem by storing intermediate results is a very important aspect of calculator problem solving rules that must be learned by the user. For the ALG interface the use of parentheses and the hierarchy of operations are the most important rules. The use of parentheses is the primary means of storing intermediate results. RPN does not have a list of hierarchy rules. ALG may six or more, in some cases as many as 11, depending on how many operators the machine has. RPN uses an automatic memory stack, see below, to store intermediate results. Neither ALG nor RPN is able to solve all problems exactly as they are written on paper. A more accurate form of using mixed data and operators utilizes a user interface that is predominantly found on the higher end calculators, the Graphing Calculators, and it is called Command Line Interface, CLI. All graphics calculators (and computers) use CLI. Arithmetic, ATH, is the user interface found on adding machines. It is the simplest and it doesnt have any rules except that it does the operators such as +, -, , & when the keys are pressed.

The RPN Stack The ENTER key, , and a group of four data storage registers, called an automatic RPN stack, is what makes RPN very powerful and simple to use. The stack may be visualized as a stack of shoe boxes as shown in figure 1. Each stack shoe box, register, is vertically related to each other. The contents of the X register are shown in the display. The original RPN machine, the HP-35A, utilized a single line display. More recent machines that have a multiline display may also show additional stack registers in the order of Y to T. Register Name T Z Y X Stack Register
Fig. 1 Automatic four high stack. Basic RPN Aside from the general rule we learned in school that is used for all moderately complex problem solving perform the operations by solving the parentheses from the inside out the basic RPN rule is: if an operation is indicated, do it. The example given above: (1+2) x 3 =? is 1 ENTER 2 + 3. Pressing the ENTER key performs three operations. (1) Pressing ENTER terminates the number so the calculator knows that the complete number has been keyed into the calculator. (2) Pressing ENTER also makes a copy of the X register data by also storing it on the stack in the Y register. The contents of the registers above are pushed up i.e. the contents of the Y resister are copied into the Z register. The contents of the Z register are copied into the T register and the contents of the top register, T, are lost. (3) Pressing ENTER also prepares the machine for accepting additional data or a keyboard operation. RPN Stack Operators (see Table 1) Using and controlling the RPN stack utilizes the five RPN operators. RPN utilizes postfix notation and just important is that RPN implies the automatic stack as shown in figure one above. Postfix notation, the use of a stack, and five stack operators is what defines RPN as a user interface. The five basic RPN Stack operators are: 1. ENTER,, is the most important RPN operator. See the description under Basic RPN above. ENTER is never a shifted operator on the RPN keyboard. 2. is the second most important RPN operator. exchanges the contents of the X and Y registers. is never a shifted operator on the RPN keyboard.
3. Roll down, R : The third most important RPN operator. Roll down rotates the (shoe box) stack downward. The contents of X are copied into T, T into Z, X into Y, and Y into X. Roll down is a primary operator on all RPN models except the HP-34C, 37E, and 38E/C. 4. LAST X is the very important error correction operator. LAST X recalls the value of the X Register prior to the most recent operation performed. LAST X is a shifted operator on all RPN models except the six early models that do not have it the HP-35A, 21, 22, 37E, 70A, and 80A. 5. Roll up, R , rotates the stack upward. The contents of X are copied into Y, Y into Z, Z into T, and T into X. R is a shifted operator and it is not found on 24 of the 43 RPN models (58%). The roll down, R , operator allows a quick verification of the stack contents by pressing the primary stack roll key four times in succession. 5

RPN Stack Diagrams In the days before RPN calculators were programmable and the basic RPN calculator was the optimum choice for complex problem solving, it was useful to make a stack diagram of the keystrokes used to repeatedly solve very complex problems. This was especially useful for iterative solutions and engineers carried 3 x 5 cards in their shirt pockets with these keystroke solutions written on them. To illustrate a stack diagram lets show the stack to make a calculation involving three values, each used twice. The best approach is one that utilizes the stack in such a way that the values are keyed only once to avoid errors. An example problem is shown below. Where: A =4 B =3 C = -2
The stack registers are identified at the left. The tilde symbol indicates any (dont care) value. The press row indicates the inputs that are keyed to solve the problem. These are the keystrokes that are recorded to solve the problem. The stack diagram is primarily used for illustration and analysis. The values involved in the example are shown in both symbolic and numerical form (where practical) so the user may solve the problem by pressing the inputs as indicated and see the stack values. The solution steps are numbered above the stack diagram. Step 9 T Z Y ~ ~ ~ ~ ~ B=3 B=3 ~ B=3 B=3 B=18 T Z Y X press C=-2 C=-2 B=3 (A+B)(A-C)=C=-2 C=-2 B=3 R 42 C=-2 B=3 C=-42 C=-2 B-C=42 B-C=5 C=-2 C=-42 B-C=5 R C=-2 C=-3 B=3 B=3 B=3 A=4 A=4 B=3 B=3 B=3 A+B=7 + 5 B=3 B=3 A+B=7 A=4 LASTX B=3 A+B=7 A=4 C=-2 C=-7 B=3 B=3 A+B=7 A-C=B=3 A+B=7 A-C=6 C=-2 LASTX 8 C=-2 B=3 A+B=7 A-C=6 R

B=3 B=3 B=3 B=3

X B=3 press B=3 Step 17
The reader is encouraged to solve the problem for practice. The stack diagram is useful for the following reasons. (1) A stack diagram illustrates how the stack works. (2) Solving complex problems repeatedly is faster if an optimum keystroke solution is known. What is the value of the above expression if: A = 1.41421357348, B = -1.73205091868, and 6
C = 2.23606808860? Solve the problem without looking at the stack diagram. See the correct answer at the end. Did you get this answer on your first try without using the stack diagram? (3) A stack diagram shows the values needed for future use in order to determine the best solution sequence especially in preparation for reason 4. (4) The optimum solution for a program that may solve the problem multiple times and program solution speed is an issue. The stack diagram is a powerful optimization tool. RPN Stack Operator Table Table 1 below shows all RPN models to date (April 2008) grouped into their series in approximate introduction order. It also shows the classic RPN stack operators with a weight value assigned to each of them. The weight value is used to compare the various models to each other for their RPNness in terms of the RPN keyboard and the included operators. The colors provide a quick overview of the model and the operator use on that model, i.e. if it is used, and if it is a shifted operator. The ideal RPN machine would have an index of 9. Obviously there are many keyboard tradeoffs that must be made for the intended class of problem the model has to solve. Table 1 HP RPN Calculator Stack Operators And Keystroke Index y = full value, s or ALPHA = half value, n = no value 1 Ideal Index = 9 Ks LASTx Index n 7 s 7.5 s 7.5 * = on top row when s 6 undefined * = on top row when s 6 undefined n 7 n 7 * = key marked SAVE n 7 s 7.5 s 7.5 n 7 s 7.5 s 7.5 s 7.5 s 7.5 s 7 n 6 s 7.5 s 8 s 7.5 s 8 s 7.5 s 8 s 8 s 7.5

Weight

3 ENTER y y y y y y y* y y y y y y y y y y y y y y y y y y
2 R y y y s* s* y y y y y y y y y y s s s y y y y y y y
2 X<>Y y y y s* s* y y y y y y y y y y y y y y y y y y y y
1 R n n n s s n n n n n n n n n n s n n ALPHA n s n s s n
35A 45A 55A Classic Series 65A 67A 70A 80A 29C 31E 32E 33E/C 34C 37E 38E/C 41C/CV/CX 10C 11C 12C 15C 16C 12Cp

Voyager Series

Spice/Spike Series

Woodstoc k

Pioneer Series

95C 97 19C 32S 32SII 42S

y y y y y y y y y* y* y* y y* y*
y y y y y y y y y y y y y y
y y y s n n s ALPHA n n s s n n
s s s s s s s s s s s s s s

8.5 8.5 8.7.5 7.8

Topcat Series
17BII 19BII 33S 35S 17BII+ Gold 03 17BII+ Silver 07 Total: 43 machines
* = SMALL key marked 7.5 INPUT 7.5 * = key marked INPUT 8 * = SMALL ENTER key 8 7.5 * = key marked INPUT 7.5 * = key marked INPUT Table by Jake Schwartz 1/08
Color legend: Green indicates primary operator. Red indicates stack operator not on the machine. Yellow indicates shifted operator. RPN And ALG All of the four basic user interfaces mentioned above (ATH, ALG, RPN, and CLI) have been used in HP calculators e.g. the 10 model number has been used five times since 1972. One was ATH, two were ALG, and two were RPN. All graphing machines are CLI. Each interface has its advantages and disadvantages. Over all, RPN is the most effective (requires less thinking, which is hard work, is faster, and usually requires fewer keystrokes) for a very large class of problems. ALG often seems easier because your problem is often expressed in an algebraic form. If you understand RPN and you dont use your calculator frequently you will always be able to remember the rules of how to solve problems because the rules are very few and they are obvious. There are two primary reasons that RPN is more effective compared to ALG. The first is the basic structure of RPN. The second reason is that you are able to see the problem unfold as work your way through it. All too often ALG keeps parts of the problem hidden until the final EQUAL. RPN users do not need to clear anything (the stack, or the X register) before starting a problem. In fact, the vast majority of RPN machines do not have a clear stack operator. You just start keying in the data. ALG users, however, know that if they dont start with a clean slate so to speak, they will get wrong answers and may have to do the problem all over again which usually means doing it over twice for insurance that they have the correct answer. ALG users soon learn to clear their machines before starting and if you watch an ALG user solve a problem you will see them pressing clear multiple times, just to be sure. In a similar way you will notice an inexperienced RPN user press zero and ENTER three times to clear the stack, just to be sure. The user interface of choice is often the one we first learned. Taking the time to master RPN, however, will save you endless key pressing hours. If, however, you are an HP calculator user it you may select the either ALG or RPN on many models. The HP-42S Owners Manual provides a very nice list of the advantages of RPN.

Advantages of RPN Remember: This method of entering numbers, called Reverse Polish Notation (RPN), is unambiguous and therefore does not need parentheses. It has the following advantages: You never work with more than two numbers at a time. Pressing a function key immediately executes that function so there i s n o n e e d f o r a n = key. Intermediate results appear as they are calculated, so you can check each step as you go. Intermediate results are automatically stored. They reappear as they are needed for the calculationthe last result stored is the first to come back out. You can calculate in the same order as you would with pencil and p a p e r. If you make a mistake during a complicated calculation, you don't have to start over (Correcting mistakes is covered in chapter 2,) Calculations with other types of data (such as complex numbers and matrices) follow the same rules. Calculations in programs follow the same steps as when you execute them manually. Keyboard or Program Solution Many RPN calculators are programmable. A program is simply a memorized list of keys pressed to solve a problem. In addition to the normal operators provided on the keyboard a programmable model provides a means of naming, starting, and stopping, a program. Another feature of a program is a format for inputting the data and running the program to get an answer. Once a program is proved to solve a complex problem (debugged) it may be run whenever the problem needs to be solved. This may be long after you have forgotten how to solve the problem. Another very important aspect of solving your problem using a program is that the program may test values and make decisions based on these tests. This feature is what makes running a program so powerful. Still another advantage of a program is executing the same series of keystrokes multiple times. This is called looping. Looping is an important feature of a programmable calculator. The straightforward logic and structure of RPN makes keystroke programming easy to learn and use. In some instances the program may make many calculations in a loop and speed becomes an important consideration. RPN saves keystrokes and speeds up programs. Conclusion No matter what midrange calculator you use, all problems are not solvable as you write them on paper. It takes a high end graphing calculator with its Command Line Interface to be able to do this. You may find examples where one of the simpler machines will solve the problem correctly, but as you increase the complexity of the problem you will have to know more of the user interface rules. The Arithmetic, ATH, user interface has no built in logic. This leaves ALG and RPN. Of these two, RPN has fewer rules and is always consistent in the process in which you solve problems. Like any well thought out system there are many ways in which a problem may be solved. The HP Solve newsletter series of RPN Tips provides techniques to better understand and improve your RPN skills. Answer to three variable, A,B,C problem: Click here to learn more about RPN.

Lets make music with math!
Previous Article Next It may come as a surprise to some that music and mathematics are intimately connected. Both are consequences of human creativity, both involve intricate patterns, both can be at once beautiful and daunting, and both can be appreciated more deeply through conscientious study. Pleasant harmonious patterns of music reveal themselves in exquisite numerical patterns of mathematics. This activity focuses on some of the connections between mathematics and music. Exercise 1: Start the MUSICMATH aplet on the HP39gs. You are presented with two columns of numbers in a table. The numbers in column C1 are used to designate different musical notes. The numbers in column C2 are the frequencies. The chart below is provided for those with some musical background, and relates the numbers in C1 to the names of different musical notes. The frequencies are repeated here, rounded to the nearest hundredth, for your convenience. Number 13 Note A4 A# B4 C5 C# D5 D# E5 F5 F# G5 G# A5 Frequency (hz) 440 466.16 493.88 523.25 554.37 587.33 622.25 659.26 698.46 739.99 783.99 830.61 880
The 39gs has a built-in speaker, and a command to play any note for any length of time. To play the first note in the table (A4) for 2 seconds, go to the HOME screen and enter the command BEEP 440;2 as shown in the screen below.
Find another student with a 39gs. Have your partner play the last note (A5 or 880 hz) at the same time you play the first note (A4 or 440 hz). Hold the calculators close together, so the notes blend. Try playing two E's together by executing BEEP 660;2 on one calculator, and BEEP 1320;2 on another at the same time. (Notice that 660 is pretty close to the frequency for E5).
The harmonies you just heard results from playing the same note in two different octaves (an octave consists of one complete set of twelve notes). These are the simplest type of harmonies. 1. What relationship do you notice between the frequencies of these harmonious notes (440 with 880; 660 with 1320)? Create a second list of frequencies one octave above those in column C2 by executing the command shown in the screen shot below.

You can play the entire scale by using the column name with the BEEP command. Try it!
Playing octaves on two calculators Then try playing the scale in C2 while your partner plays the scale one octave higher, in C3, as shown above on the right. Try to start the scales simultaneously. Now try playing 440 (A4) and 660 (E5) together. 2. Do they sound harmonious to you? (If the speakers on the 39gs are not adequate to hear the harmonies, try the first web resource, or a piano!). 3. Can you see any connection between the frequencies 440 hz and 660 hz? Legend has it that Pythagoras (the same guy from the Pythagorean Theorem) was one of the first to recognize the harmonious tone that results from playing notes such as these together. 4. Try playing an entire scale where the frequencies are in the ratio 3:2 by multiplying C2 by 1.5 and storing the result in C3. Use two calculators to play the resulting frequencies along with the originals. Then repeat after multiplying C2 by a nastier ratio like 17/13. What do you find? Exercise 2: What's So Special About Harmonious Notes In Exercise 1, you saw (or rather heard!) that certain musical notes sound nice when played together. What attributes do these notes have that produces the pleasing sound? Let's look at ratios. 1. What is the ratio of the frequency of A5 to the frequency of A4? 2. What is the ratio of the frequency of E5 to A4? 11
Let's look for notes whose frequencies are in small integer ratios. The 39gs can help in the hunt! First, we'll round the frequencies we have to integers. Then we'll multiply the rounded frequencies by small integers. Finally, we'll search the table of frequencies for matches. Execute the commands shown in the screen shots:
3. Look around in the table for frequencies that are approximately the same. Then try playing them together and listen for the harmony. Record your results, and compare with other groups. Harmonies occur when the notes are in "nice" integer ratios, such as 2:1 or 3:2. Notes in the ratio 2:1 are called an octave. Those in the ratio 3:2 are called a fifth. Ratios of 4:3 are called a fourth. C and F produce a fourth, since 4 523.25 = 2093 , while 3 698.46 = 2095.38. Pretty close. Try playing notes in the ratio 1:3 (such as A3, 220, and E5, 660) at the same time. Notes in the ratios 1:2, 1:3, 1:4, 1:5, etc are called harmonics. You may encounter the harmonic series in Precalculus or Calculus. It is the sum
+ + + +. , where the ellipsis means the sum goes on forever. 4
Sometimes, three notes played together produce a pleasant sound. Find another partner and play A, C#, and E together. This chord is called a major triad. 4. Can you spot the ratio of the frequencies? Hint: Round the frequency of C# to 550. Exercise 3: A Musical Model Take another look at the scale on the first page. Notice as we go from 440 hz (A4) to the next higher A, 880 hz (A5), we've doubled the frequency. We saw in Exercise 1 that pairs of notes that are in the ratio 1:2 produce a certain pleasant harmony. But how do the notes progress within one scale? What choices could there be? One approach would be to divide the scale so that the arithmetic difference between adjacent notes was a constant. For example, 880 hz 440 hz = 440 hz. Divide this into 12 equal steps. 1. If the scale were made this way, what would the "step" between each note be? 2. Fill in the table for the notes from 440 hz to 880 hz according to this "arithmetic" scale. Does this arithmetic scale match the scale from the table given in Exercise 1. Number 12 Note A4 A# B4 C5 C# D5 D# E5 F5 F# Frequency (hz) 440

G5 G# A5

Since harmonies occur when the ratio of notes are small whole numbers, it makes more sense to arrange the scale so that the ratios of adjacent notes is the same. Let's calculate these ratios using the values in the MUSICMATH aplet. In the 39gs, the first value in column C2 is referred to as C2(1). Calculate the ratio C2(2)/C2(1), as shown.
3. What is the ratio? After you've calculated this ratio, press the up arrow key, highlight the expression C2(2)/C2(1), and press the soft COPY key. Then edit (using the DEL key and inserting replacement digits) to produce the ratio C2(3)/C2(2). Repeat for C2(4)/C2(3). 4. What are C2(3)/C2(2) and C2(4)/C2(3) ? Clearly, the ratios of adjacent notes are the same through the scale. If we call this ratio r, then all we do to get the next note in the scale is to multiply the previous note by r. That is, with A4 = 440 hz, A# is 440r. Store the ratio you calculated into the calculator variable R, as shown. Just press the STO> soft key on the HOME screen right after you calculated one of the ratios, then press ALPHA R.
Now to get the next note above A4, calculate 440 R as shown. Again, use ALPHA R to type the letter R:
The next note is always the previous note multiplied by R. To get B4, we could enter 440 R R. (The COPY feature will help you out!).
This quickly becomes tedious. Mathematicians use exponents as a shorthand for repeated multiplications like this. For example 440 R R = 440 R , 440 R R R = 440 R , and so on. On the 39gs, exponents
are entered using the key labeled XY. Use this key to evaluate 440 R and 440 R.
440 R11 = ______________________ and 440 R12 = _____________________.
6. So, when R is multiplied by itself 12 times, what do you get? That is, what is R12 ? We call the number R the twelfth root of 2 and write it like this: R = 12 2. This number, like irrational. Overview This exploration into the connection between music and mathematics begins with listening to octaves, which are notes with frequencies in the ratio of 2:1. Most, but not all, westerners sense the harmonies from these octaves. Also in exercise 1 is a 3:2 harmony. This exercise is largely qualitative. Exercise 2 extends and quantifies the observations from Exercise 1. By making a large table of frequencies that are small integer multiples of each other, and searching for values that are close, students find more chords that produce harmonious tones. Exercise 2 also includes a triad (cord with 3 notes). In Exercise 3, students explore the relationship between frequencies within one scale. In modern western music, the scale is based on "equal temperament", where the ratio between adjacent notes is exactly the same through the scale. Since there are 12 divisions within the scale, and the next octave has frequencies twice those in the previous, each adjacent frequency pair have a ratio of

2 and , is

The last observations in Exercise 3 lead to a mathematical model for an equal temperament scale, and this is explored in Extension 1. The extension should be used with students who have some experience graphing equations. Students graph the frequencies of the notes, enter the model, and see that the model fits the frequencies. They also create a scale based on equal differences between adjacent notes, and se that the graph of these leads to a linear model (though no attempt is made to create the symbolic form of the model). Note that Exercise 3 stands by itself, and could be presented without doing Exercise 1 or 2 at all. Exercises 1 and 2 should be done together. Extension 1 fits with Exercise 3. Extension 2 fits with Exercises 1 and 2. Extension 1: Enter the notes created by the arithmetic scale in Exercise 3 into column C4 of the MUSICMATH aplet. Then press SYMB. Define two scatter plots as shown below.
Press VIEWS and select Autoscale. 1. Describe the difference in the shapes of the two graphs.
2. Evaluate the expression 12 when X = 1 and when X = 2. Recall that the first value (i.e. when X = 1) in our column C2 of notes is 440, and that the second is about 466.16.
Back in the SYMBolic view, enter 12 for S1FIT. Go to the PLOT, press the soft MENU key, then press the soft FIT key. You should see that S1FIT hits all the points on the S1 scatter plot! Extension 2: Research the musical term dissonance, and find out what it has to do with this activity. Extension 3: Find music for a song and distribute a measure to each student. Have them put the notes into a list, and organize themselves to play back the song. Teacher Notes The following TEKS (and others) can be found in this activity set: 6.1C, 6.1F, 6.2C, 6.3A, 6.5, 6.11A, 6.11D, 6.13, 7.2B, 7.2D, 7.14A, 7.15, 8.1C, 8.2D, 8.4, 8.14A, 8.15A, 8.16A Here are the TEKS organized by Exercise and Extension: Table 5: TEKS Covered, by Activity Activity TEKS Covered Exercise 1 6.1C, 6.1F, 6.2C, 6.11A, 6.11D, 7.2B, 8.2D, 8.14A Exercise 2 6.1C, 6.1F, 6.2C, 6.11A, 6.11D, 7.2B, 8.14A, 8.16A Exercise 3 6.3A, 6.13, 7.2B, 7.2D, 8.1C, 8.2D, 8.14A Extension 1 6.3A, 6.5, 7.14A, 7.15A, 7.15B, 8.4, 8.14A, 8.15A Extension 2 6.11A, 8.14A Web Resources Select and play combinations of notes and see their wave forms superimposed on this musical scale. Read the Wikipedia entry on Consonance and Dissonance. See the selection of questions and answers about music and math from Dr Math.

Answers Exercise 1 1. The frequencies are in the ratio 1:2 2. Most students should answer yes. To some, the harmony is not apparent. 3. There's no guarantee students will see what we want here. The relevant connection is the 2:3 ratio. 4. The scales in the ratio 3:2 sound harmonious to most people; the scales in the ratio 17:13 sound less pleasant. Exercise 2 15
1. 2:1 2. 3:2 3. One possibility is C and F, in the ratio 4:3. A and D are also in the ratio 4:3. In general, notes that are seven rows apart in the table are in the ratio 3:2. Notes five rows apart are in the ratio 4:3. 4. 4:5:6 Exercise 3 1. 440/12 = 36.67 2. Number 13 3. 1.059 4. Both are 1.059 5. 830.61 and 880, which are the frequencies of G# and A5 6. 2 Note A4 A# B4 C5 C# D5 D# E5 F5 F# G5 G# A5 Frequency (hz) 440 476.67 513.586.67 623.696.67 733.806.67 843.33 880
Extension 1 1. S1 is curved; S2 is linear 2. 440 and 466.15, the first two notes in our scale. Extension 2 According to Wikipedia, dissonance refers to harmonies which create more complex acoustical interactions (called 'beats'). Consonant harmonies are made up of tones that complement and increase each other's resonance. Generally speaking, dissonant intervals involve frequencies with ratios that are larger integers. (Or, at least, that's what this non-music theorist believes!) However, this notion seems to be culturally dependent.
HPs most powerful financial calculator is now sleeker than ever
Previous Article Next The new HP 17bII+, HPs most powerful financial calculator, makes a great gift for Mothers Day this May. The attractive, sleek design makes it very desirable for professionals in real estate, finance, accounting, economics and business. It is the only HP calculator with menu keys which is easy to use and allows you to customize and program your keys. The HP 17bII+ is easy to use and approved for many courses and exams. The calculator is approved by the CFP and MFA and is extremely popular among finance and business professionals with over 250 functions. With this powerful financial calculator, you can quickly calculate loan payments, interest rates and conversions, standard deviation, percent, TVM, NPV, IRR, cash flows, bonds and more. It even has special unique features such as RPN and Algebraic data entry, clock, appointments, calendar, HP solve application, menu prompts and messages. Now you can solve all your financial calculations and more with one calculator. HP provides FREE calculator training specific to our calculator models. Check out the links below: CLICK HERE FOR FREE HP 17bII+ CBT TRAINING CLICK HERE FOR FREE HP 17bII+ INTRO TO FINANCE VIDEO TRAINING CLICK HERE FOR FREE HP 17bII+ LEARNING MODULES

The 17bII A Financial Analysts Best Friend
Previous Article The HP 17bII+ financial calculator satisfies the demands of higher education students as well as business professionals in the fields of real estate, investment and finance. It is capable of handling everything from amortization to the zero coupon bond. It can keep track of more than 1,000 cash flows and will calculate four types of depreciation. In all, there are more than 250 built-in functions in the 17bII+, making it the most powerful financial calculator in HP's current line-up. The hp 17bII+ financial calculator's power is easy to use because it has a menudriven interface that simplifies each step in a calculation. Users can switch between traditional algebraic input mode and Reverse Polish Notation (RPN). RPN is HP's efficient input method that dramatically reduces the number of keystrokes required. The device also includes a comprehensive 310-page manual that shows how to perform and solve what-if analyses on hundreds of business calculations.
In addition, the hp 17bII+ includes the HP Solve application that permits the user to enter and store complex equations using complete words for variables. The two-line, 22-character LCD screen has plenty of room to display long equations and relevant labels during the input process. The hp17bII+ is capable of solving a vast array of problems that real estate, mortgage lending, banking, investment and financial professionals encounter every day. Thats what it is often referred to as a financial analysts best friend.