HP 39G Graphing Calculator
Graphing - HP
Product Type: Products - Weight: 1.80
Brand: Hewlett Packard
Part Number: HPC39GS
Here you can find all about HP 39G Graphing Calculator, for example manual and review. You can also write a review. [ Report abuse or wrong photo | Share your HP 39G Graphing Calculator photo ]
HP 39G Graphing Calculator, size: 9.8 MB
HP 39G Graphing Calculator Annexe 1
HP 39G Graphing Calculator 39g+ (39G & 40g)_mastering The 39g+__e_f2224-90010.pdf
HP 39G Graphing Calculator
HP 39GS graphing calculator with StreamSmart
User reviews and opinions
|75martin||4:42pm on Wednesday, October 6th, 2010|
|Great Calculator I bought this for my 16 year old son and he loves it. It is a lifesaver for most AP courses in high school.|
|pascal||5:55am on Thursday, April 8th, 2010|
|Good value for money Very nice calculator, especially considering the price. Good value for money Very nice calculator, especially considering the price.|
|taxwork||4:46am on Tuesday, March 30th, 2010|
|i dont really like this calculator this calculator is kind of confusing. it works but if you are taking a class that needs both radian and percentage,...|
Comments posted on www.ps2netdrivers.net are solely the views and opinions of the people posting them and do not necessarily reflect the views or opinions of us.
Meaning (Continued) Displays a menu of all available characters. To type one, use the arrow keys to highlight it, and press. To select multiple characters, select each and press , then press.
There are two shift keys that you use to access the operations and characters printed above the keys: and. Key Description Press the key to access the operations printed in blue above the keys. For instance, to access the Modes screen, press , then press. (MODES is labeled in blue above the key). You do not need to hold down when you press HOME. This action is depicted in this manual as press MODES. To cancel a shift, press again.
The alphabetic keys are also shifted keystrokes. For instance, to type Z, press Z. (The letters are printed in orange to the lower right of each key.) To cancel Alpha, press again. For a lower case letter, press. For a string of letters, hold down while typing.
The hp 39g+ built-in help is available in HOME only. It provides syntax help for built-in math functions. Access the HELPWITH command by pressing SYNTAX and then the math key for which you require syntax help.
Note: Remove the left parenthesis from built-in functions such as sine, cosine, and tangent before invoking the HELPWITH command.
) is the place to do calculations.
Keyboard keys. The most common operations are available from the keyboard, such as the arithmetic (like ) and trigonometric (like ) functions. Press to complete the operation: 256 displays 16.
MATH menu. Press to open the MATH menu. The MATH menu is a comprehensive list of math functions that do not appear on the keyboard. It also includes categories for all other functions and constants. The functions are grouped by category, ranging in alphabetical order from Calculus to Trigonometry. The arrow keys scroll through the list ( , ) and move from the category list in the left column to the item list in the right column ( , ). Press to insert the selected command onto the edit line. Press to dismiss the MATH menu without selecting a command. Pressing displays the list of Program Constants. You can use these in programs that you develop.
To save aplet configuration
The most commonly used math operations are available from the keyboard. Access to the rest of the math functions is via the MATH menu ( ).
CMDS. To access programming commands, press See Programming commands on page 16-13 for further information.
Where to start
The home base for the calculator is the HOME view ( ). You can do all calculations here, and you can access all operations. Enter an expression into the hp 39g+ in the same leftto-right order that you would write the expression. This is called algebraic entry. To enter functions, select the key or MATH menu item for that function. You can also enter a function by using the Alpha keys to spell out its name.
Press to evaluate the expression you have in the edit line (where the blinking cursor is). An expression can contain numbers, functions, and variables.
8 Calculate --------------------------- ln ( 45 ) : 3 45
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.
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
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
When you retrieve a number from ANS, you obtain the result to its full precision. When you retrieve a number from the HOMEs display history, you obtain exactly what was displayed. Pressing evaluates (or re-evaluates) the last input, ANS copies the last result (as ANS) whereas pressing into the edit line.
Storing a value in a variable
You can save an answer in a variable and use the variable in later calculations. There are 27 variables available for storing real values. These are A to Z and. See Chapter 12, Variables and memory management for more information on variables. For example: 1. Perform a calculation. 3
2. Store the result in the A variable. A
3. Perform another calculation using the A variable. A
Accessing the display history
Pressing enables the highlight bar in the display history. While the highlight bar is active, the following menu and keyboard keys are very useful: Key , Function Scrolls through the display history. Copies the highlighted expression to the position of the cursor in the edit line. Displays the current expression in standard mathematical form. Deletes the highlighted expression from the display history, unless there is a cursor in the edit line.
Clears all lines of display history and the edit line.
Clearing the display history
Its a good habit to clear the display history ( CLEAR) whenever you have finished working in HOME. It saves calculator memory to clear the display history. Remember that all your previous inputs and results are saved until you clear them.
To work with fractions in HOME, you set the number format to Fractions, as follows:
Setting Fraction mode
1. In HOME, open the HOME MODES input form.
2. Select Number Format, press options, and highlight Fraction.
to display the
3. Press to select the Number Format option, then move to the precision value field.
4. Enter the precision value that you want to use, and press to set the precision. Press to HOME. to return
See Setting fraction precision below for more information.
Setting fraction precision
The fraction precision setting determines the precision in which the hp 39g+ converts a decimal value to a fraction. The greater the precision value that is set, the closer the fraction is to the decimal value. By choosing a precision of 1 you are saying that the fraction only has to match 0.234 to at least 1 decimal place (3/13 is 0.23076.). The fractions used are found using the technique of continued fractions. When converting recurring decimals this can be important. For example, at precision 6 the decimal 0.6666 becomes 3333/5000 (6666/10000) whereas at precision 3, 0.6666 becomes 2/3, which is probably what you would want. For example, when converting.234 to a fraction, the precision value has the following effect:
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 ).
Getting started with the Parametric aplet
The following example uses the parametric equations x ( t ) = 3 sin t y ( t ) = 3 cos t Note: This example will produce a circle. For this example to work, the angle measure must be set to degrees.
Open the Parametric aplet
1. Open the Parametric aplet. Select Parametric
2. Define the expressions. 3 3
Set angle measure
3. Set the angle measure to degrees.
4. Display the graphing options.
The Plot Setup input form has two fields not included in the Function aplet, TRNG and TSTEP. TRNG specifies the range of t values. TSTEP specifies the step value between t values. 5. Set the TRNG and TSTEP so that t steps from 0 to 360 in 5 steps. 360 5
Plot the expression
6. Plot the expression.
7. To see all the circle, press
8. Plot a triangle graph over the existing circle graph.
Select Overlay Plot
A triangle is displayed rather than a circle (without changing the equation) because the changed value of TSTEP ensures that points being plotted are 120 apart instead of nearly continuous. You are able to explore the graph using trace, zoom, split screen, and scaling functionality available in the Function aplet. See Exploring the graph on page 27 for further information.
Display the numbers
9. Display the table of values.
You can highlight a t-value, type in a replacement value, and see the table jump to that value. You can also zoom in or zoom out on any t-value in the table. You are able to explore the table using , , build your own table, and split screen functionality availablfe in the Function aplet. See Exploring the table of numbers on page 2-18 for further information.
Getting started with the Polar aplet
Open the Polar aplet
1. Open the Polar aplet. Select Polar Like the Function aplet, the Polar aplet opens in the Symbolic view.
Define the expression
2. Define the polar equation r = 2 cos ( 2 ) cos ( ). 2 2
Specify plot settings
3. Specify the plot settings. In this example, we will use the default settings, except for the RNG fields.
4. Plot the expression.
Explore the graph
5. Display the Plot view menu key labels.
The Plot view options available are the same as those found in the Function aplet. See Exploring the graph on page 2-7 for further information.
6. Display the table of values for and R1.
The Numeric view options available are the same as those found in the Function aplet. See Exploring the table of numbers on page 2-18 for further information.
About the Sequence aplet
The Sequence aplet allows you to explore sequences. You can define a sequence named, for example, U1: in terms of n in terms of U1(n1) in terms of U1(n2) in terms of another sequence, for example, U2(n) in any combination of the above.
The Sequence aplet allows you to create two types of graphs: A Stairsteps graph plots n on the horizontal axis and Un on the vertical axis. A Cobweb graph plots Un1 on the horizontal axis and Un on the vertical axis.
Getting started with the Sequence aplet
The following example defines and then plots an expression in the Sequence aplet.
Open the Sequence aplet
1. Open the Sequence aplet. Select Sequence The Sequence aplet starts in the Symbolic view.
2. Define the Fibonacci sequence, in which each term (after the first two) is the sum of the preceding two terms: U 1 = 1 , U 2 = 1 , U n = U n 1 + U n 2 for n > 3. In the Symbolic view of the Sequence aplet, highlight the U1(1) field and begin defining your sequence. 1 1
Note: You can use the , , , , and menu keys to assist in the entry of equations.
3. In Plot Setup, first set the SEQPLOT option to Stairstep. Reset the default plot settings by clearing the Plot Setup view.
Plot the sequence
4. Plot the Fibonacci sequence.
5. In Plot Setup, set the SEQPLOT option to Cobweb.
Display the table
6. Display the table of values for this example.
About the Solve aplet
The Solve aplet solves an equation or an expression for its unknown variable. You define an equation or expression in the symbolic view, then supply values for all the variables except one in the numeric view. Solve works only with real numbers. Note the differences between an equation and an expression: An equation contains an equals sign. Its solution is a value for the unknown variable that makes both sides have the same value. An expression does not contain an equals sign. Its solution is a root, that is, a value for the unknown variable that makes the expression have a value of zero.
Sort data values
Defining a regression model
The Symbolic view includes an expression (Fit1 through Fit5) that defines the regression model, or fit, to use for the regression analysis of each two-variable data set. There are three ways to select a regression model: Accept the default option to fit the data to a straight line. Select one of the available fit options in Symbolic Setup view. Enter your own mathematical expression in Symbolic view. This expression will be plotted, but it will not be fitted to the data points.
You can ignore the angle measurement mode unless your Fit definition (in Symbolic view) involves a trigonometric function. In this case, you should specify in the mode screen whether the trigonometric units are to be interpreted in degrees, radians, or grads. 1. In Numeric view, make sure is set.
To choose the fit
2. Press SETUP-SYMB to display the Symbolic Setup view. Highlight the Fit number (S1FIT to S5FIT) you want to define. 3. Press and select from the list. Press when done. The regression formula for the fit is displayed in Symbolic view.
Eight fit models are available: Fit model Lineair Meaning (Default.) Fits the data to a straight line, y = mx+b. Uses a least-squares fit. Fits to a logarithmic curve, y = m lnx + b. Fits to an exponential curve, y = bemx. Fits to a power curve, y = bxm.
Logaritmisc h Macht
Fit model Quadratic
Meaning (Continued) Fits to a quadratic curve, y = ax2+bx+c. Needs at least three points. Fits to a cubic curve, y = ax3+bx2+cx+d. Needs at least four points. Fits to a logistic curve, L y = -------------------------) , ( bx 1 + ae where L is the saturation value for growth. You can store a positive real value in L, orif L=0let L be computed automatically.
Define your own expression (in Symbolic view.) is set.
To define your own fit
1. In Numeric view, make sure 2. Display the Symbolic view.
3. Highlight the Fit expression (Fit1, etc.) for the desired data set. 4. Type in an expression and press.
The hp 39g+ has approximately 232K of user memory. The calculator uses this memory to store variables, perform computations, and store history. A variable is an object that you create in memory to hold data. The hp 39g+ has two types of variables, home variables and aplet variables. Home variables are available in all aplets. For example, you can store real numbers in variables A to Z and complex numbers in variables Z0 to Z9. These can be numbers you have entered, or the results of calculations. These variables are available within all aplets and within any programs. Aplet variables apply only to a single aplet. Aplets have specific variables allocated to them which vary from aplet to aplet.
You use the calculators memory to store the following objects: copies of aplets with specific configurations new aplets that you download aplet variables home variables variables created through a catalog or editor, for example a matrix or a text note programs that you create.
You can use the Memory Manager ( MEMORY) to view the amount of memory available. The catalog views, which are accessible via the Memory Manager, can be used to transfer variables such as lists or matrices between calculators.
Variables and memory management
Storing and recalling variables
You can store numbers or expressions from a previous input or result into variables.
A number stored in a variable is always stored as a 12digit mantissa with a 3-digit exponent. Numeric precision in the display, however, depends on the display mode (Standard, Fixed, Scientific, Engineering, or Fraction). A displayed number has only the precision that is displayed. If you copy it from the HOME view display history, you obtain only the precision displayed, not the full internal precision. On the other hand, the variable Ans always contains the most recent result to full precision. 1. On the command line, enter the value or the calculation for the result you wish to store. 2. Press 3. Enter a name for the variable. 4. Press.
To store a value
To store the results of a calculation
If the value you want to store is in the HOME view display history, for example the results of a previous calculation, you need to copy it to the command line, then store it. 1. Perform the calculation for the result you want to store. 8 6
M0 to M9 can store matrices or vectors. For example, [[1,2],[3,4]] M0.
Modes variables store the modes settings that you can configure using MODES. Notepad variables store notes. Program variables store programs. A to Z and. For example, 7.45 A.
Notepad Program Real
Aplet variables store values that are unique to a particular aplet. These include symbolic expressions and equations (see below), settings for the Plot and Numeric views, and the results of some calculations such as roots and intersections. See the Reference Information chapter for more information about aplet variables. Category Function Parametric Available names F0 to F9 (Symbolic view). See Function aplet variables on page R-7. X0, Y0 to X9, Y9 (Symbolic view). See Parametric aplet variables on page R-8. R0 to R9 (Symbolic view). See Polar aplet variables on page R-9. U0 to U9 (Symbolic view). See Sequence aplet variables on page R-10. E0 to E9 (Symbolic view). See Solve aplet variables on page R-11. C0 to C9 (Numeric view). See Statistics aplet variables on page R-12.
To access an aplet variable
1. Open the aplet that contains the variable you want to recall. 2. Press to display the VARS menu. to access the variables
3. Use the arrow keys to select a variable category in the left column, then press in the right column.
4. Use the arrow keys to select a variable in the right column. 5. To copy the name of the variable onto the edit line, press.( is the default setting.)
6. To copy the value of the variable into the edit and line, press press.
You can use the Memory Manager to determine the amount of available memory on the calculator. You can also use Memory Manager to organize memory. For example, if the available memory is low, you can use the Memory Manager to determine which aplets or variables consume large amounts of memory. You can make deletions to free up memory. Example 1. Start the Memory Manager. A list of variable categories is displayed.
Free memory is displayed in the top right corner and the body of the screen lists each category, the memory it uses, and the percentage of the total memory it uses. 2. Select the category with which you want to work and press. Memory Manager displays memory details of variables within the category.
3. To delete variables in a category: Press to delete the selected variable.
CLEAR to delete all variables in the Press selected category.
You can perform matrix calculations in HOME and in programs. The matrix and each row of a matrix appear in brackets, and the elements and rows are separated by commas. For example, the following matrix: is displayed in the history as: [[1,2,3],[4,5,6]] (If the Decimal Mark mode is set to Comma, then separate each element and each row with a period.) You can enter matrices directly in the command line, or create them in the matrix editor.
Your work is automatically saved. Press any view key ( , , the Notes view.
Notes and sketches
Note edit keys
Key Meaning Space key for text entry. Displays next page of a multi-page note. Alpha-lock for letter entry. Lower-case alpha-lock for letter entry. Backspaces cursor and deletes character. Deletes current character. Starts a new line.
Erases the entire note. Menu for entering variable names, and contents of variables. Menu for entering math operations, and constants.
Menu for entering program commands. Displays special characters. To type one, highlight it and press. To copy a character without closing the CHARS screen, press.
Aplet sketch view
You can attach pictures to an aplet in its Sketch view ( SKETCH). Your work is automatically saved with the aplet. Press any other view key or to exit the Sketch view
Key Meaning Stores the specified portion of the current sketch to a graphics variable (G1 through G0). Adds a new, blank page to the current sketch set. Displays next sketch in the sketch set. Animates if held down. Opens the edit line to type a text label. Displays the menu-key labels for drawing. Deletes the current sketch.
Erases the entire sketch set. Toggles menu key labels on and off. If menu key labels are hidden, or any menu key, redisplays the menu key labels.
To draw a line
for the Sketch view.
2. In Sketch view, press and move the cursor to where you want to start the line 3. Press. This turns on line-drawing. , , , keys. 4. Move the cursor in any direction to the end point of the line by pressing the 5. Press to finish the line.
To draw a box
1. In Sketch view, press and move the cursor to where you want any corner of the box to be. 2. Press. 3. Move the cursor to mark the opposite corner for the box. You can adjust the size of the box by moving the cursor. 4. Press to finish the box.
To draw a circle
1. In Sketch view, press and move the cursor to where you want the center of the circle to be. 2. Press 4. Press. This turns on circle drawing. to draw the circle. 3. Move the cursor the distance of the radius.
Key Meaning Dot on. Turns pixels on as the cursor moves. Dot off. Turns pixels off as the cursor moves. Draws a line from the cursors starting position to the cursors current position. Press when you have finished. You can draw a line at any angle. Draws a box from the cursors starting position to the cursors current position. when you have finished. Press Draws a circle with the cursors starting position as the center. The radius is the distance between the cursors starting and ending position. Press to draw the circle.
1. Press 2. Press.
to open the Program catalog.
The hp 39g+ prompts you for a name.
A program name can contain special characters, such as a space. However, if you use special characters and then run the program by typing it in HOME, you must enclose the program name in double quotes (" "). Don't use the " symbol within your program name. 3. Type your program name, then press.
When you press , the Program Editor opens. 4. Enter your program. When done, start any other activity. Your work is saved automatically.
Until you become familiar with the hp 39g+ commands, the easiest way to enter commands is to select them from the Commands menu from the Program editor. You can also type in commands using alpha characters. 1. From the Program editor, press the Program Commands menu.
2. On the left, use
to highlight a command
category, then press to access the commands in the category. Select the command that you want.
3. Press editor.
to paste the command into the program
Edit a program
1. Press PROGRM to open the Program catalog.
2. Use the arrow keys to highlight the program you want to edit, and press. The hp 39g+ opens the Program Editor. The name of your program appears in the title bar of the display. You can use the following keys to edit your program.
The editing keys are: Key Meaning Inserts the editing point. character at the
Inserts space into text. Displays previous page of the program. Displays next page of the program. Moves up or down one line. Moves right or left one character. Alpha-lock for letter entry. Press A.Z to lock lower case. Backspaces cursor and deletes character. Deletes current character. Starts a new line.
Erases the entire program. Displays menus for selecting variable names, contents of variables, math functions, and program constants.
Displays menus for selecting program conmmands. Displays all characters. To type one, highlight it and press. To enter several characters in a row, use the menu key while in the CHARS menu.
Run a program
From HOME, type RUN program_name. or From the Program catalog, highlight the program you want to run and press Regardless of where you start the program, all programs run in HOME. What you see will differ slightly depending on where you started the program. If you start the program from HOME, the hp 39g+ displays the contents of Ans (Home variable containing the last result), when the program has finished. If you start the program from the Program catalog, the hp39g+ returns you to the Program catalog when the program ends.
Replaces graphic in graphicname with bitwise-inverted graphic. GROBNOT graphicname:
Using the logical OR, superimposes graphicname2 onto graphicname1. The upper left corner of graphicname2 is placed at position. GROBOR graphicname1;position;graphicname2:
Using the logical XOR, superimposes graphicname2 onto graphicname1. The upper left corner of graphicname2 is placed at position. GROBXOR graphicname1;position;graphicname2:
Creates graphic with given width, height, and hexadecimal data, and stores it in graphicname. MAKEGROB graphicname;width;height;hexdata:
Stores the Plot view display as a graphic in graphicname. PLOT graphicname: PLOT and DISPLAY can be used to transfer a copy of the current PLOT view into the sketch view of the aplet for later use and editing. Example 1 PageNum:
PLOTPage: FREEZE: This program stores the current PLOT view to the first page in the sketch view of the current aplet and then displays the sketch as a graphic object until any key is pressed.
Puts graph from graphicname into the Plot view display. PLOT graphicname:
Replaces portion of graphic in graphicname1 with graphicname2, starting at position. REPLACE also works for lists and matrices. REPLACE graphicname1;(position);graphicname2:
Extracts a portion of the named graphic (or list or matrix), and stores it in a new variable, name. The portion is specified by position and positions. SUB name;graphicname;(position);(positions):
Creates a blank graphic with given width and height, and stores it in graphicname. ZEROGROB graphicname;width;height:
Loop hp allow a program to execute a routine repeatedly. The hp 39g+ has three loop structures. The example programs below illustrate each of these structures incrementing the variable A from 1 to 12.
Do. Until. End is a loop command that executes the loop-clause repeatedly until test-clause returns a true (nonzero) result. Because the test is executed after the loop-clause, the loop-clause is always executed at least once. Its syntax is: DO loop-clause UNTIL test-clause END A: 1 A DO A + 1 UNTIL A == 12 END
Sorting items in the aplet library menu list
Once you have entered information into an aplet, you have defined a new version of an aplet. The information is automatically saved under the current aplet name, such as Function. To create additional aplets of the same type, you must give the current aplet a new name. The advantage of storing an aplet is to allow you to keep a copy of a working environment for later use. The aplet library is where you go to manage your aplets. Press. Highlight (using the arrow keys) the name of the aplet you want to act on.
To sort the aplet list
In the aplet library, press and press.
. Select the sorting scheme
Chronologically produces a chronological order based on the date an aplet was last used. (The lastused aplet appears first, and so on.) Alphabetically produces an alphabetical order by aplet name.
To delete an aplet
You cannot delete a built-in aplet. You can only clear its data and reset its default settings. To delete a customized aplet, open the aplet library, highlight the aplet to be deleted, and press. To CLEAR. delete all custom aplets, press
R Reference information
aplet A small application, limited to one topic. The built-in aplet types are Function, Parametric, Polar, Sequence, Solve, and Statistics. An aplet can be filled with the data and solutions for a specific problem. It is reusable (like a program, but easier to use) and it records all your settings and definitions. An operation for use in programs. Commands can store results in variables, but do not display results. Arguments are separated by semicolons, such as DISP expression;line#. A number, variable, or algebraic expression (numbers plus functions) that produces a value. An operation, possibly with arguments, that returns a result. It does not store results in variables. The arguments must be enclosed in parentheses and separated with commas (or periods in Comma mode), such as CROSS(matrix1,matrix2). The basic starting point of the calculator. Go to HOME to do calculations. For aplet management: to start, save, reset, send and receive aplets.
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.
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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 . 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.  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
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
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
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.
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