User manual HP 50g graphing calculator - annexe 1

HP 50g graphing calculator user's manual H Edition 1 HP part number F2229AA-90001 Notice REGISTER YOUR PRODUCT AT: www.register.hp.com THIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN ARE PROVIDED "AS IS" AND ARE SUBJECT TO CHANGE WITHOUT NOTICE. HEWLETT-PACKARD COMPANY MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MANUAL, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY, NON-INFRINGEMENT AND FITNESS FOR A PARTICULAR PURPOSE. HEWLETT-PACKARD CO. SHALL NOT BE LIABLE FOR ANY ERRORS OR FOR INCIDENTAL OR CONSEQUENTIAL DAMAGES IN CONNECTION WITH THE FURNISHING, PERFORMANCE, OR USE OF THIS MANUAL OR THE EXAMPLES CONTAINED HEREIN. © Copyright 2003, 2006 Hewlett-Packard Development Company, L.P. Reproduction, adaptation, or translation of this manual is prohibited without prior written permission of Hewlett-Packard Company, except as allowed under the copyright laws. Hewlett-Packard Company 4995 Murphy Canyon Rd, Suite 301 San Diego,CA 92123 Printing History Edition 1 April 2006 Preface You have in your hands a compact symbolic and numerical computer that will facilitate calculation and mathematical analysis of problems in a variety of disciplines, from elementary mathematics to advanced engineering and science subjects. This manual contains examples that illustrate the use of the basic calculator functions and operations. The chapters in this user's manual are organized by subject in order of difficulty: from the setting of calculator modes, to real and complex number calculations, operations with lists, vectors, and matrices, graphics, calculus applications, vector analysis, differential equations, probability and statistics. For symbolic operations the calculator includes a powerful Computer Algebraic System (CAS), which lets you select different modes of operation, e.g., complex numbers vs. real numbers, or exact (symbolic) vs. approximate (numerical) mode. The display can be adjusted to provide textbook-type expressions, which can be useful when working with matrices, vectors, fractions, summations, derivatives, and integrals. The high-speed graphics of the calculator are very convenient for producing complex figures in very little time. Thanks to the infrared port, the USB port, and the RS232 port and cable provided with your calculator, you can connect your calculator with other calculators or computers. This allows for fast and efficient exchange of programs and data with other calculators and computers. We hope your calculator will become a faithful companion for your school and professional applications. Table of Contents Chapter 1 - Getting started Basic Operations, 1-1 Batteries, 1-1 Turning the calculator on and off, 1-2 Adjusting the display contrast, 1-2 Contents of the calculator's display, 1-3 Menus, 1-3 The TOOL menu, 1-3 Setting time and date, 1-4 Introducing the calculator's keyboard, 1-4 Selecting calculator modes, 1-6 Operating Mode, 1-7 Number Format and decimal dot or comma, 1-10 Standard format, 1-10 Fixed format with decimals, 1-10 Scientific format, 1-11 Engineering format, 1-12 Decimal comma vs. decimal point, 1-13 Angle Measure, 1-14 Coordinate System, 1-14 Selecting CAS settings, 1-15 Explanation of CAS settings, 1-16 Selecting Display modes, 1-17 Selecting the display font, 1-18 Selecting properties of the line editor, 1-18 Selecting properties of the Stack, 1-19 Selecting properties of the equation writer (EQW), 1-20 References, 1-20 Chapter 2 - Introducing the calculator Calculator objects, 2-1 Page TOC-1 Editing expressions in the stack, 2-1 Creating arithmetic expressions, 2-1 Creating algebraic expressions, 2-4 Using the Equation Writer (EQW) to create expressions, 2-5 Creating arithmetic expressions, 2-5 Creating algebraic expressions, 2-7 Organizing data in the calculator, 2-8 The HOME directory, 2-8 Subdirectories, 2-9 Variables, 2-9 Typing variable names , 2-9 Creating variables, 2-10 Algebraic mode, 2-10 RPN mode, 2-11 Checking variables contents, 2-13 Algebraic mode, 2-13 RPN mode, 2-13 Using the right-shift key followed by soft menu key labels, 2-13 Listing the contents of all variables in the screen, 2-14 Deleting variables, 2-14 Using function PURGE in the stack in Algebraic mode, 2-14 Using function PURGE in the stack in RPN mode, 2-15 UNDO and CMD functions, 2-16 CHOOSE boxes vs. Soft MENU, 2-16 References, 2-18 Chapter 3 - Calculations with real numbers Examples of real number calculations, 3-1 Using powers of 10 in entering data, 3-3 Real number functions in the MTH menu, 3-5 Using calculator menus, 3-5 Hyperbolic functions and their inverses, 3-5 Operations with units, 3-7 The UNITS menu, 3-7 Page TOC-2 Available units, 3-9 Attaching units to numbers, 3-9 Unit prefixes, 3-10 Operations with units, 3-11 Unit conversions, 3-12 Physical constants in the calculator, 3-13 Defining and using functions, 3-15 Reference, 3-16 Chapter 4 - Calculations with complex numbers Definitions, 4-1 Setting the calculator to COMPLEX mode, 4-1 Entering complex numbers, 4-2 Polar representation of a complex number, 4-3 Simple operations with complex numbers, 4-4 The CMPLX menus, 4-4 CMPLX menu through the MTH menu, 4-4 CMPLX menu in keyboard, 4-6 Functions applied to complex numbers, 4-6 Function DROITE: equation of a straight line, 4-7 Reference, 4-7 Chapter 5 - Algebraic and arithmetic operations Entering algebraic objects, 5-1 Simple operations with algebraic objects, 5-2 Functions in the ALG menu , 5-3 Operations with transcendental functions, 5-5 Expansion and factoring using log-exp functions, 5-5 Expansion and factoring using trigonometric functions, 5-6 Functions in the ARITHMETIC menu, 5-7 Polynomials, 5-8 The HORNER function, 5-8 The variable VX, 5-8 The PCOEF function, 5-8 Page TOC-3 The PROOT function, 5-9 The QUOT and REMAINDER functions, 5-9 The PEVAL function , 5-9 Fractions, 5-9 The SIMP2 function, 5-10 The PROPFRAC function, 5-10 The PARTFRAC function, 5-10 The FCOEF function, 5-10 The FROOTS function, 5-11 Step-by-step operations with polynomials and fractions, 5-11 Reference, 5-12 Chapter 6 - Solution to equations Symbolic solution of algebraic equations, 6-1 Function ISOL, 6-1 Function SOLVE, 6-2 Function SOLVEVX, 6-4 Function ZEROS, 6-4 Numerical solver menu, 6-5 Polynomial Equations, 6-6 Finding the solutions to a polynomial equation, 6-6 Generating polynomial coefficients given the polynomial's roots, 6-7 Generating an algebraic expression for the polynomial, 6-8 Financial calculations, 6-8 Solving equations with one unknown through NUM.SLV, 6-9 Function STEQ, 6-9 Solution to simultaneous equations with MSLV, 6-10 Reference, 6-11 Chapter 7 - Operations with lists Creating and storing lists, 7-1 Operations with lists of numbers, 7-1 Changing sign , 7-1 Page TOC-4 Addition, subtraction, multiplication, division, 7-2 Functions applied to lists, 7-4 Lists of complex numbers, 7-4 Lists of algebraic objects, 7-5 The MTH/LIST menu, 7-5 The SEQ function, 7-7 The MAP function, 7-7 Reference, 7-7 Chapter 8 - Vectors Entering vectors , 8-1 Typing vectors in the stack, 8-1 Storing vectors into variables in the stack, 8-2 Using the Matrix Writer (MTRW) to enter vectors, 8-3 Simple operations with vectors, 8-5 Changing sign, 8-5 Addition, subtraction, 8-5 Multiplication by a scalar, and division by a scalar, 8-6 Absolute value function, 8-6 The MTH/VECTOR menu, 8-6 Magnitude, 8-7 Dot product , 8-7 Cross product, 8-7 Reference, 8-8 Chapter 9 - Matrices and linear algebra Entering matrices in the stack, 9-1 Using the Matrix Writer, 9-1 Typing in the matrix directly into the stack, 9-2 Operations with matrices, 9-3 Addition and subtraction, 9-4 Multiplication, 9-4 Multiplication by a scalar, 9-4 Matrix-vector multiplication, 9-5 Page TOC-5 Matrix multiplication, 9-5 Term-by-term multiplication, 9-6 Raising a matrix to a real power, 9-6 The identity matrix, 9-7 The inverse matrix, 9-7 Characterizing a matrix (The matrix NORM menu), 9-8 Function DET, 9-8 Function TRACE, 9-8 Solution of linear systems, 9-9 Using the numerical solver for linear systems, 9-9 Solution with the inverse matrix, 9-11 Solution by "division" of matrices, 9-11 References, 9-12 Chapter 10 - Graphics Graphs options in the calculator, 10-1 Plotting an expression of the form y = f(x), 10-2 Generating a table of values for a function, 10-4 Fast 3D plots, 10-5 Reference, 10-7 Chapter 11 - Calculus Applications The CALC (Calculus) menu, 11-1 Limits and derivatives, 11-1 Function lim, 11-1 Functions DERIV and DERVX, 11-3 Anti-derivatives and integrals, 11-3 Functions INT, INTVX, RISCH, SIGMA and SIGMAVX, 11-3 Definite integrals, 11-4 Infinite series, 11-5 Functions TAYLR, TAYLR0, and SERIES, 11-5 Reference, 11-6 Page TOC-6 Chapter 12 - Multi-variate Calculus Applications Partial derivatives, 12-1 Multiple integrals, 12-2 Reference, 12-2 Chapter 13 - Vector Analysis Applications The del operator, 13-1 Gradient, 13-1 Divergence, 13-2 Curl, 13-2 Reference, 13-2 Chapter 14 - Differential Equations The CALC/DIFF menu, 14-1 Solution to linear and non-linear equations, 14-1 Function LDEC, 14-1 Function DESOLVE, 14-3 The variable ODETYPE, 14-3 Laplace Transforms, 14-4 Laplace transform and inverses in the calculator, 14-4 Fourier series, 14-5 Function FOURIER, 14-5 Fourier series for a quadratic function, 14-6 Reference, 14-7 Chapter 15 - Probability Distributions The MTH/PROBABILITY.. sub-menu - part 1, 15-1 Factorials, combinations, and permutations, 15-1 Random numbers, 15-2 The MTH/PROB menu - part 2, 15-3 The Normal distribution, 15-3 The Student-t distribution, 15-3 The Chi-square distribution, 15-4 The F distribution, 15-4 Page TOC-7 Reference, 15-4 Chapter 16 - Statistical Applications Entering data, 16-1 Calculating single-variable statistics, 16-2 Sample vs. population, 16-2 Obtaining frequency distributions, 16-3 Fitting data to a function y = f(x), 16-5 Obtaining additional summary statistics, 16-6 Confidence intervals, 16-7 Hypothesis testing, 16-9 Reference, 16-11 Chapter 17 - Numbers in Different Bases The BASE menu, 17-1 Writing non-decimal numbers, 17-2 Reference, 17-2 Chapter 18 - Using SD cards Inserting and removing an SD card, 18-1 Formatting an SD card, 18-1 Accessing objects on an SD card, 18-2 Storing objects on the SD card, 18-2 Recalling an object from the SD card, 18-3 Purging an object from the SD card, 18-3 Purging all objects on the SD card (by reformatting), 18-4 Specifying a directory on an SD card, 18-4 Chapter 19 - Equation Library Reference, 19-4 Limited Warranty, W-1 Service, W-3 Regulatory information, W-5 Disposal of Waste Equipment by Users in Private Household in the European Union, W-7 Page TOC-8 Chapter 1 Getting started This chapter provides basic information about the operation of your calculator. It is designed to familiarize you with the basic operations and settings before you perform a calculation. Basic Operations Batteries The calculator uses 4 AAA (LR03) batteries as main power and a CR2032 lithium battery for memory backup. Before using the calculator, please install the batteries according to the following procedure. To install the main batteries a. Make sure the calculator is OFF. Slide up the battery compartment cover as illustrated. b. Insert 4 new AAA (LR03) batteries into the main compartment. Make sure each battery is inserted in the indicated direction. To install the backup battery a. Make sure the calculator is OFF. Press down the holder. Push the plate to the shown direction and lift it. Page 1-1 b. Insert a new CR2032 lithium battery. Make sure its positive (+) side is facing up. c. Replace the plate and push it to the original place. After installing the batteries, press $ to turn the power on. Warning: When the low battery icon is displayed, you need to replace the batteries as soon as possible. However, avoid removing the backup battery and main batteries at the same time to avoid data lost. Turning the calculator on and off The $ key is located at the lower left corner of the keyboard. Press it once to turn your calculator on. To turn the calculator off, press the rightshift key @ (first key in the second row from the bottom of the keyboard), followed by the $ key. Notice that the $ key has a OFF label printed in the upper right corner as a reminder of the OFF command. Adjusting the display contrast You can adjust the display contrast by holding the $ key while pressing the + or - keys. The $(hold) + key combination produces a darker display The $(hold) - key combination produces a lighter display Page 1-2 Contents of the calculator's display Turn your calculator on once more. At the top of the display you will have two lines of information that describe the settings of the calculator. The first line shows the characters: RAD XYZ HEX R= 'X' For details on the meaning of these symbols see Chapter 2 in the calculator's user's guide. The second line shows the characters { HOME } indicating that the HOME directory is the current file directory in the calculator's memory. At the bottom of the display you will find a number of labels, namely, @EDIT @VIEW @@RCL@@ @@STO@ !PURGE !CLEAR associated with the six soft menu keys, F1 through F6: ABCDEF The six labels displayed in the lower part of the screen will change depending on which menu is displayed. But A will always be associated with the first displayed label, B with the second displayed label, and so on. Menus The six labels associated with the keys A through F form part of a menu of functions. Since the calculator has only six soft menu keys, it only display 6 labels at any point in time. However, a menu can have more than six entries. Each group of 6 entries is called a Menu page. To move to the next menu page (if available), press the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard. The TOOL menu The soft menu keys for the default menu, known as the TOOL menu, are associated with operations related to manipulation of variables (see section on variables in this Chapter): @EDIT A EDIT the contents of a variable (see Chapter 2 in this guide and Chapter 2 and Appendix L in the user's guide for more information on editing) @VIEW B VIEW the contents of a variable Page 1-3 @@RCL@ C ReCaLl the contents of a variable @@STO@ D STOre the contents of a variable !PURGE E PURGE a variable @CLEAR F CLEAR the display or stack These six functions form the first page of the TOOL menu. This menu has actually eight entries arranged in two pages. The second page is available by pressing the L (NeXT menu) key. This key is the third key from the left in the third row of keys in the keyboard. In this case, only the first two soft menu keys have commands associated with them. These commands are: @CASCM A CASCMD: CAS CoMmanD, used to launch a command from the CAS (Computer Algebraic System) by selecting from a list @HELP B HELP facility describing the commands available in the calculator Pressing the L key will show the original TOOL menu. Another way to recover the TOOL menu is to press the I key (third key from the left in the second row of keys from the top of the keyboard). Setting time and date See Chapter 1 in the calculator's user's guide to learn how to set time and date. Introducing the calculator's keyboard The figure on the next page shows a diagram of the calculator's keyboard with the numbering of its rows and columns. Each key has three, four, or five functions. The main key function correspond to the most prominent label in the key. Also, the left-shift key, key (8,1), the right-shift key, key (9,1), and the ALPHA key, key (7,1), can be combined with some of the other keys to activate the alternative functions shown in the keyboard. Page 1-4 For example, the P key, key(4,4), has the following six functions associated with it: P ,,´ ...N ~p ~,,p Main function, to activate the SYMBolic menu Left-shift function, to activate the MTH (Math) menu Right-shift function, to activate the CATalog function ALPHA function, to enter the upper-case letter P ALPHA-Left-Shift function, to enter the lower-case letter p Page 1-5 ~...p ALPHA-Right-Shift function, to enter the symbol Of the six functions associated with a key only the first four are shown in the keyboard itself. The figure in next page shows these four labels for the P key. Notice that the color and the position of the labels in the key, namely, SYMB, MTH, CAT and P, indicate which is the main function (SYMB), and which of the other three functions is associated with the leftshift ,,(MTH), right-shift ...(CAT ), and ~ (P) keys. For detailed information on the calculator keyboard operation refer to Appendix B in the calculator's user's guide. Selecting calculator modes This section assumes that you are now at least partially familiar with the use of choose and dialog boxes (if you are not, please refer to appendix A in the user's guide). Press the H button (second key from the left on the second row of keys from the top) to show the following CALCULATOR MODES input form: Press the !!@@OK#@ soft menu key to return to normal display. Examples of selecting different calculator modes are shown next. Page 1-6 Operating Mode The calculator offers two operating modes: the Algebraic mode, and the Reverse Polish Notation (RPN) mode. The default mode is the Algebraic mode (as indicated in the figure above), however, users of earlier HP calculators may be more familiar with the RPN mode. To select an operating mode, first open the CALCULATOR MODES input form by pressing the H button. The Operating Mode field will be highlighted. Select the Algebraic or RPN operating mode by either using the \ key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOS soft menu key. If using the latter approach, use up and down arrow keys, -- ~, to select the mode, and press the !!@@OK#@ soft menu key to complete the operation. To illustrate the difference between these two operating modes we will calculate the following expression in both modes: 3.0 5.0 - 1 3.0 3.0 2.5 +e 3 23.0 To enter this expression in the calculator we will first use the equation writer, ,O. Please identify the following keys in the keyboard, besides the numeric keypad keys: !@.#*+-/R Q¸Ü,OsTM~--` The equation writer is a display mode in which you can build mathematical expressions using explicit mathematical notation including fractions, derivatives, integrals, roots, etc. To use the equation writer for writing the expression shown above, use the following keystrokes: ,OR3.*!Ü5.1./3.*3. ---------- /23.Q3TMTM+!¸2.5` After pressing ` the calculator displays the expression: (3.*(5.-1/(3.*3.))/23.^3+EXP(2.5)) Pressing ` again will provide the following value (accept Approx mode on, if asked, by pressing !!@@OK#@): Page 1-7 You could also type the expression directly into the display without using the equation writer, as follows: R!Ü3.*!Ü5.1/3.*3.TM /23.Q3+!¸2.5` to obtain the same result. Change the operating mode to RPN by first pressing the H button. Select the RPN operating mode by either using the \ key, or pressing the @CHOOS soft menu key. Press the @@OK#@ soft menu key to complete the operation. The display, for the RPN mode looks as follows: Notice that the display shows several levels of output labeled, from bottom to top, as 1, 2, 3, etc. This is referred to as the stack of the calculator. The different levels are referred to as the stack levels, i.e., stack level 1, stack level 2, etc. What RPN means is that, instead of writing an operation such as 3 + 2 by pressing 3+2` we write the operands first, in the proper order, and then the operator, i.e., 3`2+ As you enter the operands, they occupy different stack levels. Entering 3` puts the number 3 in stack level 1. Next, entering 2 pushes the 3 upwards to occupy stack level 2. Finally, by pressing +, we are telling the calculator to apply the operator, +, to the objects occupying levels 1 and 2. The result, 5, is then placed in level 1. Page 1-8 Let's try some other simple operations before trying the more complicated expression used earlier for the algebraic operating mode: 123/32 42 3 123`32/ 4`2Q 27R3@» (27) Note the position of the y and x in the last two operations. The base in the exponential operation is y (stack level 2) while the exponent is x (stack level 1) before the key Q is pressed. Similarly, in the cubic root operation, y (stack level 2) is the quantity under the root sign, and x (stack level 1) is the root. Try the following exercise involving 3 factors: (5 + 3) × 2 5`3+ Calculates (5 +3) first. 2X Completes the calculation. Let's try now the expression proposed earlier: 3 5 - 23 3` 5` 3` 3* Y * Enter 3 in level 1 3 33 2.5 +e 1 Enter 5 in level 1, 3 moves to level 2 Enter 3 in level 1, 5 moves to level 2, 3 to level 3 Place 3 and multiply, 9 appears in level 1 1/(3×3), last value in lev. 1; 5 in level 2; 3 in level 3 5 - 1/(3×3) , occupies level 1 now; 3 in level 2 3 × (5 - 1/(3×3)), occupies level 1 now. Enter 3, calculate 233 into level 1. 14.666 in lev. 2. (3 × (5-1/(3×3)))/233 into level 1 e2.5, goes into level 1, level 2 shows previous value. 23`Enter 23 in level 1, 14.66666 moves to level 2. 3Q / 2.5Enter 2.5 level 1 !¸ Page 1-9 + R (3 × (5 - 1/(3 × 3)))/233 + e2.5 = 12.18369, into lev. 1. ((3 × (5 - 1/(3×3)))/233 + e2.5) = 3.4905156, into 1. To select between the ALG vs. RPN operating mode, you can also set/ clear system flag 95 through the following keystroke sequence: H @FLAGS! 9~~~~ ` Number Format and decimal dot or comma Changing the number format allows you to customize the way real numbers are displayed by the calculator. You will find this feature extremely useful in operations with powers of tens or to limit the number of decimals in a result. To select a number format, first open the CALCULATOR MODES input form by pressing the H button. Then, use the down arrow key, ~, to select the option Number format. The default value is Std, or Standard format. In the standard format, the calculator will show floating-point numbers with no set decimal placement and with the maximum precision allowed by the calculator (12 significant digits)."To learn more about reals, see Chapter 2 in this guide. To illustrate this and other number formats try the following exercises: Standard format This mode is the most used mode as it shows numbers in the most familiar notation. Press the !!@@OK#@ soft menu key, with the Number format set to Std, to return to the calculator display. Enter the number 123.4567890123456 (with16 significant figures). Press the ` key. The number is rounded to the maximum 12 significant figures, and is displayed as follows: Fixed format with decimals Press the H button. Next, use the down arrow key, ~, to select the option Number format. Press the @CHOOS soft menu key, and select the option Fixed with the arrow down key ~. Page 1-10 Press the right arrow key, TM, to highlight the zero in front of the option Fix. Press the @CHOOS soft menu key and, using the up and down arrow keys, --~, select, say, 3 decimals. Press the !!@@OK#@ soft menu key to complete the selection: Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Notice how the number is rounded, not truncated. Thus, the number 123.4567890123456, for this setting, is displayed as 123.457, and not as 123.456 because the digit after 6 is > 5. Scientific format To set this format, start by pressing the H button. Next, use the down arrow key, ~, to select the option Number format. Press the @CHOOS soft menu key, and select the option Scientific with the arrow down key ~. Page 1-11 Keep the number 3 in front of the Sci. (This number can be changed in the same fashion that we changed the Fixed number of decimals in the example above). Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: This result, 1.23E2, is the calculator's version of powers-of-ten notation, i.e., 1.235 × 102. In this, so-called, scientific notation, the number 3 in front of the Sci number format (shown earlier) represents the number of significant figures after the decimal point. Scientific notation always includes one integer figure as shown above. For this case, therefore, the number of significant figures is four. Engineering format The engineering format is very similar to the scientific format, except that the powers of ten are multiples of three. To set this format, start by pressing the H button. Next, use the down arrow key, ~, to select the option Number format. Press the @CHOOS soft menu key, and select the option Engineering with the arrow down key ~. Keep the number 3 in front of the Eng. (This number can be changed in the same fashion that we changed the Fixed number of decimals in an earlier example). Page 1-12 Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Because this number has three figures in the integer part, it is shown with four significative figures and a zero power of ten, while using the Engineering format. For example, the number 0.00256, will be shown as: Decimal comma vs. decimal point Decimal points in floating-point numbers can be replaced by commas, if the user is more familiar with such notation. To replace decimal points for commas, change the FM option in the CALCULATOR MODES input form to commas, as follows (Notice that we have changed the Number Format to Std): Press the H button. Next, use the down arrow key, ~, once, and the right arrow key, TM, highlighting the option __FM,. To select commas, press the soft menu key. The input form will look as follows: Press the !!@@OK#@ soft menu key return to the calculator display. The number 123.4567890123456, entered earlier, now is shown as: Page 1-13 Angle Measure Trigonometric functions, for example, require arguments representing plane angles. The calculator provides three different Angle Measure modes for working with angles, namely: · Degrees: There are 360 degrees (360°) in a complete circumference. · Radians: There are 2 radians (2 r) in a complete circumference. · Grades: There are 400 grades (400 g) in a complete circumference. The angle measure affects the trig functions like SIN, COS, TAN and associated functions. To change the angle measure mode, use the following procedure: · Press the H button. Next, use the down arrow key, ~, twice. Select the Angle Measure mode by either using the \ key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOS soft menu key. If using the latter approach, use up and down arrow keys, --~, to select the preferred mode, and press the !!@@OK#@ soft menu key to complete the operation. For example, in the following screen, the Radians mode is selected: Coordinate System The coordinate system selection affects the way vectors and complex numbers are displayed and entered. To learn more about complex numbers and vectors, see Chapters 4 and 8, respectively, in this guide. There are three coordinate systems available in the calculator: Rectangular (RECT), Cylindrical (CYLIN), and Spherical (SPHERE). To change coordinate system: · Press the H button. Next, use the down arrow key, ~, three times. Select the Coord System mode by either using the \ key (second from left in the fifth row from the keyboard bottom), or pressing the @CHOOS soft menu key. If using the latter approach, use up and down arrow keys, --~, to select the preferred mode, and press the !!@@OK#@ Page 1-14 soft menu key to complete the operation. For example, in the following screen, the Polar coordinate mode is selected: Selecting CAS settings CAS stands for Computer Algebraic System. This is the mathematical core of the calculator where the symbolic mathematical operations and functions are programmed. The CAS offers a number of settings can be adjusted according to the type of operation of interest. To see the optional CAS settings use the following: · Press the H button to activate the CALCULATOR MODES input form. · To change CAS settings press the @@CAS@@ soft menu key. The default values of the CAS setting are shown below: · To navigate through the many options in the CAS MODES input form, use the arrow keys: sTM~--. · To select or deselect any of the settings shown above, select the underline before the option of interest, and toggle the soft menu key until the right setting is achieved. When an option is selected, a check mark will be shown in the underline (e.g., the Rigorous and Simp Page 1-15 Non-Rational options above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Numeric, _Approx, _Complex, _Verbose, _Step/Step, _Incr Pow options above). · After having selected and unselected all the options that you want in the CAS MODES input form, press the @@@OK@@@ soft menu key. This will take you back to the CALCULATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more. Explanation of CAS settings · Indep var: The independent variable for CAS applications. Typically, VX = `X'. · Modulo: For operations in modular arithmetic this variable holds the modulus or modulo of the arithmetic ring (see Chapter 5 in the calculator's user's guide). · Numeric: If set, the calculator produces a numeric, or floating-point result, in calculations. Note that constants will always be evaluated numerically. · Approx: If set, Approximate mode uses numerical results in calculations. If unchecked, the CAS is in Exact mode, which produces symbolic results in algebraic calculations. · Complex: If set, complex number operations are active. If unchecked the CAS is in Real mode, i.e., real number calculations are the default. See Chapter 4 for operations with complex numbers. · Verbose: If set, provides detailed information in certain CAS operations. · Step/Step: If set, provides step-by-step results for certain CAS operations. Useful to see intermediate steps in summations, derivatives, integrals, polynomial operations (e.g., synthetic division), and matrix operations. · Incr Pow: Increasing Power, means that, if set, polynomial terms are shown in increasing order of the powers of the independent variable. · Rigorous: If set, calculator does not simplify the absolute value function |X| to X. · Simp Non-Rational: If set, the calculator will try to simplify non-rational expressions as much as possible. Page 1-16 Selecting Display modes The calculator display can be customized to your preference by selecting different display modes. To see the optional display settings use the following: · First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to display the DISPLAY MODES input form. · To navigate through the many options in the DISPLAY MODES input form, use the arrow keys: sTM~--. · To select or deselect any of the settings shown above, that require a check mark, select the underline before the option of interest, and toggle the soft menu key until the right setting is achieved. When an option is selected, a check mark will be shown in the underline (e.g., the Textbook option in the Stack: line above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Small, _Full page, and _Indent options in the Edit: line above). · To select the Font for the display, highlight the field in front of the Font: option in the DISPLAY MODES input form, and use the @CHOOS soft menu. · After having selected and unselected all the options that you want in the DISPLAY MODES input form, press the @@@OK@@@ soft menu key. This will take you back to the CALCULATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more. Page 1-17 Selecting the display font First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to display the DISPLAY MODES input form. The Font: field is highlighted, and the option Ft8_0: system 8 is selected. This is the default value of the display font. Pressing the @CHOOS soft menu key will provide a list of available system fonts, as shown below: The options available are three standard System Fonts (sizes 8, 7, and 6) and a Browse.. option. The latter will let you browse the calculator memory for additional fonts that you may have created or downloaded into the calculator. Practice changing the display fonts to sizes 7 and 6. Press the OK soft menu key to effect the selection. When done with a font selection, press the @@@OK@@@ soft menu key to go back to the CALCULATOR MODES input form. To return to normal calculator display at this point, press the @@@OK@@@ soft menu key once more and see how the stack display change to accommodate the different font. Selecting properties of the line editor First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to display the DISPLAY MODES input form. Press the down arrow key, ~, once, to get to the Edit line. This line shows three properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small _Full page _Indent Changes font size to small Allows to place the cursor after the end of the line Auto indent cursor when entering a carriage return Instructions on the use of the line editor are presented in Chapter 2 in the user's guide. Page 1-18 Selecting properties of the Stack First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key (D) to display the DISPLAY MODES input form. Press the down arrow key, ~, twice, to get to the Stack line. This line shows two properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small Changes font size to small. This maximizes the amount of information displayed on the screen. Note, this selection overrides the font selection for the stack display. Displays mathematical expressions in graphical mathematical notation _Textbook To illustrate these settings, either in algebraic or RPN mode, use the equation writer to type the following definite integral: ,O...Á0TM,,èTM,,¸\xTMx` In Algebraic mode, the following screen shows the result of these keystrokes with neither _Small nor _Textbook are selected: With the _Small option selected only, the display looks as shown below: With the _Textbook option selected (default value), regardless of whether the _Small option is selected or not, the display shows the following result: Page 1-19 Selecting properties of the equation writer (EQW) First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key to display the DISPLAY MODES input form. Press the down arrow key, ~, three times, to get to the EQW (Equation Writer) line. This line shows two properties that can be modified. When these properties are selected (checked) the following effects are activated: _Small Changes font size to small while using the equation editor _Small Stack Disp Shows small font in the stack after using the equation editor Detailed instructions on the use of the equation editor (EQW) are presented elsewhere in this manual. For the example of the integral 0 e - X dX , presented above, selecting the _Small Stack Disp in the EQW line of the DISPLAY MODES input form produces the following display: References Additional references on the subjects covered in this Chapter can be found in Chapter 1 and Appendix C of the calculator's user's guide. Page 1-20 Chapter 2 Introducing the calculator In this chapter we present a number of basic operations of the calculator including the use of the Equation Writer and the manipulation of data objects in the calculator. Study the examples in this chapter to get a good grasp of the capabilities of the calculator for future applications. Calculator objects Some of the most commonly used objects are: reals (real numbers, written with a decimal point, e.g., -0.0023, 3.56), integers (integer numbers, written without a decimal point, e.g., 1232, -123212123), complex numbers (written as an ordered pair, e.g., (3,-2)), lists, etc. Calculator objects are described in Chapters 2 and 24 in the calculator's user guide. Editing expressions in the stack In this section we present examples of expression editing directly into the calculator display or stack. Creating arithmetic expressions For this example, we select the Algebraic operating mode and select a Fix format with 3 decimals for the display. We are going to enter the arithmetic expression: 1.0 7.5 5.0 3.0 - 2.0 3 1.0 + To enter this expression use the following keystrokes: 5.*,,Ü1.+1/7.5TM/ ,,ÜR3.-2.Q3 The resulting expression is: 5*(1+1/7.5)/( 3-2^3). Press ` to get the expression in the display as follows: Page 2-1 Notice that, if your CAS is set to EXACT (see Appendix C in user's guide) and you enter your expression using integer numbers for integer values, the result is a symbolic quantity, e.g., 5*,,Ü1+1/7.5TM/ ,,ÜR3-2Q3 Before producing a result, you will be asked to change to Approximate mode. Accept the change to get the following result (shown in Fix decimal mode with three decimal places ­ see Chapter 1): In this case, when the expression is entered directly into the stack, as soon as you press `, the calculator will attempt to calculate a value for the expression. If the expression is preceded by a tickmark, however, the calculator will reproduce the expression as entered. For example: ³5*,,Ü1+1/7.5TM/ ,,ÜR3-2Q3` The result will be shown as follows: To evaluate the expression we can use the EVAL function, as follows: µ,,î` Page 2-2 If the CAS is set to Exact, you will be asked to approve changing the CAS setting to Approx. Once this is done, you will get the same result as before. An alternative way to evaluate the expression entered earlier between quotes is by using the option ...ï. We will now enter the expression used above when the calculator is set to the RPN operating mode. We also set the CAS to Exact, the display to Textbook, and the number format to Standard. The keystrokes to enter the expression between quotes are the same used earlier, i.e., ³5*,,Ü1+1/7.5TM/ ,,ÜR3-2Q3` Resulting in the output Press ` once more to keep two copies of the expression available in the stack for evaluation. We first evaluate the expression by pressing: µ!î` or @ï` This expression is semi-symbolic in the sense that there are floating-point components to the result, as well as a 3. Next, we switch stack locations [using TM] and evaluate using function NUM, i.e., TM...ï. This latter result is purely numerical, so that the two results in the stack, although representing the same expression, seem different. To verify that they are not, we subtract the two values and evaluate this difference using function EVAL: -µ. The result is zero (0.). For additional information on editing arithmetic expressions in the display or stack, see Chapter 2 in the calculator's user's guide. Page 2-3 Creating algebraic expressions Algebraic expressions include not only numbers, but also variable names. As an example, we will enter the following algebraic expression: 2L 1 + x R +2L R+ y b We set the calculator operating mode to Algebraic, the CAS to Exact, and the display to Textbook. To enter this algebraic expression we use the following keystrokes: ³2*~l*R,,Ü1+~,,x/ ~rTM/,,Ü~r+~,,yTM+2*~l/ ~,,b Press ` to get the following result: Entering this expression when the calculator is set in the RPN mode is exactly the same as this Algebraic mode exercise. For additional information on editing algebraic expressions in the calculator's display or stack see Chapter 2 in the calculator's user's guide. Page 2-4 Using the Equation Writer (EQW) to create expressions The equation writer is an extremely powerful tool that not only let you enter or see an equation, but also allows you to modify and work/apply functions on all or part of the equation. The Equation Writer is launched by pressing the keystroke combination ,O (the third key in the fourth row from the top in the keyboard). The resulting screen is the following. Press L to see the second menu page: The six soft menu keys for the Equation Writer activate functions EDIT, CURS, BIG, EVAL, FACTOR, SIMPLIFY, CMDS, and HELP. Detailed information on these functions is provided in Chapter 3 of the calculator's user's guide. Creating arithmetic expressions Entering arithmetic expressions in the Equation Writer is very similar to entering an arithmetic expression in the stack enclosed in quotes. The main difference is that in the Equation Writer the expressions produced are written in "textbook" style instead of a line-entry style. For example, try the following keystrokes in the Equation Writer screen: 5/5+2 The result is the expression: The cursor is shown as a left-facing key. The cursor indicates the current edition location. For example, for the cursor in the location indicated above, type now: *,,Ü5+1/3 The edited expression looks as follows: Page 2-5 Suppose that you want to replace the quantity between parentheses in the denominator (i.e., 5+1/3) with (5+2/2). First, we use the delete key () delete the current 1/3 expression, and then we replace that fraction with 2/2, as follows: ,,ìQ2 When hit this point the screen looks as follows: In order to insert the denominator 2 in the expression, we need to highlight the entire 2 expression. We do this by pressing the right arrow key (TM) once. At that point, we enter the following keystrokes: /2 The expression now looks as follows: Suppose that now you want to add the fraction 1/3 to this entire expression, i.e., you want to enter the expression: 5 5 + 2 (5 + 2 + ) 1 3 2 Page 2-6 First, we need to highlight the entire first term by using either the right arrow (TM) or the upper arrow (--) keys, repeatedly, until the entire expression is highlighted, i.e., seven times, producing: NOTE: Alternatively, from the original position of the cursor (to the right of the 2 in the denominator of 2/2), we can use the keystroke combination ,--, interpreted as (, ` ). Once the expression is highlighted as shown above, type +1/ 3 to add the fraction 1/3. Resulting in: Creating algebraic expressions An algebraic expression is very similar to an arithmetic expression, except that English and Greek letters may be included. The process of creating an algebraic expression, therefore, follows the same idea as that of creating an arithmetic expression, except that use of the alphabetic keyboard is included. To illustrate the use of the Equation Writer to enter an algebraic equation we will use the following example. Suppose that we want to enter the expression: 2 3 + e - µ LN x + 2 µ y 1/ 3 Use the following keystrokes: 2/R3TMTM*~,n+,,¸\~,m TMTM*,¹~,,x+2*~,m*~,c Page 2-7 ~,,y------/~,tQ1/3 This results in the output: In this example we used several lower-case English letters, e.g., x (~,,x), several Greek letters, e.g., (~,n), and even a y combination of Greek and English letters, namely, (~,c~,,y). Keep in mind that to enter a lower-case English letter, you need to use the combination: ~,, followed by the letter you want to enter. Also, you can always copy special characters by using the CHARS menu (...±) if you don't want to memorize the keystroke combination that produces it. A listing of commonly used ~, keystroke combinations is listed in Appendix D of the user's guide. For additional information on editing, evaluating, factoring, and simplifying algebraic expressions see Chapter 2 of the calculator's user's guide. Organizing data in the calculator You can organize data in your calculator by storing variables in a directory tree. The basis of the calculator's directory tree is the HOME directory described next. The HOME directory To get to the HOME directory, press the UPDIR function (,,§) -- repeat as needed -- until the {HOME} spec is shown in the second line of the display header. Alternatively, use ,, (hold) §. For this example, the HOME directory contains nothing but the CASDIR. Pressing J will show the variables in the soft menu keys: Page 2-8 Subdirectories To store your data in a well organized directory tree you may want to create subdirectories under the HOME directory, and more subdirectories within subdirectories, in a hierarchy of directories similar to folders in modern computers. The subdirectories will be given names that may reflect the contents of each subdirectory, or any arbitrary name that you can think off. For details on manipulation of directories see Chapter 2 in the calculator's user's guide. Variables Variables are similar to files on a computer hard drive. One variable can store one object (numerical values, algebraic expressions, lists, vectors, matrices, programs, etc). Variables are referred to by their names, which can be any combination of alphabetic and numerical characters, starting with a letter (either English or Greek). Some non-alphabetic characters, such as the arrow () can be used in a variable name, if combined with an alphabetical character. Thus, `A' is a valid variable name, but `' is not. Valid examples of variable names are: `A', `B', `a', `b', `', `', `A1', `AB12', ` A12', 'Vel', 'Z0', 'z1', etc. A variable can not have the same name as a function of the calculator. Some of the reserved calculator variable names are the following: ALRMDAT, CST, EQ, EXPR, IERR, IOPAR, MAXR, MINR, PICT, PPAR, PRTPAR, VPAR, ZPAR, der_, e, i, n1,n2, ..., s1, s2, ..., DAT, PAR, , . Variables can be organized into sub-directories (see Chapter 2 in the calculator's user's guide). Typing variable names To name variables, you will have to type strings of letters at once, which may or may not be combined with numbers. To type strings of characters you can lock the alphabetic keyboard as follows: ~~ locks the alphabetic keyboard in upper case. When locked in this fashion, pressing the ,, before a letter key produces a lower case letter, while pressing the , key before a letter key produces a special character. If the alphabetic keyboard is already locked in upper case, to lock it in lower case, type, ,,~. ~~,,~ locks the alphabetic keyboard in lower case. When locked in this fashion, pressing the ,, before a letter key produces an upper case letter. To unlock lower case, press ,,~. Page 2-9 To unlock the upper-case locked keyboard, press ~. Try the following exercises: ~~math` ~~m,,a,,t,,h` ~~m,,~at,,h` The calculator display will show the following (left-hand side is Algebraic mode, right-hand side is RPN mode): Creating variables The simplest way to create a variable is by using the K. The following examples are used to store the variables listed in the following table (Press J if needed to see variables menu): Name A12 Q R z1 p1 Contents -0.25 3×105 `r/(m+r)' [3,2,1] 3+5i << r '*r^2' >> Type real real algebraic vector complex program Algebraic mode To store the value of ­0.25 into variable : 0.25\K ~,a. AT this point, the screen will look as follows: Press ` to create the variable. The variable is now shown in the soft menu key labels when you press J: Page 2-10 The following are the keystrokes for entering the remaining variables: A12: 3V5K~a12` Q: ~,,r/,,Ü ~,,m+~,,rTMTMK~q` R: ,,Ô3,í2,í1TMK~r` z1: 3+5*,,¥K~,,z1` (Accept change to Complex mode if asked). p1: å,é~,,r³,,ì* ~,,rQ2TMTMTMK~,,p1`. The screen, at this point, will look as follows: You will see six of the seven variables listed at the bottom of the screen: p1, z1, R, Q, A12, a. RPN mode (Use H\@@OK@@ to change to RPN mode). Use the following keystrokes to store the value of ­0.25 into variable : .25\`³ ~,a`. At this point, the screen will look as follows: With ­0.25 on the level 2 of the stack and '' on the level 1 of the stack, you can use the K key to create the variable. The variable is now shown in the soft menu key labels when you press J: Page 2-11 To enter the value 3×105 into A12, we can use a shorter version of the procedure: 3V5³~a12`K Here is a way to enter the contents of Q: Q: ~,,r/,,Ü ~,,m+~,,rTMTM³~q`K To enter the value of R, we can use an even shorter version of the procedure: R: ,,Ô3#2#1TM ³~rK Notice that to separate the elements of a vector in RPN mode we can use the space key (#), rather than the comma (,í) used above in Algebraic mode. z1: ³3+5*,,¥³~,,z1K p1: ,å,é~,,r³,,ì* ~,,rQ2TMTMTM³~,,p1TM`K. The screen, at this point, will look as follows: You will see six of the seven variables listed at the bottom of the screen: p1, z1, R, Q, A12, . Page 2-12 Checking variables contents The simplest way to check a variable content is by pressing the soft menu key label for the variable. For example, for the variables listed above, press the following keys to see the contents of the variables: Algebraic mode Type these keystrokes: J@@z1@@ ` @@@R@@ `@@@Q@@@ `. At this point, the screen looks as follows: RPN mode In RPN mode, you only need to press the corresponding soft menu key label to get the contents of a numerical or algebraic variable. For the case under consideration, we can try peeking into the variables z1, R, Q, A12, , created above, as follows: J@@z1@@ @@@R@@ @@@Q@@ @@A12@@ @@»@@ At this point, the screen looks like this: Using the right-shift key followed by soft menu key labels In Algebraic mode, you can display the content of a variable by pressing J@ and then the corresponding soft menu key. Try the following examples: J,@@p1@@ , @@z1@@ , @@@R@@ ,@@@Q@@ , @@A12@@ NOTE: In RPN mode, you don't need to press @ (just J and then the corresponding soft menu key.) Page 2-13 This produces the following screen (Algebraic mode in the left, RPN in the right) Notice that this time the contents of program p1 are listed in the screen. To see the remaining variables in this directory, press L. Listing the contents of all variables in the screen Use the keystroke combination ,~ to list the contents of all variables in the screen. For example: Press $ to return to normal calculator display. Deleting variables The simplest way of deleting variables is by using function PURGE. This function can be accessed directly by using the TOOLS menu (I), or by using the FILES menu ,,¡@@OK@@ . Using function PURGE in the stack in Algebraic mode Our variable list contains variables p1, z1, Q, R, and . We will use command PURGE to delete variable p1. Press I @PURGE@ J @@p1@@ `. The screen will now show variable p1 removed: Page 2-14 You can use the PURGE command to erase more than one variable by placing their names in a list in the argument of PURGE. For example, if now we wanted to purge variables R and Q, simultaneously, we can try the following exercise. Press : I @PURGE@ ,,ä³J @@@R!@@ TM,í³J @@@Q!@@ At this point, the screen will show the following command ready to be executed: To finish deleting the variables, press `. The screen will now show the remaining variables: Using function PURGE in the stack in RPN mode Assuming that our variable list contains the variables p1, z1, Q, R, and . We will use command PURGE to delete variable p1. Press ³ @@p1@@ ` I @PURGE@. The screen will now show variable p1 removed: To delete two variables simultaneously, say variables R and Q, first create a list (in RPN mode, the elements of the list need not be separated by commas as in Algebraic mode): J ,,ä³ @@@R!@@ TM³ @@@Q!@@ ` Then, press I@PURGE@ use to purge the variables. Additional information on variable manipulation is available in Chapter 2 of the calculator's user's guide. Page 2-15 UNDO and CMD functions Functions UNDO and CMD are useful for recovering recent commands, or to revert an operation if a mistake was made. These functions are associated with the HIST key: UNDO results from the keystroke sequence ,¯, while CMD results from the keystroke sequence ,,®. CHOOSE boxes vs. Soft MENU In some of the exercises presented in this chapter we have seen menu lists of commands displayed in the screen. These menu lists are referred to as CHOOSE boxes. Herein we indicate the way to change from CHOOSE boxes to Soft MENUs, and vice versa, through an exercise. Although not applied to a specific example, the present exercise shows the two options for menus in the calculator (CHOOSE boxes and soft MENUs). In this exercise, we use the ORDER command to reorder variables in a directory. The steps are shown for Algebraic mode. ,,°~ Show PROG menu list and select MEMORY @@OK@@ ~~~~ Show the MEMORY menu list and select DIRECTORY @@OK@@ ---- Show the DIRECTORY menu list and select ORDER Page 2-16 @@OK@@ activate the ORDER command There is an alternative way to access these menus as soft MENU keys, by setting system flag 117. (For information on Flags see Chapters 2 and 24 in the calculator's user's guide). To set this flag try the following: H@FLAGS! -------------- The screen shows flag 117 not set (CHOOSE boxes), as shown here: Press the soft menu key to set flag 117 to soft MENU. The screen will reflect that change: Press @@OK@@ twice to return to normal calculator display. Now, we'll try to find the ORDER command using similar keystrokes to those used above, i.e., we start with ,,°. Notice that instead of a menu list, we get soft menu labels with the different options in the PROG menu, i.e., Page 2-17 Press B to select the MEMORY soft menu ()@@MEM@@). The display now shows: Press E to select the DIRECTORY soft menu ()@@DIR@@) The ORDER command is not shown in this screen. To find it we use the L key to find it: To activate the ORDER command we press the C(@ORDER) soft menu key. NOTE: most of the examples in this user manual assume that the current setting of flag 117 is its default setting (that is, not set). If you have set the flag but want to strictly follow the examples in this manual, you should clear the flag before continuing. References For additional information on entering and manipulating expressions in the display or in the Equation Writer see Chapter 2 of the calculator's user's guide. For CAS (Computer Algebraic System) settings, see Appendix C in the calculator's user's guide. For information on Flags see, Chapter 24 in the calculator's user's guide. Page 2-18 Chapter 3 Calculations with real numbers This chapter demonstrates the use of the calculator for operations and functions related to real numbers. The user should be acquainted with the keyboard to identify certain functions available in the keyboard (e.g., SIN, COS, TAN, etc.). Also, it is assumed that the reader knows how to change the calculator's operating system (Chapter 1), use menus and choose boxes (Chapter 1), and operate with variables (Chapter 2). Examples of real number calculations To perform real number calculations it is preferred to have the CAS set to Real (as opposed to Complex) mode. Exact mode is the default mode for most operations. Therefore, you may want to start your calculations in this mode. Some operations with real numbers are illustrated next: · Use the \ key for changing sign of a number. For example, in ALG mode, \2.5`. In RPN mode, e.g., 2.5\. · Use the Ykey to calculate the inverse of a number. For example, in ALG mode, Y2`. In RPN mode use 4Y. · For addition, subtraction, multiplication, division, use the proper operation key, namely, +-*/. Examples in ALG mode: 3.7+5.2` 6.3-8.5` 4.2*2.5` 2.3/4.5` Examples in RPN mode: 3.7` 5.2+ 6.3` 8.54.2` 2.5* 2.3` 4.5/ Alternatively, in RPN mode, you can separate the operands with a space (#) before pressing the operator key. Examples: 3.7#5.2+ Page 3-1