There are various ways in which calculators interpret keystrokes. These can be categorized into two main types:
- On a single-step or immediate-execution calculator, the user presses a key for each operation, calculating all the intermediate results, before the final value is shown.[1][2][3]
- On an expression or formula calculator, one types in an expression and then presses a key, such as '=' or 'Enter', to evaluate the expression.[4][5][6] There are various systems for typing in an expression, as described below.
- 1Immediate execution
Immediate execution[edit]
The immediate execution mode of operation (also known as single-step, algebraic entry system (AES)[7] or chain calculation mode) is commonly employed on most general-purpose calculators. In most simple four-function calculators, such as the Windows calculator in Standard mode and those included with most early operating systems, each binary operation is executed as soon as the next operator is pressed, and therefore the order of operations in a mathematical expression is not taken into account. Scientific calculators, including the Scientific mode in the Windows calculator and most modern software calculators, have buttons for brackets and can take order of operation into account. Also, for unary operations, like √ or x2, the number is entered first, then the operator; this is largely because the display screens on these kinds of calculators are generally composed entirely of seven-segment characters and thus capable of displaying only numbers, not the functions associated with them. This mode of operation also make it impossible to change the expression being input without clearing the display entirely.
Formula | keystrokes | keystroke count |
---|---|---|
2×3+1= 1+2×3= | 6 6 | |
30SIN×30COS= SIN30xCOS30= | 8 | |
5−3= | 4 | |
15+10+10+10= | 12 |
The first and second example have been given twice. The first version is for simple calculators, showing how it is necessary to rearrange operands in order to get the correct result. The second version is for scientific calculators, where operator precedence is observed.
Immediate-execution calculators are based on a mixture of infix and postfix notation: binary operations are done as infix, but unary operations are postfix. Because operators are applied one-at-a-time, the user must work out which operator key to use at each stage, and this can lead to problems.[8][9] When discussing these problems, Professor Harold Thimbleby has pointed out that button-operated calculators 'require numbers and operation signs to be punched in a certain order, and mistakes are easy to make and hard to spot'.[10]
Problems can occur because, for anything but the simplest calculation, in order to work out the value of a written formula, the user of a button-operated calculator is required to:
- Rearrange the formula so that the value can be calculated by pressing buttons one at a time, while taking operator precedence and parentheses into account.
- Use memory buttons to ensure that operations are applied in the correct order.
- Use the special buttons ± and 1/x, that do not correspond to operations in the formula, for non-commutative operators.
Mistakes can be hard to spot because:
- For the above reasons, the sequence of button presses may bear little resemblance to the original formula.
- The operation carried out when a button is pressed is not always the same as the button, but could be a previously entered operation.
Examples of difficulties[edit]
The simplest example given by Professor Thimbleby of a possible problem when using an immediate-execution calculator is 4 × (−5).[11] As a written formula the value of this is −20 because the minus sign is intended to indicate a negative number, rather than a subtraction, and this is the way that it would be interpreted by a formula calculator.
On an immediate-execution calculator, depending on which keys are used and the order in which they are pressed, the result for this calculation may be different. Also there are differences between calculators in the way a given sequence of button presses is interpreted.[12] The result can be:
- −1: If the subtraction button − is pressed after the multiplication ×, it is interpreted as a correction of the × rather than a minus sign, so that 4 − 5 is calculated.
- 20: If the change-sign button ± is pressed before the 5, it isn't interpreted as −5, and 4 × 5 is calculated.
- −20: To get the right answer, ± must be pressed last, even though the minus sign isn't written last in the formula.[13]
The effects of operator precedence, parentheses and non-commutative operators, on the sequence of button presses, are illustrated by:
- 4 − 5 × 6: The multiplication must be done first, and the formula has to be rearranged and calculated as −5 × 6 + 4. So ± and addition have to be used rather than subtraction. When + is pressed, the multiplication is performed.
- 4 × (5 + 6): The addition must be done first, so the calculation carried out is (5 + 6) × 4. When × is pressed, the addition is performed.
- 4 / (5 + 6): One way to do this is to calculate (5 + 6) / 4 first and then use the 1/x button, so the calculation carried out is 1/[(5 + 6)/4].
- 4 × 5 + 6 × 7: The two multiplications must be done before the addition, and one of the results must be put into memory.[13]
These are only simple examples, but immediate-execution calculators can present even greater problems in more complex cases. In fact, Professor Thimbleby claims that users may have been conditioned to avoid them for all but the simplest calculations.[14]
Declarative and imperative tools[edit]
The potential problems with immediate-execution calculators stem from the fact that they are imperative.[15] This means that the user must provide details of how the calculation has to be performed.
Professor Thimbleby has identified the need for a calculator that is more automatic and therefore easier to use, and he states that such a calculator should be more declarative.[16] This means that the user should be able to specify only what has to be done, not how, and in which order, it has to be done.
Formula calculators are more declarative because the typed-in formula specifies what is to be done, and the user does not have to provide any details of the step-by-step order in which the calculation has to be performed. Film bokep dewasa tanpa sensor.
Declarative solutions are easier to understand than imperative solutions,[16][17] and there has been a long-term trend from imperative to declarative methods.[18][19] Formula calculators are part of this trend.
Many software tools for the general user, such as spreadsheets, are declarative.[20] Formula calculators are examples of such tools.
Using the full power of the computer[edit]
Software calculators that simulate hand-held, immediate execution calculators do not use the full power of the computer: 'A computer is a far more powerful device than a hand-held calculator, and thus it is illogical and limiting to duplicate hand-held calculators on a computer.' (Haxial Software Pty Ltd)[21] Formula calculators use more of the computer's power because, besides calculating the value of a formula, they work out the order in which things should be done.
Infix notation[edit]
Infix notation is a method where unary operations are input into the calculator in the same order as they are written on paper. Different forms of this input scheme exist. In the algebraic entry system with hierarchy (AESH),[7] the precedence of basic mathematical operators is taken into account,[7] whereas calculators with algebraic entry system with parentheses (AESP)[7] support the entry of parentheses.[7] An input scheme known as algebraic operating system (AOS)[7] combines both.[7]
Calculators that use infix notation tend to incorporate a dot-matrix display to display the expression being entered, frequently accompanied by a seven-segment display for the result of the expression. Because the expression is not evaluated until it is fully entered, there is provision for editing the entered expression at any point prior to evaluation, as well as replaying entered expressions and their answers from memory.
Most graphing calculators by Casio and Texas Instruments use this method. On its scientific calculators, Sharp calls this method Direct Algebraic Logic (D.A.L.),[22] and Casio calls this method the Visually Perfect Algebraic Method (V.P.A.M.).[23]
Formula | keystrokes | keystroke count |
---|---|---|
1+2×3= | 6 | |
SIN30×COS30= | 8 | |
(1+2)×(3+4)= | 12 | |
15+10+10+10= | 12 |
Reverse Polish notation[edit]
In reverse Polish notation,[7] also known as postfix notation, all operations are entered after the operands on which the operation is performed. Reverse Polish notation is parenthesis-free, which usually leads to fewer button presses needed to perform an operation. By the use of a stack, one can enter formulas without the need to rearrange operands.
Hewlett-Packard's calculators are well-known examples among calculators which use RPN. Early models, such as the HP-35, used RPN entirely without any alternative methods. Later models, such as the HP 35s, also had infix notation and could conveniently allow users to switch between it and RPN.
Formula | keystrokes | keystroke count |
---|---|---|
1↵ Enter2↵ Enter3×+ 2↵ Enter3×1+ | 7 6 | |
30SIN30COS× | 7 | |
1↵ Enter2+3↵ Enter4+× | 9 | |
15↵ Enter10+10+10+ 15↵ Enter10↵ Enter↵ Enter↵ Enter+++ 15ENTER^10ENTER^ENTER^+++ | 12 11 (RPL and Entry RPN)[24] 10 (Classical RPN)[24] |
Note: The first example illustrates one of the few cases where reverse Polish notation does not use the fewest button presses – provided one does not rearrange operands. If one would do so then only six keystrokes would be needed.
BASIC notation[edit]
BASIC notation is a particular implementation of infix notation where functions require their parameters to be in brackets.
This method was used from the 1980s to the 1990s in BASIC programmable calculators and pocket computers. Texas Instruments would later implement the method in many of its graphing calculators, including the TI-83 and TI-84 Plus series. Most computer algebra systems also use this as the default input method.
In BASIC notation, the formula is entered as it would be entered in BASIC, using the PRINT
command – the PRINT
command itself being optional. On pressing 'ENTER' or '=', the result would be displayed. As with standard infix notation, typing mistakes in the entered formula could be corrected using the same editor function as the one used when programming the calculator.
Formula | keystrokes | keystroke count |
---|---|---|
1+2×3↵ Enter | 6 | |
SIN(30)×COS(30)↵ Enter SIN(30)×COS(30)↵ Enter | 12 16 |
For the second example, two options are given depending on if the BASIC programmable pocket computers have dedicated trigonometric keys[25] or not.[26]
Ten key notation[edit]
The ten-key notation input method first became popular with accountants' paper tape adding machines. It generally makes the assumption that entered numbers are being summed, although other operations are supported. Each number entered is followed by its sign (+/−), and a running total is kept. An assumption is made that the last operand can be implicitly used next, so by just entering another + (for example), one will reuse the most recent operand. Ten key input mode is available in printing calculators from companies such as Sharp,[27] and in software calculators like Judy's TenKey[28] used by accounting firms. Online tenkey training and certification tools are available as well,[29][30] and some businesses use ten key typing speed as an employment criterion.
Formula | keystrokes | keystroke count |
---|---|---|
1+2×3=+T | 8 | |
30SIN×30COS= | 8 | |
5+3-T | 5 | |
15+10+++T | 9 |
Mathematical display[edit]
Modern computer algebra systems, as well as many scientific and graphing calculators, allow for 'pretty-printing', that is, entry of equations such that fractions, surds and integrals, etc. are displayed in the way they would normally be written. Such calculators are generally similar in appearance to those using infix notation, but feature a full dot-matrix display and templates for entering expressions, which are navigated using arrow keys on the calculator. The templates contain spaces for values or expressions to be entered, and empty values would typically result in a syntax error, making it more cumbersome to navigate than standard infix notation; standard infix notation is often an option on such calculators as well.
Casio used to call this feature Natural Display or Natural textbook display,[31][32] but now uses Natural-VPAM.[33] Sharp calls this WriteView[34] on its scientific calculators and simply Equation Editor on its graphing calculators.[35] HP calls this its Textbook display setting,[36] which can be used in both RPN and Algebraic mode and in both the Stack and in the Equation Writer application.[37]Mathematica calls this Semantic-Faithful Typesetting.[38]Mathcad calls this standard math notation.[39]Maple has a Math Equation Editor,[40] but does not have a special name for this input method. Texas Instruments calls it MathPrint,[41] incorporating it in its high-end calculators, such as the TI-Nspire series, and in 2011 added the feature to its TI-84 series with the 2.55 OS update.[42]
Formula | keystrokes | keystroke count |
---|---|---|
1+2×3↵ Enter | 6 | |
SIN30→×COS30↵ Enter SIN(30)×COS(30)↵ Enter | 9 12 | |
5−3↵ Enter | 4 | |
15+10+10+10↵ Enter | 12 |
For the second example, two options are given, depending on whether the calculators will automatically insert needed parentheses or not. Machines equipped with an alphanumeric display will display SIN(30)×COS(30) before ↵ Enter is pressed.
See also[edit]
References[edit]
- ^Microsoft’s Windows Operating System Calculator Accessory; 2001. Available on a Windows PC at: Start/All Programs/Accessories/Calculator.
- ^MotionNET Calculator page on the Internet; 2006.
- ^Flow Simulation Ltd Virtual Calc98 page on the Internet; 2008.
- ^Formula Calculators Pty Ltd [Home page on the Internet]; 2009.
- ^Moisey Oysgelt JavaScript Formula Calculator page on the Internet; 2000.
- ^Haxial Software Pty Ltd Calculator Product page on the Internet; 2001
- ^ abcdefghBall, John A. (1978). Algorithms for RPN calculators (1 ed.). Cambridge, Massachusetts, USA: Wiley-Interscience, John Wiley & Sons, Inc.ISBN0-471-03070-8.
- ^Harold Thimbleby (September 1998). 'A new calculator and why it is necessary'(PDF). Computing Science, Middlesex University, London, UK. Archived from the original(PDF) on 2007-02-07. Retrieved 2009-05-04.
- ^Neville Holmes. Truth and Clarity in Arithmetic, University of Tasmania; 2003.
- ^Professor devises easier calculator; www.physorg.com. June 2005.
- ^Reference 11, section 2.
- ^References 4, 5 and 6[clarify].
- ^ abReference 4[clarify].
- ^Reference 11[clarify], section 3.2, second paragraph.
- ^Reference 11[clarify], sections 1 and 10.
- ^ abReference 11[clarify].
- ^Roy E. Furman (July 2006). 'Declarative Programming – Strategies for Solving Software Problems'. Archived from the original on 2012-07-23. Retrieved 2009-05-04.
- ^David A. Watt. Programming language concepts and paradigms, Prentice Hall; 1990. Citation 13 at http://citeseer.ist.psu.edu/context/14802/0.
- ^Tatsuru Matsushita. Expressive Power of Declarative Programming Languages, PhD thesis, Department of Computer Science, University of York; October 1998. Citation 13 at http://citeseer.ist.psu.edu/context/14802/0.
- ^Reference 20[clarify], paragraph 6.
- ^Reference 3, second paragraph
- ^Direct algebraic logic
- ^What does VPAM stand for
- ^ abhttp://h20331.www2.hp.com/hpsub/downloads/S07%20HP%20RPN%20Evolves%20V5b.pdf
- ^The picture of the Casio FX-880P shows sin, cos and tan keys on the second row right hand side.
- ^The picture of the Sharp PC-1245 shows no trigonometric keys
- ^Assortment of physical calculators, including printing ones that use tenkey syntax
- ^Example calculator which toggles between infix, RPN, and tenkey
- ^http://www.learn2type.com/typingtest/tenkeycertificate.cfm
- ^http://www.abbyinc.com/abbyinc/KeyPro.asp
- ^Natural textbook display - Scientific calculator.
- ^Natural textbook display - Graphic calculator
- ^Natural Visually Perfect Algebraic Mode (V.P.A.M) - Scientific calculator.
- ^WriteView.
- ^Sharp Graphing equation editor.
- ^Inc., HP. 'HP Support document - HP Support Center'. h20564.www2.hp.com. Retrieved 2016-08-23.
- ^http://h20331.www2.hp.com/Hpsub/downloads/50gUsing_the_EquationWriter_Part2.pdf.
- ^Semantic-Faithful Typesetting.
- ^Mathcad.
- ^http://www.maplesoft.com/products/maple/features/feature_detail.aspx?fid=5907
- ^http://education.ti.com/educationportal/sites/US/productDetail/us_os_84plus.html
- ^'Texas Instruments releases new OS for TI-84, 2.55 MP'. Tech Powered Math. 2011-01-14. Retrieved 2018-05-12.
Casio V.P.A.M. calculators are scientific calculators made by Casio which use Casio's Visually Perfect Algebraic Method (V.P.A.M.), Natural Display or Natural V.P.A.M. input methods.
V.P.A.M. is an infix system for entering mathematical expressions, used by Casio in most of its current scientific calculators. In the infix notation the precedence of mathematical operators is taken into account. According to Casio, in V.P.A.M. calculations can be input exactly as they are normally written. Functions, operators and symbols are shown on the calculator display and calculations are performed according to operator precedence.
- 2List of calculators
- 2.2S-V.P.A.M. (Two-line, Multi-replay)
History[edit]
The V.P.A.M. name was first introduced in 1994 with the introduction of the fx-991S scientific calculator in Japan. In 1998, the Casio fx-991W model used a two-tier (multi-line) display and the system was termed as S-V.P.A.M. (Super V.P.A.M.). The model featured a 5x6-dot LCD matrix cells on the top line of the screen and a 7-segment LCD on the bottom line of the screen that had been used in Casio fx-4500P programmable calculators.[1] The S-V.P.A.M. system was also used in the other W series models and also the MS series of calculators that followed. V.P.A.M. is similar to the Direct Algebraic Logic (D.A.L.) used by Sharp in some of their scientific calculators.
The fx-82ES introduced by Casio in 2004 was the first calculator to incorporate the Natural Textbook Display (or Natural Display) system. It allowed the display of expressions of fractions, exponents, logarithms, powers and square roots etc. as they are written in a standard textbook. Natural Display uses natural representation of mathematical expressions and formulas through a 96 × 31 dot matrixLCD display. Casio uses the term Natural V.P.A.M. for the fx-ES Plus series of calculators which are the upgraded version of the fx-ES series.
In early 2015, Casio introduced a new line of calculators called CLASSWIZ for different markets, featuring a high resolution (192 × 63) dot matrix Natural Textbook Display and incorporating spreadsheet functions in some models. This series of calculators is called the fx-EX series and it succeeds the fx-ES Plus series of calculators.
List of calculators[edit]
Note: Italic figures in parentheses indicate the year of introduction, e.g.: (c. 1994)
V.P.A.M. (Visually perfect algebraic method)[edit]
Casio Vpam
Models:
- fx-991S / 570S / 911S (c. 1994)
- fx-115S / 100S / 122S (c. 1995)
- fx-993S / 992S (c. 1996)
- fx-300S (a simplified 8+2 digit version)
S-V.P.A.M. (Two-line, Multi-replay)[edit]
W-series[edit]
Changes to S-series calculators include:
- 2-line LCD display featuring 5x6-dot matrix cells top line and 7-segment LCD bottom line that had been used in Casio fx-4500P programmable calculators (used 5x7-dot matrix cells).[2]
- fx-991W / 570W / 115W / 100W / 115WA (c. 1998-99)
- fx-82TL / 83W / 85W / 85WA / 300W / 350TL / 83WA / 270W (c. 1998-99)
MS-series[edit]
Revised variants of W Series of calculators including new functions such as:
- Relocation of multiple functions into menus, previously featured in Casio fx-5500LA,[3] but function is selected by numeric keypad
- Multiple statement recall
Non-programmable models:
- fx-991MS / 570MS / 115MS / 100MS / 95MS / 85MS / 350MS / 82MS (early 2000s)
- fx-912MS (Japanese version of the fx-115MS)
- fx-300MS (U.S. model)
- OH-300MS (Overhead projection model), fx-100AU (Australia specific)
- fx-500MS (Vietnam)
- fx-290 (Japan)
- fx-220 PLUS (International)
Programmable models:
- fx-3650/3950P (International)
- fx-3650P II (International)
- fx-50F PLUS (International)
- fx-50FH (Hong Kong)
- fx-50FH II (Hong Kong)
- fx-72F (Japan)
Natural Textbook Display[edit]
Revised version of the MS Series including the following changes:
- Mutli-line 96x31 dot matrix displays, but character cells are connected by dots like graphing calculators
- The ability to display and edit mathematical formulae in more visual formats
- Revised design of function menu interface, previously featured in Casio fx-4800P[4]
- No engineering symbol (SI prefixes) input or display facility, which was available in certain MS / W / S series models (fx-100, 115, 570 and 991MS / W / S).
Model-specific features include:
- Numerical integration uses Gauss–Kronrod quadrature formula
Models:
- fx-991ES / 570ES / 115ES (c. 2004)
- fx-500ES / 350ES / 300ES / 85ES / 83ES / 82ES (c. 2004)
- OH-300ES (Overhead projection model)
- FC-100/200V (financial functions)
Natural V.P.A.M.[edit]
Models:
- fx-82AU PLUS II (Australia)
- fx-82ES PLUS A (China, simplified Chinese messages)
- fx-82 / 95 / 350 / 570 / 991ES PLUS (c. 2008-09)
- fx-115 / 300ES PLUS (U.S.)
- fx-991ES PLUS C (Canada version of fx-115ES PLUS)
- fx-82 / 85 / 87 / 991DE PLUS (Germany)
- fx-85GT/fx-83GT Plus (EU specific)
- fx-82 / 991ZA Plus (South Africa)
- fx-373 / 913 / 573 / 993ES (2008) (Japan)
- fx-375 / 915 / 995ES (2012) (Japan)
- fx-500 / 570VN PLUS (Vietnam)
- fx-95 / 96SG PLUS (Singapore)
- OH-300ES PLUS (Overhead projection model)
- fx-FD10 Pro (International civil engineering model)
The CPU found in those models is based on the nX-U8/100 architecture.[5]
ClassWiz (High-resolution Natural Textbook Display)[edit]
Changes to the ES PLUS series include:
- Standard 192×63 display
- Icon-based mode menu previously used on Casio fx-9700GE graphing calculators
- Mode-specific functions are accessible via a unified function key as in fx-4800P
Model-specific features include:
- Scientific constants and conversion functions are grouped into categories
- Multilingual messages (excluding EX models and DE X models)
- QR code export (excluding CN X models, VN X models and fx-530AZ STUDY CAL)
- Multiline display supports up to 6 rows (excluding JP series, fx-530AZ STUDY CAL and 580VN. x models)
- Vector mode now supports maximum 4-variable vectors and 4 user-defined vector storage
- Matrix mode now supports maximum 4×4 matrices and 4 user-defined matrix storage
- New spreadsheet mode (excluding CN X and VN X models)
- Simultaneous linear equation solver supports up to 4 variables
- Polynomial equation solver supports up to 4th degree equations and inequalities
- Engineering symbols display and entry previously found in MS / W / S / D-series calculators
Models:
- fx-82 / 85 / 350 / 570 / 991EX (2015 1Q) (International)[6]
- fx-JP500 / 700 / 900 and fx-530AZ STUDY CAL (late 2014) (Japan)
- fx-87 / 991DE X (Germany)
- fx-82 / 350 / 570 / 991CE X (Central Europe)
- fx-85GT X (2019) (EU, UK)
- fx-92 Spéciale Collège (France)
- fx-92B Spéciale Collège (Belgium)
- fx-82 / 350 / 570 / 991SP X II Iberia / fx-85SP X II Iberia (Portugal, Spain)
- fx-82 / 350 / 570 / 991LA X (Latin America)
- fx-82 / 95 / 570 / 991AR X Iberia, fx-95AR X, fx-570AR X, fx-991AR X (2015) (Middle East, Arabic Africa)[7]
- fx-82 / 95 / 350 / 991CN X 中文版 (Chinese Edition) (2014) (China)[8]
- fx-580VN X (Vietnam)
References[edit]
- ^Casio fx-4500P: Nice looking pocket machine made in China
- ^Casio fx-4500P: Nice looking pocket machine made in China
- ^Kyoro's Room Blog CASIO fx-5500LA
- ^Kyoro's Room Blog CASIO fx-4800P
- ^https://www.oki.com/jp/Home/JIS/Books/KENKAI/n203/pdf/203_R06.pdf
- ^Casio to Release World’s First Standard Scientific Calculator Equipped with a Spreadsheet Function
- ^Casio to Release World’s First Standard Scientific Calculator Which Displays Arabic Script on the Screen
- ^【ClassWiz联测】卡西欧旗舰型号991CNX/JP900/991DEX/991SPX对比