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Reverse Polish Calculator - By Circle Software - © Jan 1995 =========================================================== This program is shareware. Please read the file Licence in this directory. Introduction. ============ The Reverse Polish system is different from the way most calculators work, and although this is simple, you should not try to use it without first reading these notes. The nature and origin of Reverse Polish notation is explained at the end of this file. Note that the !Help facility is fully implemented. Select Help from the iconbar menu. This manual is divided into sections. Conventions: Describes the nomenclature used in the text. A Quick Start: Gives simple instructions without getting technical. How to use RPCalc: Explains the finer points of using the program. Reverse Polish: Describes the origin and nature of Reverse Polish. The Functions: Describes the use af all calculator functions. More Complex Examples. To find any function, search for the button as [name] e.g. [+] or [Sine] etc. N.B. To correctly display some button legends, your text editor should be set to display standard system font. e.g. [⇩] should appear as [down arrow]. Conventions =========== Throughout this manual, references to button icons on the calculator are shown between square brackets, e.g. [Enter]. The terms SELECT and ADJUST refer to the first and third mouse buttons, as normally defined, and may be used to prefix a function button. When not specifically stated, you should always use the SELECT button, as in most cases, ADJUST will cause an alternative action. A Quick Start ============= First, see how easy it is to use by doing a simple addition calculation, before going into details. Press the following keys in order - [3] [Enter] [4] [+] The value shown in the lower display is the result of adding 3 + 4, (7). To multiply the result by 5, press - [5] [x] And we get 35.00 as expected. Its as simple as that! Using this method you can always see exactly what values are about to be operated on, unlike ordinary calculators. You can also now see why it is called 'Reverse' Polish. On this calculator you press these buttons in the reverse order, compared to a normal one. However, this simple example may illustrate how to use the machine, but does not show the real power of the system. For that, read on. How to use RPCalc. ================= The Displays. The calculator has four display registers rather than the usual one. The lower register, labelled 'X', is equivalent to the single display found on ordinary calculators. Above this, the Y register is used to hold a second value to be operated on, while the remainder will display intermediate calculation results, where appropriate. These registers constitute a 'stack', where numbers are placed until needed. When you click on the [+] operator button, for example, the value displayed in the X register will be added to the value in Y, and the result displayed in X. A few functions produce two results, and in these cases the second result will be displayed in the Y register. Entering Values. You enter values into the X register in the normal way, by clicking on the number buttons, or by using the keyboard number pad. (In the later case, the calculator must have the input focus. If not, click in the calculator background, or the iconbar icon.) To get a value into the Y register, first enter it into X, as above, and then use the [Enter] key. This will move the value in the X register up to the Y, and clear X ready for a new value. When [Enter] is used, any values in higher registers will also be moved up, and any value in the top register will be lost. This should rarely happen. Adding Two Numbers. Thus, to add two values, enter the first value into X, click [Enter] to push this up to Y, enter the second number into X, then click [+]. This simple example does not show the advantages of Reverse Polish over the more conventional calculator, so let's try something more complex. Evaluating an Expression. To evaluate the expression ( 2 + 4 ) * ( 7 - 3 ) follow the button clicks in the first column below, and check the actions listed. In particular note the display contents as given in the last column. Button Action Stack ====== ================================== ======== X Y [2] '2' appears in X. 2 [Enter] The 2 moves up to Y, and X clears. 2 [4] '4' appears in X. 4 2 [+] '4' and '2' are unstacked, and 6 placed in X 6 [7] '7' appears in X, the 6 moves up to Y 7 6 [Enter] The 7 moves up to Y, and X clears. 7 6 [3] '3' appears in X 3 7 6 [-] '3' and '7' are unstacked, and 4 placed in X 4 6 [*] '4' and '6' are unstacked, and 24 placed in X 24 Thus X contains the final result, which we can use in further calculations if required. Note that it is not necessary to write down, remember, or re-enter either intermediate result 6 or 4, as these remain on the stack until needed. This calculation required 3 of the 4 registers provided, which you will find sufficient for the most complex calculations. Another Example Suppose you needed to calculate the VAT due and the total price of an item costing £123.50 We will use a property of the [Enter] button invoked by clicking with the ADJUST button. When you do this, the value in X will be moved up as before, but X will not be cleared, so its value may be used again. Press the buttons shown in the first column, as before - ( Remember to use the ADJUST button for [Enter] ). Buttons Action Stack ======= ================================== ============= X Y [1][2][3][.][5] 123.5 appears in X 123.50 [Enter] Use the ADJUST mouse button 123.50 123.50 [1][7][.][5] Stack rises & 17.5 appears in X 17.5 123.50 123.50 [%] X % of Y is computed in X 21.61 123.50 (This is the VAT due on 123.50) [+] 21.61 + 123.50 is computed in X 145.11 (This is the total due.) Note how the final total was computed with a single key stroke, without the need to re-enter any values at all. This was because we had duplicated the original 123.5 by clicking [Enter] with the ADJUST button. The Storage Registers In addition to the stack, the value in X may be stored in any of 10 special registers numbered from 0 to 9. To do this click on [Store] followed by the single digit indicating the required store number. To retrieve the stored number click on 'Store' with the Adjust button, followed by the store number. Try this - Buttons Action Stack ======= ================================== ============= X Y [pi] 3.14 appears in X 3.14 [Store] The store button stays 'in'. 3.14 [7] The value is stored. 3.14 [Clear] The X register is cleared 0.00 Now prove the value is stored by (Use the ADJUST button) - [Store] Use the Adjust button to 'fetch' 0.00 [7] Value stored in store 7 is shown 3.14 In the VAT example above, we could have stored the VAT rate, 17.5, in one of these storage registers, and used it in repeated VAT calculations. The Display Format In the above example, the value of 'pi' was shown as 3.14 because the calculator is initially set to display only 2 decimal places. The full value of pi (to about 15 significant figures) is of course used internally, and may be displayed by increasing the decimal places using the adjuster buttons at the top right of the calculator. Other display modes are also available. !Help RPCalc supports the Acorn !help facility, which may be used to explore all the functions on the calculator. To invoke this, select 'Help' from the iconbar menu, and move the pointer over the function buttons provided. You will find that most of the buttons provide two separate functions. The function displayed on the button is invoked using the SELECT mouse button, while another, often the inverse function, is invoked using the ADJUST button. Reverse Polish ============== Reverse Polish is a notation system for mathematical expressions invented by a Polish gentleman by the name of Lukasiewicz. Unfortunately for him, nobody could pronounce his name so the notation became known as Reverse Polish, and his name is all but forgotten. It is possible that even the notation itself would have been forgotten were it not for the arrival of computers, where it has become probably one of the most often used systems, not for writing expressions, but for storing them for later computation, as for example in compiled code. The Forth programming language is based upon the Reverse Polish system, and Postscript also works this way. The tag 'reverse' simply refers to the fact that operators, such as '+' and '-' are written after the values on which they operate rather than before, as in more conventional notation. The advantage of the notation is that it permits an expression to be written without the use of parentheses, while remaining totally un-ambiguous. For example, the expression - ( 2 + 4 ) * ( 7 - 3 ) would be written, or stored as - 2 4 + 7 3 - * To evaluate this, the expression is scanned from left to right, and any values found removed and placed aside on a stack (i.e. a heap or pile). When an operator is found, the required number of values are retrieved from the stack (last on, first off), the operation carried out on them, and the result put back on the stack. This system is continued to the end of the expression. Thus in the above case, 2 and then 4 are placed on the stack before the '+' operator is found. These values are then retrieved and the addition carried out. The result, 6, is then placed back on the stack. Then 7 and 3 are found in turn and placed on top of the 6. At this point a '-' operator is found, so the last two values, 7 and 3, are retrieved and the result of the subtraction, 4, is placed back on the stack, as before. At this point the stack now has a 4 on top of 6, so when the last operator is found, the '*', these two numbers are multiplied together and the result, now the final answer, 24, is placed back on the stack. Notice that whenever an operator is found, only the required number of values are unstacked, so that the system works equally well for unary operators such as x! or sin(x), as for binary ('+' '/' etc) or other operators. The system may be extended to include multi-parameter functions, such as function( a, b, c ). Which is simply written - a b c function It is also possible to include programming constructs, such as if-then-else, repeat, etc, as is done in Forth and Postscript. Note also that the notation does not require any knowledge of operator priority. Operators are always executed when found, using however many values are needed. The Functions. ============= This section gives full descriptions of each available function, for each calculator button. In all cases, X, or X and Y where appropriate, are unstacked, and the result placed in X, unless otherwise stared. Use of the SELECT mouse button is assumed where not stated. The stack manipulations. [Enter] Causes the stack to rise. SELECT Clears the X register ADJUST Preserves the X register. [⇩] SELECT Rotates the stack downwards. ADJUST Rotates the stack upwards. In both cases end values are wrapped around. [X⇧ Y⇩] Swaps the X and Y register values. The stack does not rise. Simple Arithmetic. [-] Subtract. Computes Y - X [+] Add. Computes Y + X [×] Multiply. Computes Y * X [÷] Divide. Computes Y ÷ X Functions of X. [x!] Factorial. Computes x * (x-1) * (x-2) * (x-3) ... * 2 * 1 e.g. to compute factorial 10, key in - [10] [x!] result 3628800 [1/x] Reciprocal. Computes 1 ÷ x e.g. to compute 1 / e, key in - [e] [1/x] result 0.37 [x^y] Powers. ( Actual legend not reproducible here) SELECT Computes X the the power of Y ADJUST Computes Y the the power of X [x²] Squares. SELECT Computes x * x ADJUST Computes the square root of x Miscellaneous Functions. [±] Changes the sign of X. i.e. -x becomes x, and x becomes -x [Int] Integer. SELECT Computes the integer part of x, removing fractions. ADJUST Computes the fractional part of x. [Abs] Absolute Computes |x| i.e. Removes any -ve sign. Logarithms. [Log] SELECT Computes the log to base 10. ADJUST Computes the anti-log to base 10. [Ln] SELECT Computes the log to base E. ADJUST Computes the anti-log to base E. Trig Functions. [Sin] SELECT Computes sine(x) ADJUST Computes angle whose sine is x [Cos] SELECT Computes cos(x) ADJUST Computes angle whose cosine is x [Tan] SELECT Computes tan(x) ADJUST Computes angle whose tangent is x Hyperbolic Functions. [Sinh] SELECT Computes sinh(x) ADJUST Computes inverse of sinh(x) [Cosh] SELECT Computes cosh(x) ADJUST Computes inverse of cosh(x) [Tanh] SELECT Computes tanh(x) ADJUST Computes inverse of tanh(x) Other Functions. [Polar] This function produces two result values. SELECT Computes polar from cartesian co-ordinates. X = Sqrt( x*x + y*y ) Y = angle whose tangent is Y ÷ X e.g. Given a 3, 4, 5 triangle, key in - [3] [Enter] [4] [Polar] results in - X = 5, the (hypotenuse) Y = 36.87° ADJUST Computes cartesian from polar co-ordinates. X = x * cos(y) Y = x * sin(y) [H.MS] Hours, Minutes and Seconds Conversion. The display should be set to Fixed point, 4 decimal places to use this function. SELECT Computes hrs, mins, secs from decimal hrs. i.e. result X = H.MMSS where MM == Minutes, SS == Seconds. e.g. Given 12.56 hrs, key in - [12.56] [H.MMSS] results in 12.3336 i.e. 12 hrs 33 mins 36 secs. ADJUST Computes the inverse, converting hrs, mins, secs to hrs. e.g. Map reference 12° 18', key in - [12.18] ADJUST [H.MMSS] results in 12.30° [yCx] Combinations and Permutations. SELECT Computes the number of possible combinations of Y objects taken X at a time, where order is unimportant. e.g. To compute the number of ways to select 8 objects from a total of 12, key in - [12] [Enter] [8] [yCx], result 495. N.B. This is how to compute (so called) permutations on your Pools coupon. These are really Combinations, as order is not important. ADJUST Computes the number of possible permutations of Y objects taken X at a time, where order IS important. e.g. Taking the above example, key in - [12] [Enter] [8] ADJUST [yCx], result 19,958,400 More Complex Examples. ===================== 1. A laser range finder gives the distance to the top of a building as 97.66 meters at an elevation of 12° 25'. Compute the height of the building, and its horizontal distance. We must first convert the angle to decimal degrees, then enter the angular distance and convert to cartesian co-ordinates. key in - [12.25] ADJUST [H.MS] [97.66] ADJUST [POLAR] Results are X = 95.38, the horizontal distance. Y = 21.00, the vertical height of the building. 2. Compute the roots of the equation: 3x² - 9x + 6 = 0 Given the equation -b ± €(b² - 4ac) i.e. a = 3, b = -9, c = 6 ---------------- 2a This example demonstrates the use of the internal storage registers as well as some stack manipulation, all of which eliminates the need to remember intermediate results, and the necessity to enter the same value more than once. The best way of tackling a problem like this is to start in the middle, with the most complex part, like the inner root function. Buttons Action Stack ======= ====================== ====================== [9] value -b 9 [Store] [1] We will need this later 0 [x²] compute b² 81 [4] enter 4 4 81 [Enter] push the stack up 4 81 [3] value a 3 4 81 [Store] [2] We will need this again 3 4 81 [x] compute 4a 12 81 [6] value c 6 12 81 [x] compute 4ac 72 81 [-] compute b² - 4ac 9 Adj [x²] compute sqr root. 3 Adj [Enter] duplicate value 3 3 Adj [Store] [1] recall -b 9 3 3 [+] compute -b + (b² - 4ac) 12 3 Adj [Store] [2] recall a 3 12 3 [2] value 2 2 3 12 3 [x] compute 2a 6 12 3 [÷] First result = 2 2 3 [⇩] loose result 3 Adj [Store] [1] recall -b 9 3 [x⇧ y⇩] swap values over 3 9 [-] compute -b - (b² - 4ac) 6 Adj [Store] [2] recall a 3 6 [2] value 2 2 3 6 [x] compute 2a 6 6 [÷] Second result = 1 1 Hence, the roots are x = 1 and x = 2. There may be shorter ways of doing this. Note that we stored -b and a values for later use. In this simple contrived example this may have been pointless, as they could easily be re-entered. However, in a real example these values may have had 9 or more digits, so that storing them would be a real time saver. If the inner function had been negative indicating complex roots, we would have had to proceed slightly differently. End.