                      UNDERSTANDING REVERSE-POLISH NOTATION
        
        For those of us using or considering a Hewlett-Packard scientific 
        calculator or computer, the question often arises, "What the heck 
        is Reverse-Polish Notation?"
        
        The conventional mathematical notation we learned in high school 
        is known as "Algebraic Notation."  In this notation, the 
        operators are placed either between or in front of the arguments.  
        Operators designate the operation to be performed ("+", "*", 
        "sin", etc.) and arguments are the values or variables upon which 
        they act ("1", "42.5", "pi", "x", etc.).  A typical mathematical 
        expression might be:
        
          sin[123 + 45 ln(27 - 6)]
        
        The operators in this expression, from left to right, are "sin", 
        "+", "*" (implied between "45" and "ln"), "ln", and "-".  By 
        their natures, "sin" and "ln" require one argument each while 
        "+", "*", and "-" require two each.
        
        The arguments are not so easily determined.  At first glance they 
        appear to be "123", "45", "27", and "6", but this is patently 
        false!  The argument of "sin" is everything within the brackets, 
        or "123 + 45 ln(27 - 6)", those of "+" are "123" and "45 ln(27 - 
        6)", those of "*" are "45" and "ln(27 - 6)", that of "ln" is "27 
        - 6", and those of "-" are "27" and "6".
        
        The whole expression may be thought of as a series of separate 
        operations:
        
          sin[123 + 45 ln(27 - 6)] = sin(a) :: a = 123 + 45 ln(27 - 6)
          123 + 45 ln(27 - 6) = 123 + b     :: b = 45 ln(27 - 6)
          45 ln(27 - 6) = 45 * c            :: c = ln(27 - 6)
          ln(27 - 6) = ln(d)                :: d = 27 - 6
          27 - 6
        
        It is because of this that complex expressions are performed from 
        the inside out:
        
          sin[123 + 45 ln(27 - 6)]      = sin[123 + 45 ln(21)
          sin[123 + 45 ln(21)]          = sin(123 + 45 * 3.04452243772)
          sin(123 + 45 * 3.04452243772) = sin(123 + 137.003509697)
          sin(123 + 137.003509697)      = sin(260.003509697)
          sin(260.003539697)            = 0.680672740775
        
        In this manner, it can easily be seen that each operation is a 
        function of its argument or arguments, and an argument may itself 
        be a function of other arguments.  Therefore, if we were to treat 
        the operations exactly as functions of arguments in the 
        traditional (x) and (y,x) formats, the expression "27 - 6" 
        would become -(27,6), where "-" is the operator "", "27" is the 
        argument "y", and "6" is the argument "x":

          27 - 6  ÄÄ>  -(27,6)
        
        This is a two-argument function:  in any two-argument function, 
        the second argument will always act against the first argument.  
        In this case, the second argument "6" will be subtracted from the 
        first argument "27".
        
        The concept of treating all operators as functions is not as 
        strange as it first appears when you consider that a considerable 
        number of operators are already functions:  ln(x) is (x) where 
        "" is "ln".
        
        In the case of our example expression, the next function out is 
        ln(d), where "d" is itself the function -(27,6):
        
          27 - 6                   ÄÄ> -(27,6)
          ln(27 - 6)               ÄÄ> ln(-(27,6))
          45 ln(27 - 6)            ÄÄ> *(45,ln(-(27,6)))
          123 + 45 ln(27 - 6)      ÄÄ> +(123,*(45,ln(-(27,6))))
          sin[123 + 45 ln(27 - 6)] ÄÄ> sin(+(123,*(45,ln(-(27,6)))))
        
        The final function sin(+(123,*(45,ln(-(27,6))))) may at first 
        seem confusing, especially with all the parentheses, until one 
        remembers that it is the one-argument function sin(x).  Once this 
        is realized, then everything within the parenthesis following 
        "sin" must be the one and only argument of sin():  the argument 
        of sin() must be +(123,*(45,ln(-(27,6)))).
        
        In a similar manner, +(123,*(45,ln(-(27,6)))) is the two-argument 
        function +(y,x), where "y" is "123" and "x" is *(45,ln(-(27,6))).
        
        A further clarification can be made if "+(y,x)" is thought of as 
        "the sum of y and x" (just as "ln(x)" is "the natural logarithm 
        of x").  This approach allows "sin(+(123,*(45,ln(-(27,6)))))" to 
        be read as "the sine of the sum of 123 and the product of 45 and 
        the natural logarithm of the difference of 27 and 6."  This 
        phrase readily illustrates its clarity when empasized function by 
        function:
        
          sin(+(123,*(45,ln(-(27,6)))))
        
          The sine of                       sin(...)
          the sum of                        sin(+(...))
          123 and the product of            sin(+(123,*(...)))
          45 and the natural logarithm of   sin(+(123,*(45,ln(...))))
          the difference of                 sin(+(123,*(45,ln(-(...)))))
          27 and 6.                         sin(+(123,*(45,ln(-(27,6)))))
        
        Awkward as that may seem, it is much cleaner and more 
        straightforward than "sin[123+45ln(27-6)]," which is "the sine of 
        the quantity 123 plus 45 times the natural logarithm of the 
        quantity 27 minus 6."
        
        What is even more important, it is unambiguous!  If you heard 
        someone say "the quantity 123 plus 45 times the natural logarithm 
        of ...," does the speaker mean "(123+45)ln()" or 
        "(123+(45ln()))".  By defining all operations as functions there 
        is no ambiguity, ever!
        
        In algebra, a complex set of rules has been established as 
        regards order and priority of operations, and if these rules are 
        strictly followed there will be no ambiguity.  Unfortunately, few 
        of the "algebraic" calculators or software packages on the market 
        follow these rules properly, and there is a total lack of 
        consistency in which rules are overlooked or changed.
        
        What is needed with a calculator or computer is a mechanistic and 
        totally unambiguous method of operation.  Treating all arguments 
        as functions provides just such a method and is known as Polish 
        Notation after its creator, the Polish logician Jan Lukasiewicz.  
        The complex function sin(+(123,*(45,ln(-(27,6))))) has one and 
        only one possible meaning, whether written in mathematical 
        symbology or spoken aloud.
        
        Further research by the logicians at Hewlett-Packard provided a 
        slight variation on the theme.  If instead of placing the 
        function ahead of its argument(s), it is placed behind, we get:
        
          sin(+(123,*(45,ln(-(27,6)))))
          ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
          sin(                        ) ÄÄ> (                        )sin
              +(123,                 )  ÄÄ>  (123,                 )+
                    *(45,           )   ÄÄ>       (45,           )*
                         ln(       )    ÄÄ>           (       )ln
                            -(27,6)     ÄÄ>            (27,6)-
                                            ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                                            ((123,(45,((27,6)-)ln)*)+)sin
        
        This method is known as Reverse Polish Notation, or RPN, in honor 
        of the original method.  RPN, like Polish Notation, is 
        unambiguous:  the order and only the order of arguments and 
        operators will determine the result.  Since this is the case, the 
        parentheses and commas are irrelevant, and we may express our 
        function as:
        
          ((123,(45,((27,6)-)ln)*)+)sin  ÄÄ>  123 45 27 6 - ln * + sin
        
        This allows a simple and straightforward solution to calculator 
        or software mathematics:  treat both the functions and their 
        arguments alike as "objects," and process them in a last-in, 
        first-out (LIFO) storage structure, a "stack."
        
        When a function-object is entered onto the stack, it operates 
        upon the argument-object(s) already there to produce a result, 
        which replaces the argument-object(s).  The stack adjusts 
        accordingly.  This can be seen as:

          object    stack level x:                  y:    z:    t:
          ÄÄÄÄÄÄ  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÂÄÄÄÄÄÂÄÄÄÄÄÂÄÄÄÄÄ¿
          123     ³ 123                           ³     ³     ³     ³
          45      ³ 45                            ³ 123 ³     ³     ³
          27      ³ 27                            ³ 45  ³ 123 ³     ³
          6       ³ 6                             ³ 27  ³ 45  ³ 123 ³
          -       ³ -(27,6)                       ³ 45  ³ 123 ³     ³
          ln      ³ ln(-(27,6))                   ³ 45  ³ 123 ³     ³
          *       ³ *(45,ln(-(27,6)))             ³ 123 ³     ³     ³
          +       ³ +(123,*(45,ln(-(27,6))))      ³     ³     ³     ³
          sin     ³ sin(+(123,*(45,ln(-(27,6))))) ³     ³     ³     ³
                  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÁÄÄÄÄÄÁÄÄÄÄÄÁÄÄÄÄÄÙ
        
        The secret behind RPN's sophisticated yet simple power is in the 
        stack.  In computer terms, a stack is an area in which pieces of 
        information may be stored on a last-in, first-out (LIFO) basis.  
        Think of the stack as a stack of plates:  the last plate placed 
        on the stack will be the first plate used.
        
        Most simple RPN calculators use a four-level stack, with the 
        levels labeled "x", "y", "z", and "t".  Some newer, more 
        sophisticated calculators (such as the HP-28S and the HP-48SX) 
        and most software programs use an "infinite" stack:  the number 
        of stack levels is limited only by available memory.  The levels 
        in an infinite stack are usually numerical.
        
        Entering a value will cause a stack lift:  existing values will 
        be pushed up a level.
        
        Entering a function will consume the arguments of the function.  
        If this consumption causes a "hole," the values above the hole 
        will be dropped to fill the hole.
        
        Following through our example on an HP-48SX calculator:
        
                  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
             123  ³ 1:            123 ³
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                  ³ 2:            123 ³
             45   ³ 1:             45 ³
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                  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
                  ³ 3:            123 ³
                  ³ 2:             45 ³
             27   ³ 1:             27 ³
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                  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
                  ³ 4:            123 ³
                  ³ 3:             45 ³
                  ³ 2:             27 ³
             6    ³ 1:              6 ³
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                  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
                  ³ 3:            123 ³
                  ³ 2:             45 ³
             -    ³ 1:             21 ³ -(27,6)
                  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
                  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
                  ³ 3:            123 ³
                  ³ 2:             45 ³
             ln   ³ 1:  3.04452243772 ³ ln(-(27,6))
                  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
                  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
                  ³ 2:            123 ³
             *    ³ 1:  137.003509697 ³ *(45,ln(-(27,6)))
                  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
                  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
             +    ³ 1:  260.003509697 ³ +(123,*(45,ln(-(27,6))))
                  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
                  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
             sin  ³ 1:  .680672740775 ³ sin(+(123,*(45,ln(-(27,6)))))
                  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

