W. Marshall Leach, Jr.
Professor of Electrical and Computer
Engineering
Georgia Institute of Technology
Atlanta, Georgia
30332-0250
Copyright 1999
The operating system for Hewlett
Packard scientific calculators is called "reverse Polish notation," or
simply rpn. In talking to students who have HP calculators, I have not
found one who knows the rpn algorithm. This is unfortunate, because not
knowing it is a major cause for making mistakes with HP calculators. This
page explains the algorithm and gives examples. When the algorithm is
applied correctly, one can use the calculator to rapidly evaluate long
expressions without stopping to think how the terms are grouped in the
expression, even with the calculator display covered.
![[GIF Image of HP35
Calculator]](35last3.gif)
Aside from computer programmers,
not many engineers had heard of reverse Polish notation until Hewlett
Packard introduced the HP35 calculator in 1972. It was a non-programmable,
four-function scientific calculator with only one memory register. It had
a light emitting diode display that drew so much current from the
batteries that an ac charger came with the calculator. To keep the
batteries charged, most people left the calculator connected to the
charger when possible. The calculator sold for $395. In 1973, the price
was reduced to $295. It was discontinued in 1975.
The major difference between the
HP35 and calculators made by other companies was that it used reverse
Polish notation. This was the major feature that HP promoted for the
calculator. The HP35 manual had an appendix devoted to explaining the rpn
algorithm. As later calculator models were introduced, the emphasis on rpn
gradually diminished. The manuals that come with HP calculators today
barely even mention it.
Polish notation was described in
the 1920s by Polish mathematician Jan Lukasiewicz as a logical system for
the specification of mathematical equations without parentheses. There are
two versions, prefix notation and postfix notation. In prefix notation,
the operators are placed before the operand. In postfix notation, this
order is reversed. The following example illustrates the two. The
asterisk is used for the multiplication sign.
Equation with parenthesis (1 + 2) * 3 Prefix notation * 3 + 1 2 or * + 1 2 3 Postfix notation 1 2 + 3 * or 3 1 2 + *
Postfix notation has since become known as reverse Polish notation. In the HP implementation of rpn, the ENTER key is pressed between any two numbers that are not separated by an operation.
The Algorithm
The basic reverse Polish
calculator algorithm is to key in a number. If you can perform a
calculator operation do it. If not, press ENTER. Then repeat until the
complete expression is evaluated. The following flow graph summarizes the
algorithm.
![[GIF Image of Flow
Graph]](flwgrph2.gif)
Examples
If you own one of the HP
calculators that allows you to set the display in either the symbolic mode
or the numeric mode, set the mode to numeric. This is done by entering the
mode/flag menu and flagging the line that says "constants to symbols" so
that it says "constants to numbers." Otherwise, you will not obtain
numerical answers when you use symbols such as "pi" in equations.
There are two evaluation methods
given for each example. The first evaluates the expression in the order
that the numbers appear. The second evaluates an equivalent expression in
which the order of some terms is reversed so as to minimize the number of
times the ENTER key is pressed. This minimizes the chances of overflowing
the calculator stack for the models that have only four stack registers.
Letters for calculator functions are upper case, * is used for the
multiplication key, / is used for the divide key, e^x is used for the "e
to the x" key, x^2 is used for the "x squared" key, and ROOT(x) is used
for the "square root of x" key. The columns to the right show the
contents of the X, Y, Z, and T registers after each operation. Note how
the numbers move to the right when ENTER is pressed and to the left after
an operation is performed.
Example 1:
Algebraic Expression (4 + 2 * 5) / (1 + 3 * 2) Reverse Polish Expression 4 2 5 * + 1 3 2 * + / HP Calculator Implementation X Y Z T 4 4 . . . ENTER 4 4 . . 2 2 4 . . ENTER 2 2 4 . 5 5 2 4 . * 10 4 . . + 14 . . . 1 1 14 . . ENTER 1 1 14 . 3 3 1 14 . ENTER 3 3 1 14 2 2 3 1 14 * 6 1 14 . + 7 14 . . / 2 . . . Reordered Algebraic Expression (2 * 5 + 4 )/(3 * 2 + 1) Reverse Polish Expression 2 5 * 4 + 3 2 * 1 + / HP Calculator Implementation X Y Z T 2 2 . . . ENTER 2 2 . . 5 5 2 . . * 10 . . . 4 4 10 . . + 14 . . . 3 3 14 . . ENTER 3 3 14 . 2 2 3 14 . * 6 14 . . 1 1 6 14 . + 7 14 . . / 2 . . . The answer is 2 for both methods. Note that the numbers never reach the T register in Method 2.
Each time ENTER is pressed, the
numbers move up in the stack. Each time an operation is performed, the
numbers move down in the stack. Some HP calculator models have only 4
registers and it is possible to overflow the stack if ENTER is pressed too
many times without operations in between. For this reason, Method 2 is
preferred where products are evaluated before sums. Note that Method 2
contains 2 less ENTERs and the numbers never reach the T register. It
should be obvious that the number of ENTERs can be minimized if the
expression is rearranged so that multiplications and divisions are
performed before additions and subtractions. The rearrangement can easily
be done mentally without rewriting the expression.
Example 2:
Algebraic Expression [5 + 8 * sin(2 * 15)] / [2 + tan(45)] Reverse Polish Expression 5 8 2 15 * sin * + 2 45 tan + / HP Calculator Implementation X Y Z T 5 5 . . . ENTER 5 5 . . 8 8 5 . . ENTER 8 8 5 . 2 2 8 5 . ENTER 2 2 8 5 15 15 2 8 5 * 30 8 5 . SIN 0.5 8 5 . * 4 5 . . + 9 . . . 2 2 9 . . ENTER 2 2 9 . 45 45 2 9 . TAN 1 2 9 . + 3 9 . . / 3 . . . Reordered Algebraic Expression [sin(2 * 15) * 8 + 5] / [tan(45) + 2] Reverse Polish Expression 2 15 * sin 8 * 5 + 45 tan 2 + / HP Calculator Impelementation X Y Z T 2 2 . . . ENTER 2 2 . . 15 15 2 . . * 30 . . . SIN 0.5 . . . 8 8 0.5 . . * 4 . . . 5 5 4 . . + 9 . . . 45 45 9 . . TAN 1 9 . . 2 2 1 9 . + 3 9 . . / 3 . . . The answer is 3 for both methods.
Method 2 requires 3 less
ENTERs.
Example 3:
Algebraic Expression [3 * ln(e^2) + 8 * cos(60)] / [3 * 4^0.5 - 1]
Reverse Polish Expression 3 e ^2 ln * 8 60 cos * + 3 4 ^0.5 * 1 - /
HP Calculator Implementation
X Y Z T
3 3 . . .
ENTER 3 3 . .
1 1 3 . .
e^x 2.718 3 . .
x^2 7.389 3 . .
LN 2 3 . .
* 6 . . .
8 8 6 . .
ENTER 8 8 6 .
60 60 8 6 .
COS 0.5 8 6 .
* 4 6 . .
+ 10 . . .
3 3 10 . .
ENTER 3 3 10 .
4 4 3 10 .
ROOT(x) 2 10 . .
* 6 10 . .
1 1 6 10 .
- 5 10 . .
/ 2 . . .
Reordered Algebraic [ln(e^2) * 3 + cos(60) * 8] / [4^0.5 * 3 - 1]
Reverse Polish e ^2 ln 3 * 60 cos 8 * + 4 ^0.5 3 * 1 - /
HP Calculator Implementation
X Y Z T
1 1 . . .
e^x 2.718 . . .
x^2 7.389 . . .
LN 2 . . .
3 3 2 . .
* 6 . . .
60 60 6 . .
COS 0.5 6 . .
8 8 0.5 6 .
* 4 6 . .
+ 10 . . .
4 4 10 . .
ROOT(x) 2 10 . .
3 3 2 10 .
* 6 10 . .
1 1 6 10 .
- 5 10 . .
/ 2 . . .
The answer is 2 for both methods.
With a little practice, it is
possible to rapidly evaluate long expressions containing many nested
parentheses without ever stopping to think about how terms are grouped.
Always start in the innermost parentheses and work out, evaluating
products and quotients before sums and differences. Once you get the hang
of it, you can even do complex calculations with the display window
covered and get the right answers. Try that with a calculator that uses
algebraic notation, and chances are you will get a different answer every
time if you don't get lost first. That tends to happen to me with
algebraic notation calculators even with the display window uncovered. If
you own a HP calculator and think in terms of algebraic notation, go buy a
TI calculator and forget this page.
Home.
This page is
not a publication of the Georgia Institute of Technology and the Georgia
Institute of Technology has not edited or examined the content. The author
of this page is solely responsible for the content.
