Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which

operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...

''follow'' their

operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above exa ...

s, in contrast to

Polish notation Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation or simply prefix notation, is a mathematical notation in which operators ''precede'' their operands, in contrast ...

(PN), in which operators ''precede'' their operands. It does not need any parentheses as long as each operator has a fixed number of operands. The description "Polish" refers to the

nationality Nationality is a legal identification of a person in international law, establishing the person as a subject, a ''national'', of a sovereign state. It affords the state jurisdiction Jurisdiction (from Latin 'law' + 'declaration') is th ...

logician Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...

Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic His work centred on philosophical logic, mathematical logic and history of logic. ...

, who invented Polish notation in 1924. The first computer to use postfix notation, though it long remained essentially unknown outside of Germany, was

Konrad Zuse Konrad Ernst Otto Zuse (; 22 June 1910 – 18 December 1995) was a German civil engineer, pioneering computer scientist, inventor and businessman. His greatest achievement was the world's first programmable computer; the functional program- ...

's Z3 in 1941 as well as his Z4 in 1945. The reverse Polish scheme was again proposed in 1954 by

Arthur Burks Arthur Walter Burks (October 13, 1915 – May 14, 2008) was an American mathematician who worked in the 1940s as a senior engineer on the project that contributed to the design of the ENIAC, the first general-purpose electronic digital computer. ...

, Don Warren, and Jesse Wright and was independently reinvented by Friedrich L. Bauer and

Edsger W. Dijkstra Edsger Wybe Dijkstra ( ; ; 11 May 1930 – 6 August 2002) was a Dutch computer scientist, programmer, software engineer, systems scientist, and science essayist. He received the 1972 Turing Award for fundamental contributions to developing progra ...

in the early 1960s to reduce

computer memory In computing, memory is a device or system that is used to store information for immediate use in a computer or related computer hardware and digital electronic devices. The term ''memory'' is often synonymous with the term '' primary storage ...

access and use the

stack Stack may refer to: Places * Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group * Blue Stack Mountains, in Co. Donegal, Ireland People * Stack (surname) (including a list of people ...

to evaluate expressions. The

algorithms In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

and notation for this scheme were extended by the Australian philosopher and

computer scientist A computer scientist is a person who is trained in the academic study of computer science. Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus ( ...

Charles L. Hamblin in the mid-1950s. During the 1970s and 1980s, Hewlett-Packard used RPN in all of their desktop and hand-held calculators, and has continued to use it in some models into the 2020s. In

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...

, reverse Polish notation is used in stack-oriented programming languages such as

Forth Forth or FORTH may refer to: Arts and entertainment * ''forth'' magazine, an Internet magazine * ''Forth'' (album), by The Verve, 2008 * ''Forth'', a 2011 album by Proto-Kaw * Radio Forth, a group of independent local radio stations in Scotla ...

, STOIC,

PostScript PostScript (PS) is a page description language in the electronic publishing and desktop publishing realm. It is a dynamically typed, concatenative programming language. It was created at Adobe Systems by John Warnock, Charles Geschke, ...

, RPL and Joy.

Explanation

In reverse Polish notation, the

follow their operands; for instance, to add 3 and 4 together, one would write rather than . If there are multiple operations, operators are given immediately after their final operands (often an operator takes two operands, in which case the operator is written after the second operand); so the expression written in conventional notation would be written in reverse Polish notation: 4 is first subtracted from 3, then 5 is added to it. The concept of a stack, a last-in/first-out construct, is integral to these actions. So in the above example, the 3 is loaded onto the bottom of the stack (the visible level) and a separate special keypress (the "" button on an HP calculator, for example) terminates that entry. Without this action, the 4 would append to the 3, giving 34, which is not desired. When the 4 is entered the 3 is promoted to the second stack level; the 3 is now above the 4, currently visible. The subtraction operator acts immediately on the first two levels of the stack contents, subtracting the lower value from the upper, yielding -1 at level one. This also terminates data entry, so the 5 can be immediately entered. This automatically raises the -1 to the second level. When the user then presses (add), the first two levels are added, and the result, 4, appears in the lower. This automatic promotion (and demotion) of data among levels in the stack as each operation is performed automatically sets up successive operators just as they are needed. In HP calculators, the stack is four levels high. So it is possible to type 3, , type 4, , type 5, , and type 6. The stack now holds all four values in its four levels. One can then hit the button three times, and the sum, 18, will appear in level one. Any new data entry promotes the 18 to level two. This activity is limited only by the "height" of the stack. Careful stack management allows complex parenthesis-filled expressions to be evaluated in a simple linear fashion. It is rarely required that intermediate results be stored and retrieved, as is required usually of algebraic notation systems. This advantage of reverse Polish notation is that it removes the need for parentheses that are required by

infix notation Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—"infixed operators"—such as the plus sign in . Usage Binary relations are ...

, since the stack holds all arguments in a last-in, first out progression. For example, to compute the expression , one would type 3, press , and type 4. Upon pressing (multiply), the intermediate product 12 appears visually at the bottom of the stack. Then one types 5, , and 6. The intermediate result 12 has been promoted to level three, with the 5 at level two and the 6 visible at level one. It is only required to press and then in succession. The intermediate product, 30, appears first in level one, and the final result, 42 appears at level one since the 12 at level two has now been added. Rare occasions of the need of storing intermediate results and how complex an expression can be solved without it can be read in Hans Klavers RPN Tutorial (see Further reading).

Practical implications

In comparison, testing of reverse Polish notation with algebraic notation, reverse Polish has been found to lead to faster calculations, for two reasons. The first reason is that reverse Polish calculators do not need expressions to be parenthesized, so fewer operations need to be entered to perform typical calculations. Additionally, users of reverse Polish calculators made fewer mistakes than for other types of calculators. Later research clarified that the increased speed from reverse Polish notation may be attributed to the smaller number of keystrokes needed to enter this notation, rather than to a smaller cognitive load on its users. However, anecdotal evidence suggests that reverse Polish notation is more difficult for users to learn than algebraic notation.

Converting from infix notation

invented the shunting-yard algorithm to convert infix expressions to postfix expressions (reverse Polish notation), so named because its operation resembles that of a railroad shunting yard. There are other ways of producing postfix expressions from infix expressions. Most operator-precedence parsers can be modified to produce postfix expressions; in particular, once an

abstract syntax tree In computer science, an abstract syntax tree (AST), or just syntax tree, is a tree representation of the abstract syntactic structure of text (often source code) written in a formal language. Each node of the tree denotes a construct occurring ...

has been constructed, the corresponding postfix expression is given by a simple post-order traversal of that tree.

Implementations

History

The first computer implementing a form of reverse Polish notation (but without the name), was

's Z3, which he started to construct in 1938 and demonstrated publicly on 12 May 1941. In dialog mode, it allowed operators to enter two operands followed by the desired operation. It was destroyed on 21 December 1943 in a bombing raid. With Zuse's help a first replica was built in 1961. The 1945 Z4 also added a

. Other early computers to implement architectures enabling reverse Polish notation were the English Electric Company's

KDF9 KDF9 was an early British 48-bit computer designed and built by English Electric (which in 1968 was merged into International Computers Limited (ICL)). The first machine came into service in 1964 and the last of 29 machines was decommissioned ...

machine, which was announced in 1960 and commercially available in 1963, and the Burroughs B5000, announced in 1961 and also delivered in 1963: Presumably, the KDF9 designers drew ideas from Hamblin's GEORGE (General Order Generator), an autocode programming system written for a

DEUCE Deuce, Deuces, or The Deuce may refer to: Arts and entertainment Fictional characters * Deuce, in the '' Danger Girl'' comic book series * Deuce, a character in ''Shake It Up'' * Deuce, in the '' Wild Cards'' science fiction universe * Deuce Bi ...

computer installed at the

University of Sydney The University of Sydney (USYD), also known as Sydney University, or informally Sydney Uni, is a public university, public research university located in Sydney, Australia. Founded in 1850, it is the oldest university in Australia and is one o ...

, Australia, in 1957. One of the designers of the B5000, Robert S. Barton, later wrote that he developed reverse Polish notation independently of Hamblin sometime in 1958 after reading a 1954 textbook on symbolic logic by Irving Copi, where he found a reference to Polish notation, which made him read the works of Jan Łukasiewicz as well, and before he was aware of Hamblin's work. Friden introduced reverse Polish notation to the desktop calculator market with the EC-130, designed by Robert "Bob" Appleby Ragen, supporting a four-level stack in June 1963. The successor EC-132 added a square root function in April 1965. Around 1966, the Monroe Epic calculator supported an unnamed input scheme resembling RPN as well.

Hewlett-Packard

Hewlett-Packard engineers designed the 9100A Desktop Calculator in 1968 with reverse Polish notation with only three stack levels with working registers ''X'' ("keyboard"), ''Y'' ("accumulate") and visible storage register ''Z'' ("temporary"), a reverse Polish notation variant later referred to as ''three-level RPN''. This calculator popularized reverse Polish notation among the scientific and engineering communities. The

HP-35 The HP-35 was Hewlett-Packard's first pocket calculator and the world's first ''scientific'' pocket calculator: a calculator with trigonometric and exponential functions. It was introduced in 1972. History In about 1970 HP co-founder Bill He ...

, the world's first handheld scientific

calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-size ...

, introduced the classical ''four-level RPN'' with its specific ruleset of the so-called ''operational (memory) stack'' (later also called ''automatic memory stack'') in 1972. In this scheme, the key duplicates values into Y under certain conditions, and the top register gets duplicated on drops in order to ease some calculations and to save keystrokes. HP used reverse Polish notation on every handheld calculator it sold, whether scientific, financial, or programmable, until it introduced the HP-10

adding machine An adding machine is a class of mechanical calculator, usually specialized for bookkeeping calculations. In the United States, the earliest adding machines were usually built to read in dollars and cents. Adding machines were ubiquitous of ...

calculator in 1977. By this time, HP was the leading manufacturer of calculators for professionals, including engineers and accountants. Later calculators with LCD displays in the early 1980s, such as the

HP-10C The Hewlett-Packard Voyager series of calculators were introduced by Hewlett-Packard in 1981. All members of this series are programmable, use Reverse Polish Notation, and feature continuous memory. Nearly identical in appearance, each mode ...

HP-11C The Hewlett-Packard Voyager series of calculators were introduced by Hewlett-Packard in 1981. All members of this series are programmable, use Reverse Polish Notation, and feature continuous memory. Nearly identical in appearance, each mode ...

HP-15C The HP-15C is a high-end scientific programmable calculator of Hewlett-Packard's Voyager series produced between 1982 and 1989. Models HP-15C The HP-15C is a high-end scientific pocket calculator with a root-solver and numerical integration. ...

, HP-16C, and the financial HP-12C calculator also used reverse Polish notation. In 1988, Hewlett-Packard introduced a business calculator, the HP-19B, without reverse Polish notation, but its 1990 successor, the HP-19BII, gave users the option of using algebraic or reverse Polish notation again. Around 1987, HP introduced RPL, an object-oriented successor to reverse Polish notation. It deviates from classical reverse Polish notation by using a stack only limited by the amount of available memory (instead of three or four fixed levels) and which could hold all kinds of data objects (including symbols, strings, lists, matrices, graphics, programs, etc.) instead of just numbers. It also changed the behaviour of the stack to no longer duplicate the top register on drops (since in an unlimited stack there is no longer a top register) and the behaviour of the key so that it no longer duplicated values into Y, which had shown to sometimes cause confusion among users not familiar with the specific properties of the ''automatic memory stack''. From 1990 to 2003, HP manufactured the HP-48 series of graphing RPL calculators, and in 2006 introduced the

HP 50g The HP 49/50 series are Hewlett-Packard (HP) manufactured graphing calculators. They are the successors of the popular HP 48 series. There are five calculators in the 49/50 series of HP graphing calculators. These calculators have both ...

. As of 2011, Hewlett-Packard was offering the calculator models 12C, 12C Platinum, 17bII+, 20b, 30b, 33s, 35s, 48gII (RPL) and 50g (RPL) which support reverse Polish notation. While calculators emulating classical models continue to support classical reverse Polish notation, new reverse Polish notation models feature a variant of reverse Polish notation, where the key behaves as in RPL. This latter variant is sometimes known as ''entry RPN''. In 2013, the HP Prime introduced a ''128-level'' form of entry RPN called ''advanced RPN''. By late 2017, only the 12C, 12C Platinum, 17bii+, 35s and Prime remain active HP models supporting reverse Polish notation.

WP 31S and WP 34S

The community-developed calculators

WP 31S The HP 30b (NW238AA, variously codenamed "Big Euro", "Mid Euro" and "Fox") is a programmable financial calculator from HP which was released on 7 January 2010. The HP 30b is an advanced version of the HP's prior model HP 20b. F ...

and

WP 34S The HP 30b (NW238AA, variously codenamed "Big Euro", "Mid Euro" and "Fox") is a programmable financial calculator from HP which was released on 7 January 2010. The HP 30b is an advanced version of the HP's prior model HP 20b. F ...

, which are based on the HP 20b/HP 30b hardware platform, support Hewlett-Packard-style classical reverse Polish notation with either a four- or an eight-level stack. A seven-level stack had been implemented in the MITS 7400C scientific desktop calculator in 1972 and an eight-level stack was already suggested by John A. Ball in 1978.

Sinclair Radionics

In Britain,

Clive Sinclair Sir Clive Marles Sinclair (30 July 1940 – 16 September 2021) was an English entrepreneur and inventor, best known for being a pioneer in the computing industry, and also as the founder of several companies that developed consumer electronic ...

Sinclair Scientific The Sinclair Scientific calculator was a 12-function, pocket-sized scientific calculator introduced in 1974, dramatically undercutting in price other calculators available at the time. The Sinclair Scientific Programmable, released a year later, ...

and Scientific Programmable models used reverse Polish notation.

Commodore

In 1974, Commodore produced the Minuteman *6 (MM6) without key and the Minuteman *6X (MM6X) with key, both implementing a form of ''two-level RPN''. The SR4921 RPN came with a variant of ''four-level RPN'' with stack levels named X, Y, Z, and W (rather than T) and an key (for "entry"). In contrast to Hewlett-Packard's reverse Polish notation implementation, W filled with 0 instead of its contents being duplicated on stack drops.

Prinztronic

Prinz and Prinztronic were own-brand trade names of the British Dixons photographic and electronic goods stores retail chain, later rebranded as Currys Digital stores, and became part of DSG International. A variety of calculator models was sold in the 1970s under the Prinztronic brand, all made for them by other companies. Among these was the PROGRAM Programmable Scientific Calculator which featured reverse Polish notation.

Heathkit

The Aircraft Navigation Computer Heathkit OC-1401/ OCW-1401 used ''five-level RPN'' in 1978.

Soviet Union

Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen national ...

programmable calculators ( MK-52,

MK-61 {{No footnotes, date=September 2008 The Elektronika MK-61 is a third-generation non-BASIC, RPN programmable calculator which was manufactured in the Soviet Union during the years 1983 to 1994. Its original selling price was 85 rubles. The M ...

, B3-34 and earlier

B3-21 Elektronika B3-21 (Cyrillic: Электроника Б3-21) was the first Soviet programmable calculator. It was released in 1977 and was sold initially for 350 rubles (190 in 1980-81, and just 80 rubles at late 1981). For comparison, 120 ruble ...

models) used reverse Polish notation for both automatic mode and programming. Modern Russian calculators MK-161 and MK-152, designed and manufactured in

Novosibirsk Novosibirsk (, also ; rus, Новосиби́рск, p=nəvəsʲɪˈbʲirsk, a=ru-Новосибирск.ogg) is the largest city and administrative centre of Novosibirsk Oblast and Siberian Federal District in Russia. As of the 2021 Census, ...

since 2007 and offered by Semico, are backwards compatible with them. Their extended architecture is also based on reverse Polish notation.

Other

Existing implementations using reverse Polish notation include: *

Stack-oriented programming language Stack-oriented programming, is a programming paradigm which relies on a stack machine model for passing parameters. Stack-oriented languages operate on one or more stacks, each of which may serve a different purpose. Programming constructs in ...

s such as: **

** STOIC **

Factor Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, ...

page description language **

BibTeX BibTeX is reference management software for formatting lists of references. The BibTeX tool is typically used together with the LaTeX document preparation system. Within the typesetting system, its name is styled as . The name is a portmanteau ...

** Befunge ** Joy **

IPTSCRAE Iptscrae is a stack-oriented scripting language used to give additional functionality to The Palace software and servers. Its name comes from the pronunciation of "script" in Pig Latin The language was created by Jim Bumgardner, who in turn was ...

Lotus 1-2-3 Lotus 1-2-3 is a discontinued spreadsheet program from Lotus Software (later part of IBM). It was the first killer application of the IBM PC, was hugely popular in the 1980s, and significantly contributed to the success of IBM PC-compatible ...

and Lotus Symphony formulas ** RPL (aka Reverse Polish Language), a programming language for the

Commodore PET The Commodore PET is a line of personal computers produced starting in 1977 by Commodore International. A single all-in-one case combines a MOS Technology 6502 microprocessor, Commodore BASIC in read-only memory, keyboard, monochrome monitor, ...

around 1979/1981 ** RPL (aka Reverse Polish Lisp), a programming language for Hewlett-Packard calculators between 1984 and 2015 ** RPNL (Reverse Polish Notation Language) * Hardware calculators: ** Some Hewlett-Packard science/engineering and business/finance calculators ** Semico calculators ** SwissMicros calculators ** Some APF calculators as well can use RPN * Software calculators: ** Mac OS X Calculator ** Several Apple iPhone applications e.g. "reverse polish notation calculator" ** Several

Android Android may refer to: Science and technology * Android (robot), a humanoid robot or synthetic organism designed to imitate a human * Android (operating system), Google's mobile operating system ** Bugdroid, a Google mascot sometimes referred to ...

applications e.g. "RealCalc" ** Several Windows 10 Mobile applications e.g. "RPN9" **

Unix system Unix (; trademarked as UNIX) is a family of multitasking, multiuser computer operating systems that derive from the original AT&T Unix, whose development started in 1969 at the Bell Labs research center by Ken Thompson, Dennis Ritchie, and ot ...

calculator program dc **

Emacs Emacs , originally named EMACS (an acronym for "Editor MACroS"), is a family of text editors that are characterized by their extensibility. The manual for the most widely used variant, GNU Emacs, describes it as "the extensible, customizable, ...

lisp library package calc ** Xorg calculator ( xcalc) ** ARPCalc, a powerful scientific/engineering RPN calculator for Windows, Linux and Android that also has a web-browser based version. ** scientific/engineering calculator using the GIMP Toolkit (

GTK+ GTK (formerly GIMP ToolKit and GTK+) is a free and open-source cross-platform widget toolkit for creating graphical user interfaces (GUIs). It is licensed under the terms of the GNU Lesser General Public License, allowing both free and prop ...

) ** F-Correlatives in MultiValue dictionary items **

RRDtool RRDtool (''round-robin database tool'') aims to handle time series data such as network bandwidth, temperatures or CPU load. The data is stored in a circular buffer based database, thus the system storage footprint remains constant over time. ...

, a widely used tabulating and graphing software ** , a program for algebraic operations on

NetCDF NetCDF (Network Common Data Form) is a set of software libraries and self-describing, machine-independent data formats that support the creation, access, and sharing of array-oriented scientific data. The project homepage is hosted by the Unidata ...

grids, part of Generic Mapping Tools (GMT) suite ** , a GTK desktop calculator ** Mouseless Stack-Calculator scientific/engineering calculator including complex numbers. ** , a simple reverse polish notation calculator written in Python for Linux and MS Windows and published under the GNU GPLv2 license. ** orpie, RPN calculator for the terminal for real or complex numbers or matrices. * Class libraries ** TRURL, a class library for the construction of RPN calculators in

Object Pascal Object Pascal is an extension to the programming language Pascal that provides object-oriented programming (OOP) features such as classes and methods. The language was originally developed by Apple Computer as ''Clascal'' for the Lisa Work ...

References

External links

Rosettacode.org
providing many implementations in several programming languages. * http://rpn.codeplex.com/ Implementation of RPN with custom functions support and flexible list of operators. * https://xrjunque.nom.es/ConvertAlg2RPN_RPL.aspx Free online Algebraic expression to RPN Converter {{DEFAULTSORT:Reverse Polish Notation Calculators Mathematical notation Science and technology in Poland Operators (programming)