Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''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. Unknown operands in equalities of expressions can be found by equation solving. Example The following arithmetic expres ...

s, in contrast to

prefix A prefix is an affix which is placed before the stem of a word. Particularly in the study of languages, a prefix is also called a preformative, because it alters the form of the word to which it is affixed. Prefixes, like other affixes, can b ...

Polish notation Polish notation (PN), also known as normal Polish notation (NPN), Łukasiewicz notation, Warsaw notation, Polish prefix notation, Eastern Notation or simply prefix notation, is a mathematical notation in which Operation (mathematics), operator ...

(PN), in which operators ''precede'' their operands. The notation does not need any parentheses for as long as each operator has a fixed number of operands. The term ''postfix notation'' describes the general scheme in mathematics and computer sciences, whereas the term ''reverse Polish notation'' typically refers specifically to the method used to enter calculations into hardware or software calculators, which often have additional

side effect In medicine, a side effect is an effect of the use of a medicinal drug or other treatment, usually adverse but sometimes beneficial, that is unintended. Herbal and traditional medicines also have side effects. A drug or procedure usually use ...

s and implications depending on the actual implementation involving a

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 ...

. The description "Polish" refers to the

nationality Nationality is the legal status of belonging to a particular nation, defined as a group of people organized in one country, under one legal jurisdiction, or as a group of people who are united on the basis of culture. In international law, n ...

of logician

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 logi ...

, 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, List of pioneers in computer science, pioneering computer scientist, inventor and businessman. His greatest achievement was the world's first programm ...

's Z3 in 1941 as well as his Z4 in 1945. The reverse Polish scheme was again proposed in 1954 by Arthur Burks, Don Warren, and Jesse Wright and was independently reinvented by

Friedrich L. Bauer Friedrich Ludwig "Fritz" Bauer (10 June 1924 – 26 March 2015) was a German pioneer of computer science and professor at the Technical University of Munich. Life Bauer earned his Abitur in 1942 and served in the Wehrmacht during World War ...

and

Edsger W. Dijkstra Edsger Wybe Dijkstra ( ; ; 11 May 1930 – 6 August 2002) was a Dutch computer scientist, programmer, software engineer, mathematician, and science essayist. Born in Rotterdam in the Netherlands, Dijkstra studied mathematics and physics and the ...

in the early 1960s to reduce

computer memory Computer memory stores information, such as data and programs, for immediate use in the computer. The term ''memory'' is often synonymous with the terms ''RAM,'' ''main memory,'' or ''primary storage.'' Archaic synonyms for main memory include ...

access and use the

to evaluate expressions. The

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

s and notation for this scheme were extended by the philosopher and computer scientist Charles L. Hamblin in the mid-1950s. During the 1970s and 1980s,

Hewlett-Packard The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company. It was founded by Bill Hewlett and David Packard in 1939 in a one-car garage in Palo Alto, California ...

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, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...

, reverse Polish notation is used in

stack-oriented programming languages Stack-oriented programming is a programming paradigm that relies on one or more stack machine, stacks to manipulate data and/or pass parameters. Programming constructs in other programming languages need to be modified for use in a stack-oriente ...

such as Forth, dc, Factor, STOIC,

PostScript PostScript (PS) is a page description language and dynamically typed, stack-based programming language. It is most commonly used in the electronic publishing and desktop publishing realm, but as a Turing complete programming language, it c ...

, RPL, and Joy.

Explanation

In reverse Polish notation, the operators follow their

s. For example, to add 3 and 4 together, the expression is rather than . The conventional notation expression becomes in reverse Polish notation: 4 is first subtracted from 3, then 5 is added to it. The concept of a ''

'', a last-in/first-out construct, is integral to the left-to-right evaluation of RPN. In the example , first the 3 is put onto the stack, then the 4; the 4 is now on top and the 3 below it. The subtraction operator removes the top two items from the stack, performs , and puts the result of −1 onto the stack. The common terminology is that added items are ''pushed'' on the stack and removed items are ''popped''. The advantage of reverse Polish notation is that it removes the need for order of operations and 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 ...

and can be evaluated linearly, left-to-right. For example, the infix expression becomes in reverse Polish notation.

Practical implications

Reverse Polish notation has been compared to how one had to work through problems with a

slide rule A slide rule is a hand-operated mechanical calculator consisting of slidable rulers for conducting mathematical operations such as multiplication, division, exponents, roots, logarithms, and trigonometry. It is one of the simplest analog ...

. 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 who previously learned 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 An abstract syntax tree (AST) is a data structure used in computer science to represent the structure of a program or code snippet. It is a tree representation of the abstract syntactic structure of text (often source code) written in a formal ...

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

Implementations

Hardware calculators

Early history

The first computer implementing a form of reverse Polish notation (but without the name and also without a

), 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 2-level

. Other early computers to implement architectures enabling reverse Polish notation were the English Electric Company's KDF9 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 computer installed at the

University of Sydney The University of Sydney (USYD) is a public university, public research university in Sydney, Australia. Founded in 1850, it is the oldest university in both Australia and Oceania. One of Australia's six sandstone universities, it was one of the ...

, 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

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-si ...

, 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 (''automatic stack lift'' with ''temporary stack lift disable''), and the top register ''T'' ("top") gets duplicated on drops (''top copy on pop'' aka ''top stack level repetition'') 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. Consequently, the earliest adding machines were often designed to read in particular currencies. Adding machines were ubiquitous office ...

calculator in 1977. By this time, HP was the leading manufacturer of calculators for professionals, including engineers and accountants. Later calculators with LCDs in the early 1980s, such as the HP-10C, HP-11C, HP-15C, HP-16C, and the financial

HP-12C The HP-12C is a financial calculator made by Hewlett-Packard (HP) and its successor HP Inc. as part of the HP Voyager series, introduced in 1981. It is HP's longest and best-selling product and is considered the '' de facto standard'' among fin ...

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. In 1986, HP introduced RPL, an object-oriented successor to reverse Polish notation. It deviates from classical reverse Polish notation by using a dynamic 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. The system would display an error message when running out of memory instead of just dropping values off the stack on overflow as with fixed-sized stacks. 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, followed by the HP-49 series between 1999 and 2008. The last RPL calculator was named HP 50g, introduced in 2006 and discontinued in 2015. However, there are several community efforts like newRPL or DB48X to recreate RPL on modern calculators. 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 continued 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''. In contrast to RPL with its dynamic stack, it just drops values off the stack on overflow like other fixed-sized stacks do. However, like RPL, it does not emulate the behaviour of a classical operational RPN stack to duplicate the top register on drops. In late 2017, the list of active models supporting reverse Polish notation included only the 12C, 12C Platinum, 17bii+, 35s, and Prime. By July 2023, only the 12C, 12C Platinum, the freshly released HP 15C Collector's Edition, and the Prime remain active models supporting RPN.

Sinclair Radionics

In Britain, Clive Sinclair's Sinclair Scientific (1974) and Scientific Programmable (1975) models used reverse Polish notation.

Commodore

In 1974, Commodore produced the Minuteman *6 (MM6) without an key and the Minuteman *6X (MM6X) with an 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 / Semico

Soviet The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...

programmable calculators ( MK-52, MK-61, B3-34 and earlier B3-21 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 is the largest city and administrative centre of Novosibirsk Oblast and the Siberian Federal District in Russia. As of the 2021 Russian census, 2021 census, it had a population of 1,633,595, making it the most populous city in Siber ...

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

Others

* A seven-level stack had been implemented in the MITS 7400C scientific desktop calculator in 1972 *

National Semiconductor National Semiconductor Corporation was an United States of America, American Semiconductor manufacturing, semiconductor manufacturer, which specialized in analogue electronics, analog devices and subsystems, formerly headquartered in Santa Clara, ...

4615 and 4640 * Novus 650 Mathbox, 3500 Sliderule, 4510 Mathematician, 4515 Mathematician PRO/RG, 4520 Scientist and 4525 Scientist PR * Some APF calculators like the Mark 55 (1976) * SwissMicros (originally firming as RPN-Calc) calculators including the DM-10CC (2012), DM-11CC (2012), DM-12CC (2012), DM-15CC (2012), DM-16CC (2012), DM10 (2013), DM11 (2013), DM12 (2013), DM15 (2013), DM16 (2013), DM10L Collector's Edition (2020), DM11L (2016), DM12L (2016), DM15L (2015), DM16L (2015), DM41 (2015), DM41L (2015), DM41X (2020), DM42 (2017) and DM32 (2023).

Community-developed hardware-based calculators

An eight-level stack was suggested by John A. Ball in 1978. The community-developed calculators WP 34S (2011), WP 31S (2014) and WP 34C (2015), which are based on the HP 20b/ HP 30b hardware platform, support classical Hewlett-Packard-style reverse Polish notation supporting automatic stack lift behaviour of the key and top register copies on pops, but switchable between a four- and an eight-level operational stack. In addition to the optional support for an eight-level stack, the newer SwissMicros DM42-based WP 43S as well as the WP 43C (2019) / C43 (2022) / C47 (2023) derivatives support data types for stack objects (real numbers, infinite integers, finite integers, complex numbers, strings, matrices, dates and times). The latter three variants can also be switched between ''classical'' and ''entry RPN'' behaviour of the key, a feature often requested by the community. They also support a rarely seen significant figures mode, which had already been available as a compile-time option for the WP 34S and WP 31S. Since 2021, the HP-42S simulator Free42 version 3 can be enabled to support a dynamic RPN stack only limited by the amount of available memory instead of the classical 4-level stack. This feature was incorporated as a selectable function into the DM42 since firmware DMCP-3.21 / DM42-3.18.

Software calculators

Software calculators: *

Atari Calculator Atari Calculator (or Calculator) is a proprietary software program developed by Atari, Inc. for Atari 8-bit computers and published in 1979. It incorporates the functionality of a scientific calculator into a software calculator. It was written ...

* Mac OS X Calculator * Several Apple

iPhone The iPhone is a line of smartphones developed and marketed by Apple that run iOS, the company's own mobile operating system. The first-generation iPhone was announced by then–Apple CEO and co-founder Steve Jobs on January 9, 2007, at ...

applications e.g. "reverse polish notation calculator" * Several Android applications e.g. "RealCalc" * Several

Windows 10 Mobile Windows 10 Mobile is the fourth and final generation of Microsoft's Windows Phone mobile operating system, succeeding Windows Phone 8.1. First released in 2015, it was marketed by Microsoft as being an Windows 10 editions, edition of its Per ...

applications e.g. "RPN9" * Unix system 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, s ...

lisp library package calc * Xorg calculator (

xcalc The X.Org Foundation is a non-profit corporation chartered to research, develop, support, organize, administrate, standardize, promote, and defend a free and open accelerated graphics stack. This includes, but is not limited to, the following p ...

) * 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 software 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 proprietary s ...

) * F-Correlatives in

MultiValue A MultiValue database is a type of NoSQL and multidimensional database. It is typically considered synonymous with PICK, a database originally developed as the Pick operating system. MultiValue databases include commercial products from Rocket ...

dictionary items *

RRDtool RRDtool (''round-robin database tool'') is a data logging and graphing system for time series data. It aims to handle time series data such as computer network, network bandwidth (computing), bandwidth, temperatures or Load (computing), CPU loa ...

, 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 Unidat ...

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 Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (prog ...

for Linux and MS Windows and published under the

GNU GPLv2 The GNU General Public Licenses (GNU GPL or simply GPL) are a series of widely used free software licenses, or ''copyleft'' licenses, that guarantee end users the freedom to run, study, share, or modify the software. The GPL was the first ...

license * orpie, RPN calculator for the terminal for real or complex numbers or matrices *

Qalculate! Qalculate! is an arbitrary precision cross-platform software calculator. It supports complex mathematical operations and concepts such as derivation, integration, data plotting, and unit conversion. It is a free and open-source software relea ...

, a powerful and versatile cross-platform desktop calculator * WRPN Calculator

Programming languages

Existing implementations using reverse Polish notation include: * Stack-oriented programming languages such as: ** Forth ** dc ** STOIC ** Factor **

page description language **

BibTeX BibTeX is both a bibliographic flat-file database file format and a software program for processing these files to produce lists of references (citations). The BibTeX file format is a widely used standard with broad support by reference manage ...

style files **

Befunge Befunge is a two-dimensional stack-based, reflective, esoteric programming language. It differs from conventional languages in that programs are arranged on a two-dimensional grid. "Arrow" instructions direct the control flow to the left, ri ...

** 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-compatibles ...

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

calculators between 1986 and 2015 ** RPNL (Reverse Polish Notation Language) * 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 (programming language), Pascal that provides object-oriented programming (OOP) features such as Class (computer programming), classes and Method (computer programming), methods. T ...

Notes

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)