In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as ana ...
, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given
mathematical expression
In mathematics, an expression or mathematical expression is a finite combination of Glossary of mathematical symbols, symbols that is well-formed formula, well-formed according to rules that depend on the context. Mathematical symbols can desi ...
.
For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern
algebraic notation.
Thus, the expression is interpreted to have the value , and not . When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base.
Thus and .
These conventions exist to eliminate notational
ambiguity
Ambiguity is the type of meaning in which a phrase, statement or resolution is not explicitly defined, making several interpretations plausible. A common aspect of ambiguity is uncertainty. It is thus an attribute of any idea or statement ...
, while allowing notation to be as brief as possible. Where it is desired to override the precedence conventions, or even simply to emphasize them,
parentheses
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
( ) can be used. For example, forces addition to precede multiplication, while forces addition to precede
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
. If multiple pairs of parentheses are required in a mathematical expression (such as in the case of nested parentheses), the parentheses may be replaced by
bracket
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
s or
braces to avoid confusion, as in .
Definition
The order of operations, which is used throughout mathematics, science, technology and many computer
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
s, is expressed here:
#
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
and
root extraction
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'':
:r^n = x,
where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A roo ...
#
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
and
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
#
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
and
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
This means that if, in a mathematical expression, a subexpression appears between two
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 ...
, the operator that is higher in the above list should be applied first.
The
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
and
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations must obey the standard order of operations.
In some contexts, it is helpful to replace a division with multiplication by the reciprocal (multiplicative inverse) and a subtraction by addition of the opposite (additive inverse). For example, in
computer algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
, this allows one to handle fewer
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s, and makes it easier to use
commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
and
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
when simplifying large expressions (for more, see ). Thus ; in other words, the quotient of 3 and 4 equals the product of 3 and . Also ; in other words the difference of 3 and 4 equals the sum of 3 and −4. Thus, can be thought of as the sum of , and the three
summand
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of t ...
s may be added in any order, in all cases giving 5 as the result.
The root symbol √ is traditionally prolongated by a bar (called
vinculum) over the radicand (this avoids the need for parentheses around the radicand). Other functions use parentheses around the input to avoid ambiguity.
The parentheses can be omitted if the input is a single numerical variable or constant,
as in the case of and .
Another shortcut convention that is sometimes used is when the input is
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ...
; thus, rather than , but , because is not a monomial. This, however, is ambiguous and not universally understood outside of specific contexts.
Some calculators and programming languages require parentheses around function inputs, some do not.
Symbols of grouping can be used to override the usual order of operations.
Grouped symbols can be treated as a single expression.
Symbols of grouping can be removed using the associative and
distributive laws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal.
Examples
:
A horizontal fractional line also acts as a symbol of grouping:
:
For ease in reading, other grouping symbols, such as curly braces or square brackets , are often used along with parentheses . For example:
:
Mnemonics
Mnemonics
A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding.
Mnemonics make use of elaborative encoding, retrieval cues, and imagery ...
are often used to help students remember the rules, involving the first letters of words representing various operations. Different mnemonics are in use in different countries.
* In the
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
and in France,
the acronym ''PEMDAS'' is common. It stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
PEMDAS is often expanded to the mnemonic "Please Excuse My Dear Aunt Sally" in schools.
* Canada and New Zealand use ''BEDMAS'', standing for Brackets, Exponents, Division/Multiplication, Addition/Subtraction.
* Most common in the
UK, Pakistan, India, Bangladesh and
Australia
Australia, officially the Commonwealth of Australia, is a Sovereign state, sovereign country comprising the mainland of the Australia (continent), Australian continent, the island of Tasmania, and numerous List of islands of Australia, sma ...
and some other English-speaking countries is ''BODMAS'' meaning either Brackets, Order, Division/Multiplication, Addition/Subtraction or Brackets, Of, Division/Multiplication, Addition/Subtraction.
Nigeria and some other West African countries also use BODMAS. Similarly in the UK, ''BIDMAS'' is also used, standing for Brackets, Indices, Division/Multiplication, Addition/Subtraction.
These mnemonics may be misleading when written this way.
For example, misinterpreting any of the above rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression
:
The "Addition/Subtraction" in the mnemonics should be interpreted as that any additions and subtractions should be performed in order from left to right. Similarly, the expression might be read multiple ways, but the "Multiplication/Division" in the mnemnonic means the multiplications and divisions should be performed from left to right.
:
Additional ambiguities caused by the use of
multiplication by juxtaposition
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition ...
and using the
slash
Slash may refer to:
* Slash (punctuation), the "/" character
Arts and entertainment Fictional characters
* Slash (Marvel Comics)
* Slash (''Teenage Mutant Ninja Turtles'')
Music
* Harry Slash & The Slashtones, an American rock band
* Nash ...
to represent division are discussed
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
. In general, the surest way to avoid ambiguity is to use parentheses.
Special cases
Serial exponentiation
If
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
is indicated by stacked symbols using superscript notation, the usual rule is to work from the top down:
:
which typically is not equal to (''a''
''b'')
''c''. This convention is useful because there is
a property of exponentiation that (''a''
''b'')
''c'' = ''a''
''bc'', so it's unnecessary to use serial exponentiation for this.
However, when using operator notation with a
caret
Caret is the name used familiarly for the character , provided on most QWERTY keyboards by typing . The symbol has a variety of uses in programming and mathematics. The name "caret" arose from its visual similarity to the original proofreade ...
(^) or
arrow
An arrow is a fin-stabilized projectile launched by a bow. A typical arrow usually consists of a long, stiff, straight shaft with a weighty (and usually sharp and pointed) arrowhead attached to the front end, multiple fin-like stabilizers c ...
(↑), there is no common standard.
For example,
Microsoft Excel
Microsoft Excel is a spreadsheet developed by Microsoft for Microsoft Windows, Windows, macOS, Android (operating system), Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro (comp ...
and computation programming language
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...
evaluate
''a''^''b''^''c''
as (''a''
''b'')
''c'', but
Google Search
Google Search (also known simply as Google) is a search engine provided by Google. Handling more than 3.5 billion searches per day, it has a 92% share of the global search engine market. It is also the most-visited website in the world.
The ...
and
Wolfram Alpha
WolframAlpha ( ) is an answer engine developed by Wolfram Research. It answers factual queries by computing answers from externally sourced data.
WolframAlpha was released on May 18, 2009 and is based on Wolfram's earlier product Wolfram Mathe ...
as ''a''
(''b''''c''). Thus
4^3^2
is evaluated to 4,096 in the first case and to 262,144 in the second case.
Unary minus sign
There are differing conventions concerning the unary operator − (usually read "minus"). In written or printed mathematics, the expression −3
2 is interpreted to mean .
In some applications and programming languages, notably
Microsoft Excel
Microsoft Excel is a spreadsheet developed by Microsoft for Microsoft Windows, Windows, macOS, Android (operating system), Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro (comp ...
,
PlanMaker
PlanMaker is a spreadsheet program that is part of the SoftMaker Office suite. Versions are available on Microsoft Windows, MacOS, Linux and Apps for Android and iOS.
PlanMaker is largely similar to Microsoft Excel in function and workflow ...
(and other spreadsheet applications) and
the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus has higher precedence than exponentiation, so in those languages −3
2 will be interpreted as .
This does not apply to the binary minus operator −; for example in Microsoft Excel while the formulas
=−2^2
,
=-(2)^2
and
=0+−2^2
return 4, the formula
=0−2^2
and
=−(2^2)
return −4.
Mixed division and multiplication
In some of the academic literature,
multiplication denoted by juxtaposition
Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
(also known as
implied multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additio ...
) is interpreted as having higher precedence than division, so that equals , not .
For example, the manuscript submission instructions for the ''
Physical Review
''Physical Review'' is a peer-reviewed scientific journal established in 1893 by Edward Nichols. It publishes original research as well as scientific and literature reviews on all aspects of physics. It is published by the American Physical S ...
'' journals state that multiplication is of higher precedence than division,
and this is also the convention observed in prominent physics textbooks such as the ''
Course of Theoretical Physics
The ''Course of Theoretical Physics'' is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s.
It is said that Landau ...
'' by
Landau
Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990 ...
and
Lifshitz Lifshitz (or Lifschitz) is a surname, which may be derived from the Polish city of Głubczyce (German: Leobschütz).
The surname has many variants, including: , , Lifshits, Lifshuts, Lefschetz; Lipschitz ( Lipshitz), Lipshits, Lipchitz, Lips ...
and the ''
Feynman Lectures on Physics
''The Feynman Lectures on Physics'' is a physics textbook based on some lectures by Richard Feynman, a Nobel laureate who has sometimes been called "The Great Explainer". The lectures were presented before undergraduate students at the Californ ...
''.
This ambiguity is often exploited in
internet meme
An Internet meme, commonly known simply as a meme ( ), is an idea, behavior, style, or image that is spread via the Internet, often through social media platforms. What is considered a meme may vary across different communities on the Internet ...
s such as "".
Ambiguity can also be caused by the use of the
slash symbol, '/', for division. The ''Physical Review'' submission instructions suggest to avoid expressions of the form a/b/c; ambiguity can be avoided by instead writing (a/b)/c or a/(b/c).
Calculators
Different calculators follow different orders of operations.
Many simple calculators without a stack implement
chain input
There are various ways in which calculators interpret keystrokes. These can be categorized into two main types:
* On a single-step or immediate-execution calculator, the user presses a key for each operation, calculating all the intermediate resul ...
working left to right without any priority given to different operators, for example typing
:
1 + 2 × 3
yields 9,
while more sophisticated calculators will use a more standard priority, for example typing
:
1 + 2 × 3
yields 7.
The ''
Microsoft Calculator
Windows Calculator is a software calculator developed by Microsoft and included in Windows. In its Windows 10 incarnation it has four modes: standard, scientific, programmer, and a graphing mode. The standard mode includes a number pad and button ...
'' program uses the former in its standard view and the latter in its scientific and programmer views.
Chain input expects two
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 examp ...
s and an operator. When the next operator is pressed, the expression is immediately evaluated and the answer becomes the left hand of the next operator. Advanced calculators allow entry of the whole expression, grouped as necessary, and evaluates only when the user uses the equals sign.
Calculators may associate exponents to the left or to the right. For example, the expression
''a''^''b''^''c''
is interpreted as ''a''
(''b''''c'') on the
TI-92
The TI-92 series of graphing calculators are a line of calculators produced by Texas Instruments. They include: the TI-92 (1995), the TI-92 II (1996), the TI-92 Plus (1998, 1999) and the Voyage 200 (2002). The design of these relatively large ...
and the
TI-30XS MultiView in "Mathprint mode", whereas it is interpreted as (''a''
''b'')
''c'' on the
TI-30XII and the
TI-30XS MultiView in "Classic mode".
An expression like
1/2''x''
is interpreted as 1/(2''x'') by
TI-82
The TI-82 is a graphing calculator made by Texas Instruments. The TI-82 was designed in 1993 as a stripped down, more user friendly version of the TI-85, and as a replacement for the TI-81. It was the direct predecessor of the TI-83. It shares ...
, as well as many modern
Casio
is a Japanese multinational electronics manufacturing corporation headquartered in Shibuya, Tokyo, Japan. Its products include calculators, mobile phones, digital cameras, electronic musical instruments, and analogue and digital watches. It ...
calculators,
but as (1/2)''x'' by
TI-83 and every other TI calculator released since 1996,
as well as by all
Hewlett-Packard
The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company headquartered in Palo Alto, California. HP developed and provided a wide variety of hardware components ...
calculators with algebraic notation. While the first interpretation may be expected by some users due to the nature of
implied multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additio ...
, the latter is more in line with the rule that multiplication and division are of equal precedence.
When the user is unsure how a calculator will interpret an expression, parentheses can be used to remove the ambiguity.
Order of operations arose due to the adaptation of
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 a ...
in
standard mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathemati ...
, which can be notationally ambiguous without such conventions, as opposed to
postfix notation
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 operands, in contrast to Polish notation (PN), in whi ...
or
prefix 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 t ...
, which do not need orders of operations.
Hence, calculators utilizing
Reverse Polish notation
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 operands, in contrast to Polish notation (PN), in whi ...
(RPN) using 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 ...
to enter expressions in the correct order of precedence do not need parentheses or any possibly model-specific order of execution.
Programming languages
Some
programming languages
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
use precedence levels that conform to the order commonly used in mathematics,
though others, such as
APL,
Smalltalk
Smalltalk is an object-oriented, dynamically typed reflective programming language. It was designed and created in part for educational use, specifically for constructionist learning, at the Learning Research Group (LRG) of Xerox PARC by Alan Ka ...
,
Occam and
Mary
Mary may refer to:
People
* Mary (name), a feminine given name (includes a list of people with the name)
Religious contexts
* New Testament people named Mary, overview article linking to many of those below
* Mary, mother of Jesus, also calle ...
, have no
operator precedence rules (in APL, evaluation is strictly right to left; in Smalltalk, it is strictly left to right).
Furthermore, because many operators are not associative, the order within any single level is usually defined by grouping left to right so that
16/4/4
is interpreted as rather than ; such operators are referred to as "left associative". Exceptions exist; for example, languages with operators corresponding to the
cons
In computer programming, ( or ) is a fundamental function in most dialects of the Lisp programming language. ''constructs'' memory objects which hold two values or pointers to two values. These objects are referred to as (cons) cells, conses, ...
operation on lists usually make them group right to left ("right associative"), e.g. in
Haskell
Haskell () is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research and industrial applications, Haskell has pioneered a number of programming lan ...
,
1:2:3:4:[] 1:(2:(3:(4:[]))) [1,2,3,4]
.
Dennis Ritchie, creator of the C (programming language), C language, has said of the precedence in C (shared by programming languages that borrow those rules from C, for example,
C++
C++ (pronounced "C plus plus") is a high-level general-purpose programming language created by Danish computer scientist Bjarne Stroustrup as an extension of the C programming language, or "C with Classes". The language has expanded significan ...
,
Perl
Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was offici ...
and
PHP
PHP is a general-purpose scripting language geared toward web development. It was originally created by Danish-Canadian programmer Rasmus Lerdorf in 1993 and released in 1995. The PHP reference implementation is now produced by The PHP Group ...
) that it would have been preferable to move the
bitwise operators
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operat ...
above the
comparison operators.
Many programmers have become accustomed to this order, but more recent popular languages like Python and Ruby do have this order inversed. The relative precedence levels of
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 ...
found in many C-style languages are as follows:
Examples:
(Note: in the examples below, '≡' is used to mean "is identical to", and not to be interpreted as an actual assignment operator used as part of the example expression.)
*
!A + !B
≡
(!A) + (!B)
*
++A + !B
≡
(++A) + (!B)
*
A + B * C
≡
A + (B * C)
*
A , , B && C
≡
A , , (B && C)
*
A && B C
≡
A && (B C)
*
A & B C
≡
A & (B C)
(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 (pro ...
,
Ruby
A ruby is a pinkish red to blood-red colored gemstone, a variety of the mineral corundum ( aluminium oxide). Ruby is one of the most popular traditional jewelry gems and is very durable. Other varieties of gem-quality corundum are called sa ...
,
PARI/GP
PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. Versions 2.1.0 and higher are distributed under the GNU General Public License. It runs on most common operating systems.
System overview
The ...
and other popular languages,
A & B C
≡
(A & B) C
.)
Source-to-source compiler
A source-to-source translator, source-to-source compiler (S2S compiler), transcompiler, or transpiler is a type of translator that takes the source code of a program written in a programming language as its input and produces an equivalent sou ...
s that compile to multiple languages need to explicitly deal with the issue of different order of operations across languages.
Haxe
Haxe is an open source high-level cross-platform programming language and compiler that can produce applications and source code, for many different computing platforms from one code-base. It is free and open-source software, released under the ...
for example standardizes the order and enforces it by inserting brackets where it is appropriate.
The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.
See also
*
Common operator notation
In programming languages, scientific calculators and similar common operator notation or operator grammar is a way to define and analyse mathematical and other formal expressions. In this model a linear sequence of tokens are divided into two clas ...
(for a more formal description)
*
Hyperoperation
In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with ...
*
Operator associativity
In programming language theory, the associativity of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. If an operand is both preceded and followed by operators (for example ...
*
Operator overloading
In computer programming, operator overloading, sometimes termed ''operator ad hoc polymorphism'', is a specific case of polymorphism, where different operators have different implementations depending on their arguments. Operator overloading is ...
*
Operator precedence in C and C++
*
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 t ...
*
Reverse Polish notation
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 operands, in contrast to Polish notation (PN), in whi ...
Explanatory notes
References
Further reading
*
*
{{DEFAULTSORT:Order Of Operations
Mnemonics
Operators (programming)
Algebra