order of operation
   HOME

TheInfoList



OR:

In 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 anal ...
, 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 symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, f ...
. 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, 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 r ...
. 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 brackets 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 r ...
and root extraction # multiplication 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 and subtraction 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 of ...
and associative 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, this allows one to handle fewer binary operations, and makes it easier to use commutativity 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 ...
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 ...
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 expone ...
; 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

: \sqrt + 5 = \sqrt 4 + 5 = 2 + 5 = 7. A horizontal fractional line also acts as a symbol of grouping: : \frac + 5 = \frac + 5. For ease in reading, other grouping symbols, such as curly braces or square brackets , are often used along with parentheses . For example: : ( + 2\div + 4) + 5 = (3 \div 7) + 5


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 imager ...
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 territori ...
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 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 : a - b + c = (a - b) + c \ne a-(b+c) 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. : a \div b \times c = (a \div b) \times c \ne a \div (b \times c) 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 additio ...
and using the slash to represent division are discussed below. 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 r ...
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 (↑), there is no common standard. For example,
Microsoft Excel Microsoft Excel is a spreadsheet developed by Microsoft for Windows, macOS, Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for App ...
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, implementa ...
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 Mat ...
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 −32 is interpreted to mean . In some applications and programming languages, notably
Microsoft Excel Microsoft Excel is a spreadsheet developed by Microsoft for Windows, macOS, Android and iOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for App ...
, PlanMaker (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 −32 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 (also known as implied multiplication) is interpreted as having higher precedence than division, so that equals , not . For example, the manuscript submission instructions for the '' Physical Review'' 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 Land ...
'' 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''. This ambiguity is often exploited in internet memes 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'' 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 exam ...
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 share ...
, 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. I ...
calculators, but as (1/2)''x'' by TI-83 and every other TI calculator released since 1996, as well as by all Hewlett-Packard calculators with algebraic notation. While the first interpretation may be expected by some users due to the nature of implied multiplication, 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 in standard mathematical notation, which can be notationally ambiguous without such conventions, as opposed to postfix notation 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 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, 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 offic ...
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 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 * CA + (B * C) * A , , B && CA , , (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 ...
, PARI/GP 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 (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 i ...
* 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 ...
*
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