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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 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. 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 ( ) 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 ...
. 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 ...
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 root ...
# multiplication and division # addition and subtraction This means that if, in a mathematical expression, a subexpression appears between two operators, the operator that is higher in the above list should be applied first. The commutative 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 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 operations, 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 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 summands 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 expon ...
; 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 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 Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...
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 country comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands. With an area of , Australia is the largest country by ...
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 and using the
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to represent division are discussed
below Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred 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 ...
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 (^) or arrow (↑), 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, implementat ...
evaluate ''a''^''b''^''c'' as (''a''''b'')''c'', but
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and Wolfram Alpha 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 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 workflo ...
(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 ''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 ...
'' 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'' 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 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 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 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, 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. ...
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, 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, 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 wh ...
(RPN) using a
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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 use precedence levels that conform to the order commonly used in mathematics, though others, such as APL, Smalltalk, Occam and Mary, 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 operation on lists usually make them group right to left ("right associative"), e.g. in Haskell, 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++,
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) 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 opera ...
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 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,
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,
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 compilers 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 * Operator associativity * Operator overloading * Operator precedence in C and C++ * Polish notation *
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 wh ...


Explanatory notes


References


Further reading

* * {{DEFAULTSORT:Order Of Operations Mnemonics Operators (programming) Algebra