Order of operations

Inmathematicsandcomputer programming, theorder of operationsis a collection of rules that reflect conventions about which operations to perform first in order to evaluate a givenmathematical expression.
These rules are formalized with a ranking of the operations. The rank of an operation is called itsprecedence, and an operation with ahigherprecedence is performed before operations withlowerprecedence.Calculatorsgenerally perform operations with the same precedence from left to right,but someprogramming languagesand calculators adopt different conventions.
For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modernalgebraic notation.Thus, in the expression1 + 2 × 3, the multiplication is performed before addition, and the expression has the value1 + (2 × 3) = 7, and not(1 + 2) × 3 = 9. When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base.Thus3 + 5= 28and3 × 5= 75.
These conventions exist to avoid notationalambiguitywhile allowing notation to remain brief.Where it is desired to override the precedence conventions, or even simply to emphasize them,parentheses( ) can be used. For example,(2 + 3) × 4 = 20forces addition to precede multiplication, while(3 + 5)= 64forces addition to precedeexponentiation. 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 other types ofbracketsto avoid confusion, as in[2 × (3 + 4)] − 5 = 9.
These rules are meaningful only when the usual notation (calledinfix notation) is used. WhenfunctionalorPolish notationare used for all operations, the order of operations results from the notation itself.
The order of operations, that is, the order in which the operations in an expression are usually performed, results from a convention adopted  throughout mathematics, science, technology and many computerprogramming languages. It is summarized as:
This means that to evaluate an expression, one first evaluates any sub-expression inside parentheses, working inside to outside if there is more than one set. Whether inside parenthesis or not, the operation that is higher in the above list should be applied first. Operations of the same precedence are conventionally evaluated from left to right.
If each division is replaced with multiplication by thereciprocal(multiplicative inverse) then theassociativeandcommutativelaws of multiplication allow the factors in eachtermto be multiplied together in any order. Sometimes multiplication and division are given equal precedence, or sometimes multiplication is given higher precedence than division; see§ Mixed division and multiplicationbelow. If each subtraction is replaced with addition of theopposite(additive inverse), then the associative and commutative laws of addition allow terms to be added in any order.
Theradical symbol⁠t{\displaystyle {\sqrt {\vphantom {t}}}}⁠is traditionally extended by a bar (calledvinculum) 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 ofsinx= sin(x)andsin π = sin(π).Traditionally this convention extends tomonomials; thus,sin 3x= sin(3x)and evensin⁠1/2⁠xy= sin(xy/2), butsinx+y= sin(x) +y, becausex+yis not a monomial. However, this convention is not universally understood, and some authors prefer explicit parentheses.Some calculators and programming languages require parentheses around function inputs, some do not.
Symbols of groupingcan 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 anddistributivelaws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal.
Multiplication before addition:
Parenthetical subexpressions are evaluated first:
Exponentiation before multiplication, multiplication before subtraction:
When an expression is written as a superscript, the superscript is considered to be grouped by its position above its base:
The operand of a root symbol is determined by the overbar:
A horizontal fractional line also acts as a symbol of grouping:
Parentheses can be nested, and should be evaluated from the inside outward. For legibility, outer parentheses can be made larger than inner parentheses. Alternately, other grouping symbols, such as curly braces{} or square brackets[ ], are sometimes used along with parentheses( ). For example:
There are differing conventions concerning theunary operation'−'(usually pronounced "minus"). In written or printed mathematics, the expression −3is interpreted to mean−(3) = −9.
In some applications and programming languages, notablyMicrosoft Excel,PlanMaker(and other spreadsheet applications) andthe programming language bc, unary operations have a higher priority than binary operations, that is, the unary minus has higher precedence than exponentiation, so in those languages −3will be interpreted as(−3)= 9.This does not apply to the binary minusoperation '−';for example in Microsoft Excel while the formulas=-2^2,=-(2)^2and=0+-2^2return 4, the formulas=0-2^2and=-(2^2)return −4.
There is no universal convention for interpreting an expression containing both division denoted by '÷' and multiplication denoted by '×'. Proposed conventions include assigning the operations equal precedence and evaluating them from left to right, or equivalently treating division as multiplication by the reciprocal and then evaluating in any order;evaluating all multiplications first followed by divisions from left to right; or eschewing such expressions and instead always disambiguating them by explicit parentheses.
Beyond primary education, the symbol '÷' for division is seldom used, but is replaced by the use ofalgebraic fractions,typically written vertically with the numerator stacked above the denominator – which makes grouping explicit and unambiguous – but sometimes written inline using theslashor solidus symbol '/'.
Multiplication denoted by juxtaposition (also known asimplied multiplication) creates a visual unit and has higher precedence than most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g.1 / 2nis interpreted to mean1 / (2 ·n)rather than(1 / 2) ·n.For instance, the manuscript submission instructions for thePhysical Reviewjournals directly state that multiplication has precedence over division,and this is also the convention observed in physics textbooks such as theCourse of Theoretical PhysicsbyLandauandLifshitzand mathematics textbooks such asConcrete MathematicsbyGraham,Knuth, andPatashnik.However, some authors recommend against expressions such asa/bc, preferring the explicit use of parenthesisa/ (bc).
More complicated cases are more ambiguous. For instance, the notation1 / 2π(a+b)could plausibly mean either1 / [2π· (a+b)]or[1 / (2π)] · (a+b).Sometimes interpretation depends on context. ThePhysical Reviewsubmission instructions recommend against expressions of the forma/b/c; more explicit expressions(a/b) /cora/ (b/c)are unambiguous.
This ambiguity has been the subject ofInternet memessuch as "8 ÷ 2(2 + 2)", for which there are two conflicting interpretations: 8 ÷ [2 · (2 + 2)] = 1 and (8 ÷ 2) · (2 + 2) = 16.Mathematics education researcher Hung-Hsi Wu points out that "one never gets a computation of this type in real life", and calls such contrived examples "a kind of Gotcha! parlor game designed to trap an unsuspecting person by phrasing it in terms of a set of unreasonably convoluted rules".
Ifexponentiationis indicated by stacked symbols using superscript notation, the usual rule is to work from the top down:
which typically is not equal to (a). This convention is useful because there isa property of exponentiationthat (a)=a, so it's unnecessary to use serial exponentiation for this.
However, when exponentiation is represented by an explicit symbol such as acaret(^) orarrow(↑), there is no common standard. For example,Microsoft Exceland computation programming languageMATLABevaluatea^b^cas (a), butGoogle SearchandWolfram Alphaasa. Thus4^3^2is evaluated to 4,096 in the first case and to 262,144 in the second case.
Mnemonicacronymsare often taught in primary schools to help students remember the order of operations.The acronymPEMDAS, which stands forParentheses,Exponents,Multiplication/Division,Addition/Subtraction,is common in theUnited Statesand France.Sometimes the letters are expanded into words of a mnemonic sentence such as "Please Excuse My Dear Aunt Sally".The United Kingdom and otherCommonwealthcountries may useBODMAS(or sometimesBOMDAS), standing forBrackets,Of,Division/Multiplication,Addition/Subtraction, with "of" meaning fraction multiplication.Sometimes theOis instead expanded asOrder, meaning exponent or root,or replaced byIforIndices in the alternative mnemonicBIDMAS.In Canada and New ZealandBEDMASis common.
In Germany, the convention is simply taught asPunktrechnung vor Strichrechnung, "dot operations before line operations" referring to the graphical shapes of the taught operator signsU+00B7·MIDDLE DOT(multiplication),U+2236∶RATIO(division), andU+002B+PLUS SIGN(addition),U+2212−MINUS SIGN(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 expressiona−b+c{\displaystyle a-b+c}asa−(b+c){\displaystyle a-(b+c)}, while the correct evaluation is(a−b)+c{\displaystyle (a-b)+c}. These values are different whenc≠0{\displaystyle c\neq 0}.
Mnemonic acronyms have been criticized for not developing a conceptual understanding of the order of operations, and not addressing student questions about its purpose or flexibility.Students learning the order of operations via mnemonic acronyms routinely make mistakes,as do some pre-service teachers.Even when students correctly learn the acronym, a disproportionate focus on memorization of trivia crowds out substantive mathematical content.The acronym's procedural application does not match experts' intuitive understanding of mathematical notation: mathematical notation indicates groupings in ways other than parentheses or brackets and a mathematical expression is atree-like hierarchyrather than a linearly "ordered" structure; furthermore, there is no single order by which mathematical expressions must be simplified or evaluated and no universal canonical simplification for any particular expression, and experts fluently apply valid transformations and substitutions in whatever order is convenient, so learning a rigid procedure can lead students to a misleading and limiting understanding of mathematical notation.
Different calculators follow different orders of operations.Many simple calculators without astackimplementchain input, working in button-press order without any priority given to different operations, give a different result from that given by more sophisticated calculators. For example, on a simple calculator, typing1 + 2 × 3 =yields 9, while a more sophisticated calculator will use a more standard priority, so typing1 + 2 × 3 =yields 7.
Calculators may associate exponents to the left or to the right. For example, the expressiona^b^cis interpreted asaon theTI-92and theTI-30XS MultiViewin "Mathprint mode", whereas it is interpreted as (a)on theTI-30XIIand the TI-30XS MultiView in "Classic mode".
An expression like1/2xis interpreted as 1/(2x) byTI-82,as well as many modernCasiocalculators(configurable on some like thefx-9750GIII), but as (1/2)xbyTI-83and every other TI calculator released since 1996,as well as by allHewlett-Packardcalculators with algebraic notation. While the first interpretation may be expected by some users due to the nature ofimplied 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 ofinfix notationinstandard mathematical notation, which can be notationally ambiguous without such conventions, as opposed topostfix notationorprefix notation, which do not need orders of operations.Hence, calculators utilizingreverse Polish notation(RPN) using astackto enter expressions in the correct order of precedence do not need parentheses or any possibly model-specific order of execution.
Mostprogramming languagesuse precedence levels that conform to the order commonly used in mathematics,though others, such asAPL,Smalltalk,OccamandMary, have nooperatorprecedence 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 that16/4/4is interpreted as(16/4)/4 = 1rather than16/(4/4) = 16; such operators are referred to as "left associative". Exceptions exist; for example, languages with operators corresponding to theconsoperation on lists usually make them group right to left ("right associative"), e.g. inHaskell,1:2:3:4:[] == 1:(2:(3:(4:[]))) == [1,2,3,4].
Dennis Ritchie, creator of theC language, said of the precedence in C (shared by programming languages that borrow those rules from C, for example,C++,PerlandPHP) that it would have been preferable to move thebitwise operatorsabove thecomparison operators.Many programmers have become accustomed to this order, but more recent popular languages likePythonandRubydo have this order reversed. The relative precedence levels of operators found in many C-style languages are as follows:
Examples:
(InPython,Ruby,PARI/GPand other popular languages,A & B == Cis interpreted as(A & B) == C.)
Source-to-source compilersthat compile to multiple languages need to explicitly deal with the issue of different order of operations across languages.Haxefor 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.
The order of operations emerged progressively over centuries. The rule that multiplication has precedence over addition was incorporated into the development ofalgebraic notationin the 1600s, since the distributive property implies this as a natural hierarchy. As recently as the 1920s, the historian of mathematicsFlorian Cajoriidentifies disagreement about whether multiplication should have precedence over division, or whether they should be treated equally. The term "order of operations" and the "PEMDAS/BEDMAS" mnemonics were formalized only in the late 19th or early 20th century, as demand for standardized textbooks grew.
Ambiguity about issues such as whether implicit multiplication takes precedence over explicit multiplication and division in such expressions asa/2b, which could be interpreted asa/(2b) or (a/2) ×b, imply that the conventions are not yet completely stable.