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| − | {{short description|In mathematics and computer science, order in which operations are performed}}
| + | * [https://en.wikipedia.org/wiki/Order_of_operations Order of Operations], Wikipedia |
| − | {{Use dmy dates|date=February 2015|cs1-dates=y}}
| + | * [https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-arithmetic-operations/cc-6th-order-of-operations/v/introduction-to-order-of-operations Introduction to Order of Operations], Khan Academy |
| − | In [[mathematics]] and [[computer programming]], 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]].
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| − | 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 [[mathematical notation|algebraic notation]].<ref name="Bronstein_1987"/><ref name="Math_2000"/> Thus, the expression {{nowrap|1 + 2 × 3}} is interpreted to have the value {{nowrap|1 + (2 × 3) {{=}} 7}}, and not {{nowrap|(1 + 2) × 3 {{=}} 9}}. 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.<ref name="Bronstein_1987"/> Thus {{nowrap|3 + 5<sup>2</sup> {{=}} 28}} and {{nowrap|3 × 5<sup>2</sup> {{=}} 75}}.
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| − | 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 ( ) can be used to indicate an alternative order of [[operation (mathematics)|operation]]s (or to simply reinforce the default order of operations). For example, {{nowrap|(2 + 3) × 4 {{=}} 20}} forces addition to precede multiplication, while {{nowrap|(3 + 5)<sup>2</sup> {{=}} 64}} forces addition to precede [[exponentiation]]. 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]]s or [[Braces (punctuation)|braces]] to avoid confusion, as in {{nowrap|[2 × (3 + 4)] − 5 {{=}} 9}}.<ref>{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-08-22|website=Math Vault|language=en-US}}</ref>
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| − | ==Definition==
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| − | The order of operations, which is used throughout mathematics, science, technology and many computer [[programming language]]s, is expressed here:<ref name="Bronstein_1987"/><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Precedence|url=https://mathworld.wolfram.com/Precedence.html|access-date=2020-08-22|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|last=Stapel|first=Elizabeth|title=The Order of Operations: PEMDAS|url=https://www.purplemath.com/modules/orderops.htm|access-date=2020-08-22|website=Purplemath}}</ref>
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| − | # [[exponentiation]] and [[root extraction]]
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| − | # [[multiplication]] and [[Division (mathematics)|division]]
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| − | # [[addition]] and [[subtraction]]
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| − | This means that if, in a mathematical expression, a subexpression appears between two [[Operator (programming)|operators]], the operator that is higher in the above list should be applied first.
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| − | The [[commutative property|commutative]] and [[associative property|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.
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| − | In some contexts, it is helpful to replace a division by 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 operation]]s, and makes it easier to use [[commutativity]] and [[associativity]] when simplifying large expressions (for more, see {{slink|Computer algebra|Simplification}}). Thus {{nowrap|3 ÷ 4 {{=}} 3 × {{sfrac|1|4}}}}; in other words, the quotient of 3 and 4 equals the product of 3 and {{sfrac|1|4}}. Also {{nowrap|3 − 4 {{=}} 3 + (−4)}}; in other words the difference of 3 and 4 equals the sum of 3 and −4. Thus, {{nowrap|1 − 3 + 7}} can be thought of as the sum of {{nowrap|1 + (−3) + 7}}, and the three [[summand]]s may be added in any order, in all cases giving 5 as the result.
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| − | The root symbol √ is traditionally prolongated by a bar (called [[vinculum (symbol)|vinculum]]) over the radicand (this avoids the need for parentheses around the radicand). Other functions use parentheses around the input to avoid ambiguity.<ref name="Oldham_2009"/><ref name="NIST_2010"/><ref group="lower-alpha" name="NB3"/> The parentheses can be omitted if the input is a single numerical variable or constant<ref name="Bronstein_1987"/> (as in the case of {{nowrap|sin ''x'' {{=}} sin(''x'')}} and {{nowrap|sin π {{=}} sin(π)}}.<ref group="lower-alpha" name="NB3"/> Another shortcut convention that is sometimes used is when the input is [[monomial]]; thus, {{nowrap|sin 3''x'' {{=}} sin(3''x'')}} rather than {{nowrap|(sin(3)) ''x''}}, but {{nowrap|sin ''x'' + ''y'' {{=}} sin(''x'') + ''y''}}, because {{nowrap|''x'' + ''y''}} is not a monomial. This, however, is ambiguous and not universally understood outside of specific contexts.<ref group="lower-alpha" name="NB4"/>
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| − | Some calculators and programming languages require parentheses around function inputs, some do not.
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| − | {{anchor|Grouping}}Symbols of grouping can be used to override the usual order of operations.<ref name="Bronstein_1987"/> Grouped symbols can be treated as a single expression.<ref name="Bronstein_1987"/> Symbols of grouping can be removed using the associative and [[Distributivity|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.
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| − | ===Examples===
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| − | : <math>\sqrt{1 + 3} + 5 = \sqrt 4 + 5 = 2 + 5 = 7.</math>
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| − | A horizontal fractional line also acts as a symbol of grouping:
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| − | : <math>\frac{1 + 2}{3 + 4} + 5 = \frac{3}{7} + 5.</math>
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| − | For ease in reading, other grouping symbols, such as curly braces {{nowrap|{ }}} or square brackets {{nowrap|[ ]}}, are often used along with parentheses {{nowrap|( )}}. For example:
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| − | : <math>( [1 + 2] \div [3 + 4] ) + 5 = (3 \div 7) + 5 </math>
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| − | =={{anchor|PEMDAS|PEDMAS|BEDMAS|BEMDAS|BODMAS|BIDMAS|BIMDAS|BOMDAS|GEMDAS|BERDMAS|PERDMAS|BPODMAS}}Mnemonics==
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| − | [[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.<ref name="Rules"/><ref name="EW_2011"/><ref name="Study"/>
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| − | * In the United States, the acronym '''''PEMDAS''''' is common.<ref name="Vanderbeek_2007"/> It stands for ''P''arentheses, ''E''xponents, ''M''ultiplication/''D''ivision, ''A''ddition/''S''ubtraction.<ref name="Vanderbeek_2007"/> PEMDAS is often expanded to the mnemonic "'''Please Excuse My Dear Aunt Sally'''" in schools.<ref name="Ball_1978"/> | |
| − | * Canada and New Zealand use '''''BEDMAS''''', standing for ''B''rackets, ''E''xponents, ''D''ivision/''M''ultiplication, ''A''ddition/''S''ubtraction.<ref name="Vanderbeek_2007"/>
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| − | * Most common in the UK, Pakistan, India, Bangladesh and Australia<ref name="Syllabus_2019"/> and some other English-speaking countries is '''''BODMAS''''' meaning either ''B''rackets, ''O''rder, ''D''ivision/''M''ultiplication, ''A''ddition/''S''ubtraction or ''B''rackets, ''O''f/''D''ivision/''M''ultiplication, ''A''ddition/''S''ubtraction.<ref group="lower-alpha" name="NB2"/><ref name="Vedantu_2019"/> Nigeria and some other West African countries also use BODMAS. Similarly in the UK, '''''BIDMAS''''' is also used, standing for ''B''rackets, ''I''ndices, ''D''ivision/''M''ultiplication, ''A''ddition/''S''ubtraction.
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| − | These mnemonics may be misleading when written this way.<ref name="Ball_1978"/> For example, misinterpreting any of the above rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression<ref name="Ball_1978"/>
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| − | :<math> a - b + c = (a - b) + c \ne a-(b+c)</math>
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| − | When evaluating expressions that only include addition and subtraction, such as the expression above, addition and subtraction may be performed sequentially, working left to right, but of the six major arithmetic operations, only addition and multiplication are commutative and associative, and so professionals usually view subtraction, especially in algebra and in higher mathematics, as addition of a negative number. Either working left to right or treating the subtraction as adding a [[signed number]] will produce the correct answer; performing subtraction in the wrong order will result in the incorrect answer. Mnemonics do not reflect the grouping of addition/subtraction or multiplication/division so their use can result in this misunderstanding.
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| − | A similar ambiguity exists in the case of serial division, for example, the expression {{nowrap| a ÷ b ÷ c × d}} might be read multiple ways but they may not always come to the same answer. For example, {{nowrap| 3 ÷ 1/2}} evaluated from left to right would be three divided by one equals three, divided by two equals three halves, but using division as multiplication by the reciprocal gives three divided by one half equals three multiplied by two equals six. There are three possibilities, none universally agreed upon. Either each of the several different division symbols must be given complicated rules of precedence{{Citation needed|date=August 2021}}, or imitating the rule for subtraction as addition of the opposite, division can be defined as multiplication by the reciprocal, or parentheses can be used to avoid ambiguities.
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| − | :<math> ((a \div b) \div c )\times d \ne a \div (b \div (c \times d)) </math>
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| − | Since this is controversial, we summarize the above:
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| − | The first rule is that if there are multiple divisions in a row, the order of calculation may go from left to right:
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| − | : <math>a / b / cd = (a / b) / cd = a / (bcd)=\frac{a}{bcd} </math>
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| − | But this is not universally accepted.<ref name="Order of arithmetic operations">George Mark Bergman: [https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html Order of arithmetic operations] {{Webarchive|url=https://web.archive.org/web/20170305004813/https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html|date=2017-03-05}}</ref><ref name="The Order of Operations">Education Place: [http://eduplace.com/math/mathsteps/4/a/index.html The Order of Operations] {{Webarchive|url=https://web.archive.org/web/20170608144614/http://eduplace.com/math/mathsteps/4/a/index.html|date=2017-06-08}}</ref>
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| − | The second rule, treating division as multiplication by a reciprocal, greatly reduces the frequency of ambiguous division.
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| − | :<math> a \div b \div c \times d = a \times \frac{1}{b} \times \frac{1}{c} \times d = \frac{a}{b} \times \frac{1}{c} \times d = \frac{a}{bc} \times d = \frac{ad}{bc} </math>
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| − | But the only sure way to avoid ambiguity is to use parentheses.
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| − | ==Special cases==
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| − | ===Serial exponentiation===
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| − | <!-- Section header "Serial exponentiation" used as redirect target! -->
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| − | If [[exponentiation]] is indicated by stacked symbols using superscript notation, the usual rule is to work from the top down:<ref name="Robinson_1958"/><ref name="Bronstein_1987"/><ref name="NIST_2010"/><ref name="Zeidler_2013"/>
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| − | : {{nowrap|''a''<sup>''b''<sup>''c''</sup></sup> {{=}} ''a''<sup>(''b''<sup>''c''</sup>)</sup>}}
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| − | which typically is not equal to (''a''<sup>''b''</sup>)<sup>''c''</sup>. This convention is useful because there is [[Exponentiation#Identities and properties|a property of exponentiation]] that (''a''<sup>''b''</sup>)<sup>''c''</sup> = ''a''<sup>''bc''</sup>, so it's unnecessary to use serial exponentiation for this.
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| − | However, when using operator notation with a [[caret]] (^) or [[arrow (symbol)|arrow]] (↑), there is no common standard.<ref name="VanWinkle_2016"/> For example, [[Microsoft Excel]] and computation programming language [[MATLAB]] evaluate <code>''a''^''b''^''c''</code> as (''a''<sup>''b''</sup>)<sup>''c''</sup>, but [[Google Search]] and [[Wolfram Alpha]] as ''a''<sup>(''b''<sup>''c''</sup>)</sup>. Thus <code>4^3^2</code> is evaluated to 4,096 in the first case and to 262,144 in the second case.
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| − | ===Unary minus sign===
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| − | There are differing conventions concerning the unary operator − (usually read "minus"). In written or printed mathematics, the expression −3<sup>2</sup> is interpreted to mean {{nowrap|−(3<sup>2</sup>) {{=}} − 9}}.<ref name="Bronstein_1987"/><ref name="Angel"/>
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| − | In some applications and programming languages, notably [[Microsoft Excel]], [[PlanMaker]] (and other spreadsheet applications) and [[bc programming language|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<sup>2</sup> will be interpreted as {{nowrap|(−3)<sup>2</sup> {{=}} 9}}.<ref name="Microsoft_2015"/> This does not apply to the binary minus operator −; for example in Microsoft Excel while the formulas <code>=−2^2</code>, <code>=-(2)^2</code> and <code>=0+−2^2</code> return 4, the formula <code>=0−2^2</code> and <code>=−(2^2)</code> return −4.
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| − | ===Mixed division and multiplication===
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| − | Similarly, there can be ambiguity in the use of the [[slash (punctuation)#Arithmetic|slash symbol]] / in expressions such as 1/2''n''.<ref name="Ball_1978"/> If one rewrites this expression as {{nowrap|1 ÷ 2''n''}} and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes:
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| − | : {{nowrap|1 ÷ 2 × ''n'' {{=}} 1 × {{sfrac|1|2}} × ''n'' {{=}} {{sfrac|1|2}} × ''n''}}.
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| − | With this interpretation {{nowrap|1 ÷ 2''n''}} is equal to {{nowrap|(1 ÷ 2)''n''}}.<ref name="Bronstein_1987"/><ref name="Rules"/> However, 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 {{nowrap|1 ÷ 2''n''}} equals {{nowrap|1 ÷ (2''n'')}}, not {{nowrap|(1 ÷ 2)''n''}}.
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| − | For example, the manuscript submission instructions for the ''[[Physical Review]]'' journals state that multiplication is of higher precedence than division with a slash,<ref name="APS_2012"/> and this is also the convention observed in prominent physics textbooks such as the ''[[Course of Theoretical Physics]]'' by [[Lev Landau|Landau]] and [[Evgeny Lifshitz|Lifshitz]] and the ''[[Feynman Lectures on Physics]]''.<ref group="lower-alpha" name="NB1"/>
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| − | ==Calculators==
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| − | {{main article|Calculator input methods}}
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| − | Different calculators follow different orders of operations.<ref
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| − | name="Bronstein_1987"/> Many simple calculators without a stack implement [[chain input]] working left to right without any priority given to different operators, for example typing
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| − | : <code>1 + 2 × 3</code> yields 9,
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| − | while more sophisticated calculators will use a more standard priority, for example typing
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| − | : <code>1 + 2 × 3</code> yields 7.
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| − | The ''[[Microsoft Calculator]]'' program uses the former in its standard view and the latter in its scientific and programmer views.
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| − | Chain input expects two [[operand]]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.
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| − | Calculators may associate exponents to the left to the right. For example, the expression <code>''a''^''b''^''c''</code> is interpreted as ''a''<sup>(''b''<sup>''c''</sup>)</sup> on the [[TI-92]] and the [[TI-30#Models|TI-30XS MultiView]] in "Mathprint mode", whereas it is interpreted as (''a''<sup>''b''</sup>)<sup>''c''</sup> on the [[TI-30#Models|TI-30XII]] and the [[TI-30#Models|TI-30XS MultiView]] in "Classic mode".
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| − | An expression like <code>1/2''x''</code> is interpreted as 1/(2''x'') by [[TI-82]], as well as many modern [[Casio]] calculators,<ref name="Casio"/> but as (1/2)''x'' by [[TI-83]] and every other TI calculator released since 1996,<ref name="TI_2011_11773"/> 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 standard rule that multiplication and division are of equal precedence,<ref name="Zachary_1997_Maple"/><ref name="Zachary_1997_Mathematica"/> where 1/2''x'' is read one divided by two and the answer multiplied by ''x''.
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| − | When the user is unsure how a calculator will interpret an expression, it is a good idea to use parentheses so there is no ambiguity.
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| − | Calculators that use [[reverse Polish notation]] (RPN), also known as postfix notation, use a [[stack (data structure)|stack]] to enter expressions in the correct order of precedence without a need for parentheses or any possibly model-specific order of execution.<ref name="Ball_1978"/><ref name="Vanderbeek_2007"/>
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| − | ==Programming languages==
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| − | Some [[programming languages]] use precedence levels that conform to the order commonly used in mathematics,<ref name="VanWinkle_2016"/> though others, such as [[APL (programming language)|APL]], [[Smalltalk]], [[Occam (programming language)|Occam]] and [[Mary (programming language)|Mary]], have no [[Operator (programming)|operator]] precedence rules (in APL, evaluation is strictly right to left; in Smalltalk, it is strictly left to right).
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| − | In addition, because many operators are not associative, the order within any single level is usually defined by grouping left to right so that <code>16/4/4</code> is interpreted as {{nowrap|(16/4)/4 {{=}} 1}} rather than {{nowrap|16/(4/4) {{=}} 16}}; such operators are perhaps misleadingly 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 (programming language)|Haskell]], <code>1:2:3:4:[] == 1:(2:(3:(4:[]))) == [1,2,3,4]</code>.
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| − | The [[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]] and [[PHP]]) that it would have been preferable to move the [[Bitwise operation|bitwise operators]] above the [[Relational operator|comparison operators]].<ref name="Ritchie_1996"/> However, many programmers have become accustomed to this order. The relative precedence levels of [[Operator (programming)|operators]] found in many C-style languages are as follows:
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| − | {| class="wikitable"
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| − | |1 || () [] -> . :: || Function call, scope, array/member access
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| − | |-
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| − | |2 || ! ~ - + * & [[sizeof]] ''[[Type conversion|type cast]]'' ++ -- || (most) unary operators, [[sizeof]] and [[Type conversion|type casts]] (right to left)
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| − | |-
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| − | |3 || * / % MOD || Multiplication, division, [[modular arithmetic|modulo]]
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| − | |-
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| − | |4 || + - || Addition and subtraction
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| − | |-
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| − | |5 || << >> || Bitwise shift left and right
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| − | |-
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| − | |6 || < <= > >= || Comparisons: less-than and greater-than
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| − | |-
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| − | |7 || == != || Comparisons: equal and not equal
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| − | |-
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| − | |8 || & || Bitwise AND
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| − | |-
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| − | |9 || ^ || Bitwise exclusive OR (XOR)
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| − | |-
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| − | |10 || <nowiki>|</nowiki> || Bitwise inclusive (normal) OR
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| − | |-
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| − | |11 || && || Logical AND
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| − | |-
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| − | |12 || <nowiki>||</nowiki> || Logical OR
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| − | |-
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| − | |13 || ? : || Conditional expression (ternary)
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| − | |-
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| − | |14 || = += -= *= /= %= &= <nowiki>|=</nowiki> ^= <<= >>= || Assignment operators (right to left)
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| − | |-
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| − | |15 ||, || [[Comma operator]]
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| − | |}
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| − | Examples:
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| − | (Note: in the examples below, '≡' is used to mean "is equivalent to", and not to be interpreted as an actual assignment operator used as part of the example expression.)
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| − | * <code>!A + !B</code> ≡ <code>(!A) + (!B)</code>
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| − | * <code>++A + !B</code> ≡ <code>(++A) + (!B)</code>
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| − | * <code>A + B * C</code> ≡ <code>A + (B * C)</code>
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| − | * <code>A || B && C</code> ≡ <code>A || (B && C)</code>
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| − | * <code>A && B == C</code> ≡ <code>A && (B == C)</code>
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| − | * <code>A & B == C</code> ≡ <code>A & (B == C)</code>
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| − | [[Source-to-source compiler]]s that compile to multiple languages need to explicitly deal with the issue of different order of operations across languages. [[Haxe]] for example standardizes the order and enforces it by inserting brackets where it is appropriate.<ref name="Li_2011"/>
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| − | The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.<ref name="Jones"/>
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| − | ==See also==
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| − | * [[Common operator notation]] (for a more formal description)
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| − | * [[Hyperoperation]]
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| − | * [[Operator associativity]]
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| − | * [[Operator overloading]]
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| − | * [[Operator precedence in C and C++]]
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| − | * [[Polish notation]]
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| − | * [[Reverse Polish notation]]
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| − | ==Notes==
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| − | {{notelist|refs=
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| − | <ref group="lower-alpha" name="NB1">For example, the third edition of ''Mechanics'' by [[Landau and Lifshitz (book)|Landau and Lifshitz]] contains expressions such as ''hP''<sub>''z''</sub>/2{{pi}} (p. 22), and the first volume of the ''[[Feynman Lectures]]'' contains expressions such as 1/2{{sqrt|''N''}} [http://www.feynmanlectures.caltech.edu/I_06.html (p. 6–7)]. In both books, these expressions are written with the convention that the [[Slash (punctuation)|solidus]] is evaluated last. This also implies that an expression like 8/2(4) has solution 1 as the omission of the [[Multiplication sign| multiplication sign (x * or .)]] implies that the solidus is evaluated last even if positioned more to the left.</ref>
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| − | <ref group="lower-alpha" name="NB2">"Of" is equivalent to division or multiplication, and commonly used especially at primary school level, as in "Half ''of'' fifty".</ref>
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| − | <ref group="lower-alpha" name="NB3">Some authors deliberately avoid any omission of parentheses with functions even in the case of single numerical variable or constant arguments (f.e. {{citeref|Oldham|Myland|Spanier|2009|Oldham in ''Atlas''|style=plain}}), whereas other authors (like {{citeref|Olver|Lozier|Boisvert|Clark|2010|NIST|style=plain}}) apply this notational simplification only conditionally in conjunction with specific multi-character function names (like <code>sin</code>), but don't use it with generic function names (like <code>''f''</code>).</ref>
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| − | <ref group="lower-alpha" name="NB4">To avoid any ambiguity, this notational simplification for [[monomial]]s is deliberately avoided in works such as {{citeref|Oldham|Myland|Spanier|2009|Oldham's ''Atlas of Functions''|style=plain}} or the {{citeref|Olver|Lozier|Boisvert|Clark|2010|''NIST Handbook of Mathematical Functions''|style=plain}}.</ref>
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| − | }}
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| − | ==References==
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| − | {{reflist|refs=
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| − | <ref name="Robinson_1958">{{cite journal |title=A report on primes of the form k · 2<sup>n</sup> + 1 and on factors of Fermat numbers |author-first=Raphael Mitchel |author-last=Robinson |author-link=Raphael Mitchel Robinson |journal=[[Proceedings of the American Mathematical Society]] |volume=9 |issue=5 |date=October 1958 |orig-year=1958-04-07 |location=[[University of California]], Berkeley, California, USA |doi=10.1090/s0002-9939-1958-0096614-7 |pages=673–681 [677] |url=https://www.ams.org/journals/proc/1958-009-05/S0002-9939-1958-0096614-7/S0002-9939-1958-0096614-7.pdf |access-date=2020-06-28 |url-status=live |archive-url=https://web.archive.org/web/20200628100823/https://www.ams.org/journals/proc/1958-009-05/S0002-9939-1958-0096614-7/S0002-9939-1958-0096614-7.pdf |archive-date=2020-06-28|doi-access=free }}</ref>
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| − | <ref name="Bronstein_1987">{{cite book |title=Taschenbuch der Mathematik |language=de |trans-title=Pocketbook of mathematics |title-link=Bronstein and Semendjajew |chapter=2.4.1.1. Definition arithmetischer Ausdrücke |trans-chapter=Definition of arithmetic expressions |author-first1=Ilja Nikolaevič<!-- Nikolajewitsch --> |author-last1=Bronstein |author-link1=Ilya Nikolaevich Bronshtein<!-- 1903–1976 --> |author-first2=Konstantin Adolfovič<!-- Adolfowitsch --> |author-last2=Semendjajew |author-link2=Konstantin Adolfovic Semendyayev<!-- 1908–1988 --> |editor-first1=Günter |editor-last1=Grosche |editor-first2=Viktor |editor-last2=Ziegler<!-- 1922–1980--> |editor-first3=Dorothea |editor-last3=Ziegler |others=Weiß, Jürgen<!-- lector --> |translator-first=Viktor |translator-last=Ziegler |volume=1 |date=1987 |edition=23 |orig-year=1945 |publisher=[[Verlag Harri Deutsch]] (and [[B. G. Teubner Verlagsgesellschaft]], Leipzig) |publication-place=Thun, Switzerland / Frankfurt am Main, Germany |location=Leipzig, Germany |isbn=3-87144-492-8 |pages=115–120, 802 |quote=Regel 7: Ist ''F''(''A'') Teilzeichenreihe eines arithmetischen Ausdrucks oder einer seiner Abkürzungen und ''F'' eine Funktionenkonstante und ''A'' eine Zahlenvariable oder Zahlenkonstante, so darf ''F A'' dafür geschrieben werden. [Darüber hinaus ist noch die Abkürzung ''F''<sup>''n''</sup>(''A'') für (''F''(''A''))<sup>''n''</sup> üblich. Dabei kann ''F'' sowohl Funktionenkonstante als auch Funktionenvariable sein.]}}</ref>
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| − | <ref name="Math_2000">{{cite web |url=http://mathforum.org/library/drmath/view/52582.html |title=Ask Dr. Math |publisher=Math Forum |date=2000-11-22 |access-date=2012-03-05}}</ref>
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| − | <ref name="Angel">{{cite book |author-first=Allen R. |author-last=Angel |title=Elementary Algebra for College Students |edition=8 |at=Chapter 1, Section 9, Objective 3}}</ref>
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| − | <ref name="Microsoft_2015">{{cite web |url=https://support.microsoft.com/en-gb/kb/kbview/132686 |archive-url=https://web.archive.org/web/20150419091629/https://support.microsoft.com/en-gb/kb/kbview/132686 |url-status=dead |archive-date=2015-04-19 |title=Formula Returns Unexpected Positive Value |publisher=[[Microsoft]] |date=2005-08-15 |access-date=2012-03-05}}</ref>
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| − | <ref name="Ball_1978">{{cite book |title=Algorithms for RPN calculators |url=https://archive.org/details/algorithmsforrpn0000ball |url-access=registration |author-first=John A. |author-last=Ball |date=1978 |edition=1 |publisher=[[Wiley-Interscience]], [[John Wiley & Sons, Inc.]] |location=Cambridge, Massachusetts, USA |isbn=0-471-03070-8 |page=[https://archive.org/details/algorithmsforrpn0000ball/page/31 31]}}</ref>
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| − | <ref name="Rules">{{cite web |url=http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-rules-2009-1.pdf |title=Rules of arithmetic |website=Mathcentre.ac.uk |access-date=2019-08-02}}</ref>
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| − | <ref name="APS_2012">{{cite web |url=https://publish.aps.org/files/styleguide-pr.pdf |title=Physical Review Style and Notation Guide |publisher=[[American Physical Society]] |access-date=2012-08-05 |at=Section IV–E–2–e}}</ref>
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| − | <ref name="Vanderbeek_2007">{{cite book |title=Order of Operations and RPN |author-first=Greg |author-last=Vanderbeek |date=June 2007 |type=Expository paper |series=Master of Arts in Teaching (MAT) Exam Expository Papers |id=Paper 46 |publisher=[[University of Nebraska]] |location=Lincoln, Nebraska, USA |url=https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1045&context=mathmidexppap |access-date=2020-06-14 |url-status=live |archive-url=https://web.archive.org/web/20200614191608/https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1045&context=mathmidexppap |archive-date=2020-06-14}}</ref>
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| − | <ref name="EW_2011">{{cite web |url=http://blogs.edweek.org/teachers/coach_gs_teaching_tips/2011/01/math_teachers_please_excuse_dear_aunt_sally--forever.html |title=Please Excuse My Dear Aunt Sally (PEMDAS)--Forever! |date=2011-01-01 |work=Education Week - Coach G's Teaching Tips}}</ref>
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| − | <ref name="Study">{{cite web |url=http://study.com/academy/lesson/what-is-pemdas-definition-rule-examples.html |title=What is PEMDAS? - Definition, Rule & Examples |work=Study.com}}</ref>
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| − | <ref name="Syllabus_2019">{{cite web |url=http://syllabus.bos.nsw.edu.au/assets/global/files/maths_s3_sampleu1.doc |format=DOC |title=Order of operations |website=Syllabus.bos.nsw.edu.au |access-date=2019-08-02}}</ref>
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| − | <ref name="Vedantu_2019">{{cite web |url=https://www.vedantu.com/maths/bodmas-rule |title=Bodmas Rule - What is Bodmas Rule - Order of Operations |website=vedantu.com |access-date=2019-08-21}}</ref>
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| − | <ref name="NIST_2010">{{cite book |title=NIST Handbook of Mathematical Functions |title-link=NIST Handbook of Mathematical Functions |editor-first=Frank W. J. |editor-last=Olver |editor2-first=Daniel W. |editor2-last=Lozier |editor3-first=Ronald F. |editor3-last=Boisvert |editor4-first=Charles W. |editor4-last=Clark |date=2010 |publisher=[[National Institute of Standards and Technology]] (NIST), [[U.S. Department of Commerce]], [[Cambridge University Press]] |isbn=978-0-521-19225-5 |mr=2723248}}[http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521140638]</ref>
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| − | <ref name="Casio">{{cite web |url=https://support.casio.com/global/en/calc/manual/fx-82MS_85MS_220PLUS_300MS_350MS_en/technical_informatoin/sequence.html |title=Calculation Priority Sequence |publisher=[[Casio]] |website=support.casio.com |access-date=2019-08-01}}</ref>
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| − | <ref name="TI_2011_11773">{{cite web |title=Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators |publisher=[[Texas Instruments]] |date=2011-01-16 |id=11773 |url=http://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110 |access-date=2015-08-24 |url-status=live |archive-url=https://web.archive.org/web/20160417214906/https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110 |archive-date=2016-04-17}}</ref>
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| − | <ref name="Zachary_1997_Maple">{{cite web |title=Introduction to scientific programming - Computational problem solving using Maple and C - Operator precedence worksheet |author-first=Joseph L. |author-last=Zachary |date=1997 |url=http://www.cs.utah.edu/~zachary/isp/worksheets/operprec/operprec.html |access-date=2015-08-25}}</ref>
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| − | <ref name="Zachary_1997_Mathematica">{{cite web |title=Introduction to scientific programming - Computational problem solving using Mathematica and C - Operator precedence notebook |author-first=Joseph L. |author-last=Zachary |date=1997 |url=http://www.cs.utah.edu/~zachary/ispmma/notebooks/operprec/operprec.html |access-date=2015-08-25}}</ref>
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| − | <ref name="Ritchie_1996">{{cite book |author-first=Dennis M. |author-last=Ritchie |author-link=Dennis M. Ritchie |url=https://www.bell-labs.com/usr/dmr/www/chist.html |chapter=The Development of the C Language |title=History of Programming Languages |edition=2 |publisher=[[ACM Press]] |date=1996}}</ref>
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| − | <ref name="VanWinkle_2016">{{cite web |title=Exponentiation Associativity and Standard Math Notation |author-first=Lewis |author-last=Van Winkle |work=Codeplea - Random thoughts on programming |date=2016-08-23 |url=https://codeplea.com/exponentiation-associativity-options |access-date=2016-09-20 |url-status=live |archive-url=https://web.archive.org/web/20200628215506/https://codeplea.com/exponentiation-associativity-options |archive-date=2020-06-28}}</ref>
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| − | <ref name="Li_2011">{{cite web |url=http://blog.onthewings.net/2011/05/02/six-divided-by-two-bracket-one-plus-two/ |title=6÷2(1+2)=? |work=Andy Li's Blog |author-first=Andy |author-last=Li |date=2011-05-02 |access-date=2012-12-31}}</ref>
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| − | <ref name="Jones">{{cite journal |url=http://www.knosof.co.uk/cbook/accu06.html |title=Developer beliefs about binary operator precedence |author-first=Derek M. |author-last=Jones |journal=CVu |volume=18 |issue=4 |pages=14–21}}</ref>
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| − | <ref name="Zeidler_2013">{{cite book |title=Springer-Handbuch der Mathematik I |title-link=Springer-Handbuch der Mathematik I |volume=I |language=de |editor-first=Eberhard |editor-last=Zeidler |editor-link=:de:Eberhard Zeidler |author-last1=Zeidler |author-first1=Eberhard |author-link1=:de:Eberhard Zeidler |author-last2=Schwarz |author-first2=Hans Rudolf |author-last3=Hackbusch |author-first3=Wolfgang |author-link3=Wolfgang Hackbusch |author-last4=Luderer |author-first4=Bernd |author-link4=:de:Bernd Luderer |author-last5=Blath |author-first5=Jochen |author-last6=Schied |author-first6=Alexander |author-last7=Dempe |author-first7=Stephan |author-last8=Wanka |author-first8=Gert |author-link8=Gert Wanka |author-last9=Hromkovič |author-first9=Juraj |author-link9=Juraj Hromkovič |author-last10=Gottwald |author-first10=Siegfried |author-link10=Siegfried Gottwald |publisher=[[Springer Spektrum]], [[Springer Fachmedien Wiesbaden]] |location=Berlin / Heidelberg, Germany |edition=1 |date=2013 |orig-year=2012 |isbn=978-3-658-00284-8 |doi=10.1007/978-3-658-00285-5 |page=590 <!-- |url=https://www.springer.com/de/book/9783658002848 |access-date=2020-06-27 -->}} (xii+635 pages)</ref>
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| − | <ref name="Oldham_2009">{{cite book |title=An Atlas of Functions: with Equator, the Atlas Function Calculator | first1=Keith B. |last1=Oldham |first2=Jan C. |last2=Myland |first3=Jerome |last3=Spanier |publisher=[[Springer Science+Business Media, LLC]] |edition=2 |date=2009 |orig-year=1987 |isbn=978-0-387-48806-6 |doi=10.1007/978-0-387-48807-3 |lccn=2008937525}}</ref>
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| − | }}
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| − | | |
| − | ==Further reading==
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| − | * {{cite web |title=Order of arithmetic operations; in particular, the 48/2(9+3) question. |author-first=George Mark |author-last=Bergman |author-link=George Mark Bergman |date=2013-02-21<!-- or earlier --> |publisher=Department of Mathematics, University of California |url=https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html |access-date=2020-07-22 |url-status=live |archive-url=https://web.archive.org/web/20200520170331/https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html |archive-date=2020-05-20}}
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| − | * {{cite web |title=The Order of Operations |work=MathSteps: What is it? |date=1999 |publisher=[[Houghton Mifflin Company]] |url=http://eduplace.com/math/mathsteps/4/a/index.html |access-date=2020-07-22 |url-status=live |archive-url=https://web.archive.org/web/20200721215611/http://eduplace.com/math/mathsteps/4/a/index.html |archive-date=2020-07-21}}
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| − | | |
| − | {{DEFAULTSORT:Order Of Operations}}
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| − | [[Category:Mnemonics]]
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| − | [[Category:Operators (programming)]]
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| − | [[Category:Algebra]]
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