Editing Order of operations

{{short description|Performing order of mathematical operations}}
{{distinguish|Operations order}}
[[File:Order of operations.svg|thumb|'''Order of operations''']]
{{use dmy dates|date=February 2015|cs1-dates=y}}
{{use list-defined references|date=January 2022}}
In [[mathematics]] and [[computer programming]], the '''order of operations'''  is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given [[mathematical expression]].

These rules are formalized with a ranking of the operations. The rank of an operation is called its '''precedence''', and an operation with a ''higher'' precedence is performed before operations with ''lower'' precedence. [[Calculator]]s generally perform operations with the same precedence from left to right,{{r|Microsoft}} but some [[programming language]]s and calculators adopt different conventions.

For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern [[mathematical notation|algebraic notation]].{{r|BS}}{{r|Peterson}} Thus, in the expression {{nowrap|1 + 2 × 3}}, the multiplication is performed before addition, and the expression has 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 placed as a superscript to the right of their base.{{r|BS}} Thus {{nowrap|3 + 5<sup>2</sup> {{=}} 28}} and {{nowrap|3 × 5<sup>2</sup> {{=}} 75}}.

These conventions exist to avoid notational [[ambiguity]] while allowing notation to remain brief.{{r|Swokowski}} Where it is desired to override the precedence conventions, or even simply to emphasize them, [[Bracket#Parentheses|parentheses]] ( ) can be used. 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 other types of [[bracket]]s to avoid confusion, as in {{nowrap|[2 × (3 + 4)] − 5 {{=}} 9}}.

These rules are meaningful only when the usual notation (called [[infix notation]]) is used. When [[functional notation|functional]] or [[Polish notation]] are used for all operations, the order of operations results from the notation itself.

== Conventional order ==
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 computer [[programming language]]s. It is summarized as:{{r|BS}}{{r|Mathworld}}
# [[Bracket (mathematics)|Parentheses]]
# [[Exponentiation]] 
# [[Multiplication]] and [[Division (mathematics)|division]]
# [[Addition]] and [[subtraction]]
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 parentheses 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 the [[Multiplicative inverse|reciprocal]] (multiplicative inverse) then the [[associative property|associative]] and [[commutative property|commutative]] laws of multiplication allow the factors in each [[Addition#Terms|term]] to be multiplied together in any order. Sometimes multiplication and division are given equal precedence, or sometimes multiplication is given higher precedence than division; see {{slink||Mixed division and multiplication}} below. If each subtraction is replaced with addition of the [[additive inverse|opposite]] (additive inverse), then the associative and commutative laws of addition allow terms to be added in any order.

The [[radical symbol]] {{tmath|\sqrt{\vphantom t} }} is traditionally extended 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.{{r|OMS}}{{r|NIST}}<ref group="lower-alpha" name="NB2"/> The parentheses can be omitted if the input is a single numerical variable or constant,{{r|BS}} as in the case of {{math|sin ''x'' {{=}} sin(''x'')}} and {{math|sin ''π'' {{=}} sin(''π'')}}.<ref group="lower-alpha" name="NB2"/>  Traditionally this convention extends to [[monomial]]s; thus, {{math|sin 3''x'' {{=}} sin(3''x'')}} and even {{math|sin {{sfrac|1|2}}''xy'' {{=}} sin({{sfrac|1|2}}''xy'')}}, but {{math|sin ''x'' + ''y'' {{=}} sin(''x'') + ''y''}}, because {{math|''x'' + ''y''}} is not a monomial. However, this convention is not universally understood, and some authors prefer explicit parentheses.<ref group="lower-alpha" name="NB3"/> Some calculators and programming languages require parentheses around function inputs, some do not.

{{anchor|Grouping}} Parentheses and alternate symbols of grouping can be used to override the usual order of operations or to make the intended order explicit. Grouped symbols can be treated as a single expression.{{r|BS}}

===Examples===

Multiplication before addition:
: <math>1 + 2 \times 3 = 1 + 6 = 7 .</math>

Parenthetical subexpressions are evaluated first:
: <math>(1 + 2) \times 3 = 3 \times 3 = 9 .</math>

Exponentiation before multiplication, multiplication before subtraction:
: <math>1 - 2 \times 3 ^ 4 = 1 - 2 \times 81 = 1 - 162 = -161 .</math>

When an expression is written as a superscript, the superscript is considered to be grouped by its position above its base:
: <math>1+2^{3+4} = 1+2^7 = 1+128 = 129 .</math>

The operand of a root symbol is determined by the overbar:
: <math>\sqrt{1 + 3} + 5 = \sqrt 4 + 5 = 2 + 5 = 7.</math>

A horizontal fractional line forms two grouped subexpressions, one above divided by another below:
: <math>\frac{1 + 2}{3 + 4} + 5 = \frac{3}{7} + 5.</math>

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 {{nowrap|{ }}} or square brackets {{nowrap|[ ]}}, are sometimes used along with parentheses {{nowrap|( )}}. For example: 
: <math>\bigl[ (1 + 2) \div (3 + 4) \bigr] + 5 = (3 \div 7) + 5 </math>

==Special cases==
===Unary minus sign===
There are differing conventions concerning the [[unary operation]]&nbsp;{{nowrap|'−'}} (usually pronounced "minus"). In written or printed mathematics, the expression −3<sup>2</sup> is interpreted to mean {{nowrap|−(3<sup>2</sup>) {{=}} −9}}.{{r|BS}}{{r|ARGS}}

In some applications and programming languages, notably [[Microsoft Excel]], [[PlanMaker]] (and other spreadsheet applications) and [[bc programming language|the 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 −3<sup>2</sup> will be interpreted as {{nowrap|(−3)<sup>2</sup> {{=}} 9}}.{{r|Microsoft 2005}} This does not apply to the binary minus {{nowrap|operation '−';}} 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 formulas <code>=0-2^2</code> and <code>=-(2^2)</code> return −4.

===Mixed division and multiplication===

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;{{r|Chrystal}} evaluating all multiplications first followed by divisions from left to right; or eschewing such expressions and instead always disambiguating them by explicit parentheses.{{r|Cajori}}

Beyond primary education, the symbol '÷' for division is seldom used, but is replaced by the use of [[algebraic fraction]]s,{{r|Wu}} typically written vertically with the numerator stacked above the denominator – which makes grouping explicit and unambiguous – but sometimes written inline using the [[Division slash|slash]] or solidus symbol '/'.{{r|iso}}

{{anchor|IMF}} Multiplication denoted by juxtaposition (also known as [[implied multiplication]]) creates a visual unit and is often given 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. {{nowrap|1&hairsp;/&hairsp;2''n''}} is interpreted to mean {{nowrap|1&hairsp;/&hairsp;(2&thinsp;·&thinsp;''n'')}} rather than {{nowrap|(1&hairsp;/&hairsp;2)&thinsp;·&thinsp;''n''}}.{{r|BS}}{{r|Chrystal}}{{r|Lennes}}{{r|Strogatz}} For instance, the manuscript submission instructions for the ''[[Physical Review]]'' journals directly state that multiplication has precedence over division,{{r|APS}} and this is also the convention observed in physics textbooks such as the ''[[Course of Theoretical Physics]]'' by [[Lev Landau|Landau]] and [[Evgeny Lifshitz|Lifshitz]]<ref group="lower-alpha" name="NB1"/> and mathematics textbooks such as ''[[Concrete Mathematics]]'' by [[Ronald Graham|Graham]], [[Donald Knuth|Knuth]], and [[Oren Patashnik| Patashnik]].{{r|GKP}} However, some authors recommend against expressions such as {{nowrap|''a''&hairsp;/&hairsp;''bc''}}, preferring the explicit use of parenthesis {{nowrap|''a''&hairsp;/&hairsp;(''bc'')}}.{{r|Peterson}}

More complicated cases are more ambiguous. For instance, the notation {{nowrap|1&hairsp;/&hairsp;2''π''(''a''&thinsp;+&thinsp;''b'')}} could plausibly mean either
{{nowrap|1&hairsp;/&hairsp;[2''π''&thinsp;·&thinsp;(''a''&thinsp;+&thinsp;''b'')]}} or {{nowrap|[1&hairsp;/&hairsp;(2''π'')]&thinsp;·&thinsp;(''a''&thinsp;+&thinsp;''b'')}}.{{r|FatemanCaspi}} Sometimes interpretation depends on context. The ''Physical Review'' submission instructions recommend against expressions of the form {{nowrap|''a''&hairsp;/&hairsp;''b''&hairsp;/&hairsp;''c''}}; more explicit expressions {{nowrap|(''a''&hairsp;/&hairsp;''b'')&hairsp;/&hairsp;''c''}} or {{nowrap|''a''&hairsp;/&hairsp;(''b''&hairsp;/&hairsp;''c'')}} are unambiguous.{{r|APS}}

[[File:Precedence62xplus.jpg|thumb|6÷2(1+2) is interpreted as 6÷(2×(1+2)) by a ''[[fx-82MS]]'' (upper), and (6÷2)×(1+2) by a ''[[TI-83 Plus]]'' calculator (lower), respectively.]]
This ambiguity has been the subject of [[Internet meme]]s such as "{{nowrap|8&thinsp;÷&thinsp;2(2&thinsp;+&thinsp;2)}}", for which there are two conflicting interpretations: 8&thinsp;÷&thinsp;[2&thinsp;·&thinsp;(2&thinsp;+&thinsp;2)] = 1 and (8&thinsp;÷&thinsp;2)&thinsp;·&thinsp;(2&thinsp;+&thinsp;2) = 16.{{r|Strogatz}}{{r|Haelle}} 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".{{r|Wu}}

===Serial exponentiation===
<!-- Section header "Serial exponentiation" used as redirect target! -->
If [[exponentiation]] is indicated by stacked symbols using superscript notation, the usual rule is to work from the top down:{{r|BS}}{{r|NIST}}
: {{nowrap|''a''<sup>''b''<sup>''c''</sup></sup> {{=}} ''a''<sup>(''b''<sup>''c''</sup>)</sup>}},
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.

However, when exponentiation is represented by an explicit symbol such as a [[caret]] (^) or [[arrow (symbol)|arrow]] (↑), there is no common standard. 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.

==Mnemonics==
{{anchor|PEMDAS|PEDMAS|BEDMAS|BEMDAS|BODMAS|BIDMAS|BIMDAS|BOMDAS|GEMDAS|BERDMAS|PERDMAS|BPODMAS}}
[[Mnemonic]] [[Acronym|acronyms]] are often taught in primary schools to help students remember the order of operations.{{r|Mathcentre}}{{r|Ginsburg}} The acronym '''''PEMDAS''''', which stands for '''P'''arentheses, '''E'''xponents, '''M'''ultiplication/'''D'''ivision, '''A'''ddition/'''S'''ubtraction,{{r|Vanderbeek}} is common in the [[Mathematics education in the United States|United States]]{{r|ASAT}} and France.{{r|Micmaths}} Sometimes the letters are expanded into words of a mnemonic sentence such as "Please Excuse My Dear Aunt Sally".{{r|Ball}} The United Kingdom and other [[Commonwealth of Nations|Commonwealth]] countries may use '''''BODMAS''''' (or sometimes '''''BOMDAS'''''), standing for '''B'''rackets, '''O'''f, '''D'''ivision/'''M'''ultiplication, '''A'''ddition/'''S'''ubtraction, with "of" meaning fraction multiplication.{{r|Davies|Knight}} Sometimes the '''O''' is instead expanded as '''O'''rder, meaning exponent or root,{{r|Knight|NSW syllabus}} or replaced by '''I''' for '''I'''ndices in the alternative mnemonic '''''BIDMAS'''''.{{r|Knight|Foster}} In Canada and New Zealand '''''BEDMAS''''' is common.{{r|Naddor}}

{{anchor|Punkt vor Strich}}In Germany, the convention is simply taught as {{lang|de|[[:de:Punktrechnung vor Strichrechnung|Punktrechnung vor Strichrechnung]]}}, "dot operations before line operations" referring to the graphical shapes of the taught operator signs {{unichar|B7|note=multiplication}}, {{unichar|2236|note=division}}, and {{unichar|2B|note=addition}}, {{unichar|2212|note=subtraction}}.

These mnemonics may be misleading when written this way.{{r|Ball}} For example, misinterpreting any of the above rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression{{r|Ball}} <math>a - b + c</math> as <math>a-(b+c)</math>, while the correct evaluation is <math>(a - b) + c</math>. These values are different when <math>c\ne 0</math>.

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.{{r|Ameis}}{{r|Cheng}} Students learning the order of operations via mnemonic acronyms routinely make mistakes,{{r|LLT}} as do some pre-service teachers.{{r|Dupree}} Even when students correctly learn the acronym, a disproportionate focus on memorization of trivia crowds out substantive mathematical content.{{r|Wu}} 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 a [[binary expression tree|tree-like hierarchy]] rather 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.{{r|Taff}}

==Calculators==
{{main article|Calculator input methods}}
Different calculators follow different orders of operations.{{r|BS}} Many simple calculators without a [[stack machine|stack]] implement [[chain 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, typing <code>1 + 2 × 3 =</code> yields 9, while a more sophisticated calculator will use a more standard priority, so typing <code>1 + 2 × 3 =</code> yields 7.

Calculators may associate exponents to the left or 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-30XS MultiView]] in "Mathprint mode", whereas it is interpreted as (''a''<sup>''b''</sup>)<sup>''c''</sup> on the [[TI-30XII]] and the TI-30XS MultiView in "Classic mode".

An expression like <code>1/2''x''</code> is interpreted as 1/(2''x'') by [[TI-82]],{{r|Peterson}} as well as many modern [[Casio]] calculators{{r|Casio}} (configurable on some like the [[Casio fx-9750GIII|fx-9750GIII]]), but as (1/2)''x'' by [[TI-83]] and every other TI calculator released since 1996,{{r|TI 2011}}{{r|Peterson}} 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]],{{r|TI 1982}} the latter is more in line with the rule that multiplication and division are of equal precedence.{{r|Peterson}}

When the user is unsure how a calculator will interpret an expression, parentheses can be used to remove the ambiguity.{{r|Peterson}}

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.{{r|Simons}}{{r|KS}} Hence, calculators utilizing [[reverse Polish notation]] (RPN) using a [[stack (data structure)|stack]] to enter expressions in the correct order of precedence do not need parentheses or any possibly model-specific order of execution.{{r|Ball}}{{r|Vanderbeek}}

==Programming languages==

Most [[programming languages]] use precedence levels that conform to the order commonly used in mathematics,{{r|Henderson}} 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).

Furthermore, 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 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>.

[[Dennis Ritchie]], creator of the [[C (programming language)|C language]], 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]].{{r|Ritchie}} Many programmers have become accustomed to this order, but more recent popular languages like [[Python (programming language)|Python]]{{r|Python}} and [[Ruby (programming language)|Ruby]]{{r|Ruby}} do have this order reversed. The relative precedence levels of operators found in many C-style languages are as follows:

{| class="wikitable"
|1 || <code>()</code> &nbsp; <code>[]</code> &nbsp; <code>-&gt;</code> &nbsp; <code>.</code> &nbsp; <code>::</code> || Function call, scope, array/member access
|-
|2 || <code>!</code> &nbsp; <code>~</code> &nbsp; <code>-</code> &nbsp; <code>+</code> &nbsp; <code>*</code> &nbsp; <code>&amp;</code> &nbsp; <u>[[sizeof|<code>sizeof</code>]]</u> &nbsp; ''[[Type conversion|type cast]]'' &nbsp; <code>++</code> &nbsp; <code>--</code> &nbsp; || (most) unary operators, [[sizeof]] and [[Type conversion|type casts]] (right to left)
|-
|3 || <code>*</code> &nbsp; <code>/</code> &nbsp; <code>%</code> <code>MOD</code> || Multiplication, division, [[modular arithmetic|modulo]]
|-
|4 || <code>+</code> &nbsp; <code>-</code> || Addition and subtraction
|-
|5 || <code>&lt;&lt;</code> &nbsp; <code>&gt;&gt;</code> || Bitwise shift left and right
|-
|6 || <code>&lt;</code> &nbsp; <code>&lt;=</code> &nbsp; <code>&gt;</code> &nbsp; <code>&gt;=</code> || Comparisons: less-than and greater-than
|-
|7 || <code>==</code> &nbsp; <code>!=</code> || Comparisons: equal and not equal
|-
|8 || <code>&amp;</code> || Bitwise AND
|-
|9 || <code>^</code> || Bitwise exclusive OR (XOR)
|-
|10 || <code><nowiki>|</nowiki></code> || Bitwise inclusive (normal) OR
|-
|11 || <code>&amp;&amp;</code> || Logical AND
|-
|12 || <code><nowiki>||</nowiki></code> || Logical OR
|-
|13 || <code>?</code> <code>:</code> || Conditional expression (ternary)
|-
|14 || <code>=</code> &nbsp; <code>+=</code> &nbsp; <code>-=</code> &nbsp; <code>*=</code> &nbsp; <code>/=</code> &nbsp; <code>%=</code> &nbsp; <code>&amp;=</code> &nbsp; <code><nowiki>|=</nowiki></code> &nbsp; <code>^=</code> &nbsp; <code>&lt;&lt;=</code> &nbsp; <code>&gt;&gt;=</code> || Assignment operators (right to left)
|-
|15 || <code>,</code> || [[Comma operator]]
|}
[[File:Algol grammar expr svg.svg|thumb|500px|Simplified [[Context-free grammar|formal grammar]] for arithmetical expressions in a programming language ''(left)'',{{r|Algol}} and derivation of the example expression <code>(a+b)^2/2</code> ''(right)''. The latter corresponds to a hierarchical structure ("[[abstract syntax tree|syntax tree]]") which is unique for the given expression. The [[compiler]] generates [[machine code]] from the tree in such a way that operations originating at the lowest hierarchy level are executed first.]]Examples:
* <code>!A + !B</code> is interpreted as <code>(!A) + (!B)</code>
* <code>++A + !B</code> is interpreted as <code>(++A) + (!B)</code>
* <code>A + B * C</code> is interpreted as <code>A + (B * C)</code>
* <code>A || B && C</code> is interpreted as <code>A || (B && C)</code>
* <code>A && B == C</code> is interpreted as <code>A && (B == C)</code>
* <code>A & B == C</code> is interpreted as <code>A & (B == C)</code>
(In [[Python (programming language)|Python]], [[Ruby (programming language)|Ruby]], [[PARI/GP]] and other popular languages, <code>A & B == C</code> is interpreted as <code>(A & B) == C</code>.)

[[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.

The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.{{r|Jones}}

==History==

The order of operations emerged progressively over centuries. The rule that multiplication has precedence over addition was incorporated into the development of [[Mathematical notation|algebraic notation]] in the 1600s, since the distributive property implies this as a natural hierarchy. As recently as the 1920s, the historian of mathematics [[Florian Cajori]] identifies 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 as ''a''/2''b'', which could be interpreted as ''a''/(2''b'') or (''a''/2)&nbsp;×&nbsp;''b'', imply that the conventions are not yet completely stable.<ref name="Jensen2013"/><ref name="Peterson2000"/>

==See also==
* [[Common operator notation]] (for a more formal description)
* [[Hyperoperation]]
* [[Logical connective#Order of precedence]]
* [[Operator associativity]]
* [[Operator overloading]]
* [[Operator precedence in C and C++]]
* [[Polish notation]]
* [[Reverse Polish notation]]

== Notes ==
{{Notelist|refs=

<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''}} [https://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.</ref>

<ref group="lower-alpha" name="NB2">Some authors deliberately avoid any omission of parentheses with functions even in the case of single numerical variable or constant arguments (i.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>

<ref group="lower-alpha" name="NB3">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>

}} <!-- END NOTELIST -->

==References==
{{reflist|refs=

<ref name = ASAT>{{cite journal
 | last1 = Ali Rahman | first1 = Ernna Sukinnah
 | last2 = Shahrill | first2 = Masitah
 | last3 = Abbas | first3 = Nor Arifahwati
 | last4 = Tan | first4 = Abby
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 | pages = 373–382
 | doi-broken-date = 13 December 2024
 | url = https://files.eric.ed.gov/fulltext/EJ1148460.pdf
 | quote-page = 373
 | quote = The PEMDAS is an acronym or mnemonic for the order of operations that stands for Parenthesis, Exponents, Multiplication, Division, Addition and Subtraction. This acronym is widely used in the United States of America. Meanwhile, in other countries such as United Kingdom and Canada, the acronyms used are BODMAS (Brackets, Order, Division, Multiplication, Addition and Subtraction) and BIDMAS (Brackets, Indices, Division, Multiplication, Addition and Subtraction).
 }}</ref>

<ref name=Ameis>{{cite journal
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 | jstor = 41183631
 }}</ref>
 
<ref name = APS>{{cite web
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 }}</ref>

<ref name = ARGS>{{cite book
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<ref name = Algol>{{cite report
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 | display-authors = 1
 | title = Revised Report on the Algorithmic Language Algol 60
 | chapter = § 3.3.1: Arithmetic expressions
 | last2 = Bauer | first2 = Friedrich Ludwig | last3 = Green | first3 = Julien | last4 = Katz | first4 = Charles | last5 = McCarthy | first5 = John | last6 = Naur | first6 = Peter | last7 = Perlis | first7 = Alan Jay | last8 = Rutishauser | first8 = Heinz | last9 = Samelson | first9 = Klaus | last10 = Vauquois | first10 = Bernard | last11 = Wegstein | first11 = Joseph Henry | last12 = van Wijngaarden | first12 = Adriaan | last13 = Woodger | first13 = Michael
 | url = https://www.masswerk.at/algol60/report.htm
 | access-date = 2023-09-17 }} (CACM Vol. 6 pp. 1–17; The Computer Journal, Vol. 9, p. 349; Numerische Mathematik, Vol. 4, p. 420.)
</ref>

<ref name = Ball>{{cite book
 | last = Ball | first = John A.
 | date = 1978
 | title = Algorithms for RPN calculators | edition = 1st
 | location = Cambridge, Mass | publisher = Wiley
 | isbn = 0-471-03070-8
 | page =  31
 | url = https://archive.org/details/algorithmsforrpn0000ball/page/31
 | url-access = registration
 }}</ref>

<ref name = BS>{{cite book
 | last1 = Bronstein | first1 = Ilja Nikolaevič | author1-link = Ilya Nikolaevich Bronshtein
 | last2 = Semendjajew | first2 = Konstantin Adolfovič | author2-link = Konstantin Adolfovic Semendyayev
 | date = 1987 | orig-date = 1945
 | chapter = 2.4.1.1. Definition arithmetischer Ausdrücke | trans-chapter = Definition of arithmetic expressions
 | title = [[Bronstein and Semendjajew|Taschenbuch der Mathematik]] | language = de | trans-title = Pocketbook of mathematics
 | volume = 1 | edition = 23rd
 | editor-first1 = Günter | editor-last1 = Grosche
 | editor-first2 = Viktor | editor-last2 = Ziegler
 | editor-first3 = Dorothea | editor-last3 = Ziegler
 | translator-first = Viktor | translator-last = Ziegler
 | place = Thun, Switzerland | publisher = [[Verlag Harri Deutsch|Harri Deutsch]]
 | 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&thinsp;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>

<ref name = Cajori>{{cite book
 | last = Cajori
 | first = Florian
 | author-link = Florian Cajori
 | year = 1928
 | title = [[A History of Mathematical Notations]]
 | volume = 1
 | place = La Salle, Illinois
 | publisher = Open Court
 | at = [https://archive.org/details/b29980343_0001/page/274/ §242. "Order of operations in terms containing both ÷ and ×"], {{pgs|274}} 
 }}</ref>

<ref name = Casio>{{cite web
 | publisher = [[Casio]]
 | title = Calculation Priority Sequence
 | date =
 | website = support.casio.com
 | url = https://support.casio.com/global/en/calc/manual/fx-82MS_85MS_220PLUS_300MS_350MS_en/technical_informatoin/sequence.html
 | access-date = 2019-08-01 
 }}</ref>

<ref name = Cheng>{{cite book
 | last = Cheng | first = Eugenia | author-link = Eugenia Cheng
 | year = 2023 | title = Is Math Real? How Simple Questions Lead Us to Mathematics' Deepest Truths
 | publisher = Basic Books | isbn = 978-1-541-60182-6 | pages = 235–238
 }}</ref>

<ref name=Chrystal>{{cite book
 | last = Chrystal | first = George | author-link = George Chrystal
 | year = 1904 | orig-year = 1886
 | title = Algebra | edition = 5th | volume = 1
 | at=[https://archive.org/details/algebraelementar01chryuoft/page/14/ "Division", Ch. 1 §§19–26], {{pgs|14–20}}
 }} {{pb}}
Chrystal's book was the canonical source in English about secondary school algebra of the turn of the 20th century, and plausibly the source for many later descriptions of the order of operations. However, while Chrystal's book initially establishes a rigid rule for evaluating expressions involving '÷' and '×' symbols, it later consistently gives implicit multiplication higher precedence than division when writing inline fractions, without ever explicitly discussing the discrepancy between formal rule and common practice.
</ref>

<ref name=Davies>{{cite journal
 | last = Davies | first = Peter
 | year = 1979
 | title = BODMAS Exposed
 | journal = Mathematics in School
 | volume = 8 | number = 4 | pages = 27–28
 | jstor = 30213488
 }}</ref>

<ref name = Dupree>{{cite journal
 | last = Dupree | first = Kami M.
 | year = 2016
 | title = Questioning the Order of Operations
 | journal = Mathematics Teaching in the Middle School
 | volume = 22 | number = 3
 | pages = 152–159
 | doi = 10.5951/mathteacmiddscho.22.3.0152
 }}</ref>

<ref name = Foster>{{cite journal
 | last = Foster | first = Colin
 | year = 2008
 | title = Higher Priorities
 | journal = Mathematics in School
 | volume = 37 | number = 3 | page = 17
 | jstor = 30216129
 | url = https://www.researchgate.net/publication/261812399
 }}</ref>

<ref name = Ginsburg>{{cite news
 | last = Ginsburg | first = David
 | date = 2011-01-01
 | title = Please Excuse My Dear Aunt Sally (PEMDAS)--Forever!
 | work = Education Week - Coach G's Teaching Tips
 | url = http://blogs.edweek.org/teachers/coach_gs_teaching_tips/2011/01/math_teachers_please_excuse_dear_aunt_sally--forever.html
 | access-date = 2023-09-17 
 }}</ref>

<ref name = GKP>{{cite book 
 | last1 = Graham | first1 = Ronald L. | author1-link = Ronald L. Graham
 | last2 = Knuth | first2 = Donald E. | author2-link = Donald E. Knuth
 | last3 = Patashnik | first3 = Oren | author3-link = Oren Patashnik
 | year = 1994
 | title = Concrete Mathematics | edition = 2nd
 | location = Reading, Mass | publisher = Addison-Wesley
 | at = "A Note on Notation", {{pgs|xi}}
 | isbn = 0-201-55802-5
 | mr = 1397498
 | quote = An expression of the form {{nobr|''a''/''bc''}} means the same as {{nobr|''a''/(''bc'')}}. Moreover, {{nobr|log ''x''/log ''y'' {{=}}}} {{nobr|(log ''x'')/(log ''y'')}} and {{nobr|2''n''! {{=}} 2(''n''!)}}.
 }}</ref>

<ref name = FatemanCaspi>{{cite conference
 | last1 = Fateman | first1 = R. J.
 | last2 = Caspi | first2 = E.
 | year = 1999
 | title = Parsing TEX into mathematics
 | conference = International Symposium on Symbolic and Algebraic Computation, Vancouver, 28–31 July 1999  
 | url = https://people.eecs.berkeley.edu/~fateman/papers/parsing_tex.pdf
 }}</ref>

<ref name = Haelle>{{cite web
 | last = Haelle | first = Tara
 | title = What ''Is'' the Answer to That Stupid Math Problem on Facebook? And why are people so riled up about it?
 | date = 2013-03-12
 | website = Slate
 | url = https://slate.com/technology/2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer.html
 | access-date = 2023-09-17 
 }}</ref>

<ref name = Henderson>{{cite book
 | last = Henderson | first = Harry
 | year = 2009 | orig-year = 2003
 | title = Henderson's Encyclopedia of Computer Science and Technology | edition = Rev.
 | chapter = Operator Precedence
 | place = New York | publisher = [[Facts On File, Inc.|Facts on File]]
 | isbn = 978-0-8160-6382-6
 | page = 355
 | chapter-url = https://books.google.com/books?id=3Tla6d153uwC&pg=PA355
 | access-date = 2023-09-17
 }}</ref>

<ref name = iso>In the [[ISO/IEC 80000|ISO 80000]] standard, the division symbol '÷' is entirely disallowed in favor of a slash symbol: {{pb}} ISO 80000-2:2019, [https://www.iso.org/obp/ui/#iso:std:iso:80000:-2:ed-2:v2:en "Quantities and units – Part 2: Mathematics"]. [[International Standards Organization]].
</ref>

<ref name="Jensen2013">{{cite web |last1=Jensen |first1=Patricia |title=History and Background |url=http://5010.mathed.usu.edu/Fall2013/PJensen/History.html |website=5010.mathed.usu.edu |publisher=Utah State University |access-date=4 October 2024}}</ref>

<ref name = Jones>{{cite journal
 | last = Jones | first = Derek M.
 | date = 2008 | orig-date = 2006
 | title = Developer beliefs about binary operator precedence
 | journal = CVu
 | volume = 18 | issue = 4
 | pages = 14–21
 | url = http://www.knosof.co.uk/cbook/accu06.html
 | access-date = 2023-09-17 
 }}</ref>

<ref name=Knight>{{cite journal
 | last = Knight | first = I. S.
 | year = 1997
 | title = Why BODMAS?
 | journal = The Mathematical Gazette
 | volume = 81 | number = 492
 | pages = 426–427
 | doi = 10.2307/3619621
 | jstor = 3619621
 }}</ref>
 
<ref name = KS>{{cite journal
 | last1 = Krtolica | first1 = Predrag V.
 | last2 = Stanimirović | first2 = Predrag S.
 | date = 1999
 | title = On some properties of reverse Polish Notation
 | journal = Filomat
 | volume = 13
 | pages = 157–172
 | jstor = 43998756
 }}</ref>

<ref name=Lennes>{{cite journal
 | last = Lennes | first = N. J.
 | year = 1917
 | title = Discussions: Relating to the Order of Operations in Algebra 
 | journal = The American Mathematical Monthly
 | volume = 24 | number = 2
 | pages = 93–95
 | doi = 10.2307/2972726
 | jstor = 2972726
 }}</ref>

<ref name = LLT>{{cite journal
 | last1 = Lee | first1 = Jae Ki
 | last2 = Licwinko | first2 = Susan
 | last3 = Taylor-Buckner | first3 = Nicole
 | year = 2013
 | title = Exploring Mathematical Reasoning of the Order of Operations: Rearranging the Procedural Component PEMDAS
 | journal = [[Journal of Mathematics Education at Teachers College]]
 | volume = 4 | number = 2
 | pages = 73–78
 | doi = 10.7916/jmetc.v4i2.633
 | url = https://journals.library.columbia.edu/index.php/jmetc/article/download/633/79
 | quote-page = 73
 | quote = [...] students frequently make calculation errors with expressions which have either multiplication and division or addition and subtraction next to each other. [...]
 }}</ref>

<ref name = Mathcentre>{{cite web
 | title = Rules of arithmetic
 | website = Mathcentre.ac.uk
 | date = 2009
 | url = http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-rules-2009-1.pdf
 | access-date = 2019-08-02 
 }}</ref>

<ref name = Micmaths>{{cite web
 | title = Le calcul qui divise : 6÷2(1+2)  | language = fr
 | work = Micmaths
 | date = 17 November 2020
 | type = Video
 | url = https://www.youtube.com/watch?v=tYf3CpbqAVo
 }}</ref>

<ref name = "Microsoft 2005">{{cite web
 | title = Formula Returns Unexpected Positive Value
 | publisher = [[Microsoft]]
 | date = 2005-08-15
 | url = https://support.microsoft.com/en-gb/kb/kbview/132686
 | access-date = 2012-03-05
 | url-status = dead
 | archive-url = https://web.archive.org/web/20150419091629/https://support.microsoft.com/en-gb/kb/kbview/132686
 | archive-date = 2015-04-19
 }}</ref>

<ref name = Microsoft>{{cite web
 | publisher = [[Microsoft]]
 | date = 2023
 | title = Calculation operators and precedence: Excel
 | work = Microsoft Support
 | url = https://support.microsoft.com/en-us/office/calculation-operators-and-precedence-36de9366-46fe-43a3-bfa8-cf6d8068eacc
 | access-date = 2023-09-17
 }}</ref>

<ref name=Naddor>{{cite thesis
 | last = Naddor | first = Josh
 | year = 2020
 | title = Order of Operations: Please Excuse My Dear Aunt Sally as her rule is deceiving
 | type = MA thesis
 | publisher = University of Georgia
 | url = https://esploro.libs.uga.edu/esploro/outputs/9949365543302959
 }}</ref>

<ref name = "NSW syllabus">{{cite web
 | title = Order of operations
 | website = Syllabus.bos.nsw.edu.au
 | format = DOC
 | url = http://syllabus.bos.nsw.edu.au/assets/global/files/maths_s3_sampleu1.doc
 | access-date = 2019-08-02 
 }}</ref>

<ref name = OMS>{{cite book
 | last1 = Oldham | first1 = Keith B.
 | last2 = Myland | first2 = Jan C.
 | last3 = Spanier | first3 = Jerome
 | year = 2009 | orig-year = 1987
 | title = An Atlas of Functions: with Equator, the Atlas Function Calculator | edition = 2nd
 | publisher = Springer
 | isbn = 978-0-387-48806-6
 | doi = 10.1007/978-0-387-48807-3
 }}</ref>

<ref name="Peterson2000">{{cite web |last1=Doctor Peterson, The Math Forum |title=History of the Order of Operations |url=http://mathforum.org/library/drmath/view/52582.html |website=Ask Dr Math |access-date=4 October 2024 |archive-url=https://web.archive.org/web/20020619105005/http://mathforum.org/library/drmath/view/52582.html |archive-date=June 19, 2002 |date=November 22, 2000 |url-status=dead}}</ref>

<ref name = NIST>{{cite book
 | editor-last = Olver | editor-first = Frank W. J.
 | editor2-last = Lozier | editor2-first = Daniel W.
 | editor3-last = Boisvert | editor3-first = Ronald F.
 | editor4-last = Clark | editor4-first = Charles W.
 | date = 2010
 | title = NIST Handbook of Mathematical Functions
 | title-link = NIST Handbook of Mathematical Functions
 | publisher = [[National Institute of Standards and Technology]]
 | isbn = 978-0-521-19225-5
 | mr = 2723248
 }}</ref>

<ref name = Peterson>
Peterson, Dave (September–October 2019). ''The Math Doctors'' (blog). Order of Operations: [https://www.themathdoctors.org/order-of-operations-why/ "Why?"]; [https://www.themathdoctors.org/order-of-operations-why-these-rules/  "Why These Rules?"]; [https://www.themathdoctors.org/order-of-operations-subtle-distinctions/ "Subtle Distinctions"]; [https://www.themathdoctors.org/order-of-operations-fractions-evaluating-and-simplifying/ "Fractions, Evaluating, and Simplifying"]; [https://www.themathdoctors.org/order-of-operations-implicit-multiplication/ "Implicit Multiplication?"]; [https://www.themathdoctors.org/order-of-operations-historical-caveats/ "Historical Caveats"]. Retrieved 2024-02-11. {{br}}
Peterson, Dave (August–September 2023). ''The Math Doctors'' (blog). Implied Multiplication: [https://www.themathdoctors.org/implied-multiplication-1-not-as-bad-as-you-think/ "Not as Bad as You Think"]; [https://www.themathdoctors.org/implied-multiplication-2-is-there-a-standard/ "Is There a Standard?"]; [https://www.themathdoctors.org/implied-multiplication-3-you-cant-prove-it/ "You Can't Prove It"]. Retrieved 2024-02-11.
</ref>

<ref name = Python>{{Cite web
 | title = 6. Expressions
 | website = Python documentation
 | url = https://docs.python.org/3/reference/expressions.html
 | access-date = 2023-12-31 
 }}</ref>

<ref name = Ritchie>{{cite book
 | last = Ritchie | first = Dennis M. | author-link = Dennis M. Ritchie
 | year = 1996
 | title = History of Programming Languages | edition = 2
 | chapter = The Development of the C Language
 | publisher = [[ACM Press]]
 | url = https://www.bell-labs.com/usr/dmr/www/chist.html
 }}</ref>

<ref name = Ruby>{{Cite web
 | title = precedence - RDoc Documentation
 | url = https://docs.ruby-lang.org/en/master/syntax/precedence_rdoc.html
 }}</ref>

<ref name = Simons>{{cite encyclopedia
 | last = Simons | first = Peter Murray | author-link = Peter Murray Simons
 | date = 2021
 | title = Łukasiewicz's Parenthesis-Free or Polish Notation
 | encyclopedia = [[Stanford Encyclopedia of Philosophy]]
 | publisher = Dept. of Philosophy, Stanford University
 | url = https://plato.stanford.edu/entries/lukasiewicz/polish-notation.html
 | access-date = 2022-03-26 
 }}</ref>

<ref name=Strogatz>{{cite news
 | last = Strogatz | first = Steven | author-link = Steven Strogatz
 | date = 2019-08-02
 | title = The Math Equation That Tried to Stump the Internet
 | newspaper = The New York Times
 | url = https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.html
 | access-date = 2024-02-12
 }} In this article, Strogatz describes the order of operations as taught in middle school. However, [https://nyti.ms/32mhext#permid=101802829 in a comment], he points out, {{pb}} "Several commenters appear to be using a different (and more sophisticated) convention than the elementary PEMDAS convention I described in the article. In this more sophisticated convention, which is often used in algebra, implicit multiplication (also known as multiplication by juxtaposition) is given higher priority than explicit multiplication or explicit division (in which one explicitly writes operators like × *  / or  ÷). Under this more sophisticated convention, the implicit multiplication in 2(2 + 2) is given higher priority than the explicit division implied by the use of ÷. That’s a very reasonable convention, and I agree that the answer is 1 if we are using this sophisticated convention. {{pb}} "But that convention is not universal. For example, the calculators built into Google and WolframAlpha use the less sophisticated convention that I described in the article; they make no distinction between implicit and explicit multiplication when they are asked to evaluate simple arithmetic expressions. [...]"
</ref>

<ref name = Swokowski>{{cite book
 | last = Swokowski | first = Earl William
 | date = 1978
 | title = Fundamentals of Algebra and Trigonometry | edition = 4
 | place = Boston | publisher = Prindle, Weber & Schmidt
 | isbn = 0-87150-252-6
 | url = https://books.google.com/books?id=bZdfLRHFGw8C
 | quote-page = 1
 | quote = The ''language of algebra'' [...] may be used as shorthand, to abbreviate and simplify long or complicated statements.
 }}</ref>

<ref name=Taff>{{cite journal
 | last = Taff | first = Jason
 | year = 2017
 | title = Rethinking the Order of Operations (or What Is the Matter with Dear Aunt Sally?)
 | journal = The Mathematics Teacher
 | volume = 111 | number = 2
 | pages = 126–132
 | doi = 10.5951/mathteacher.111.2.0126
 }}</ref>

<ref name = "TI 1982">{{cite book
 | publisher = [[Texas Instruments]]
 | year = 1982<!--or 1983-->
 | title = Announcing the TI Programmable 88!
 | url = http://www.datamath.net/Leaflets/TI-88_Announcement.pdf
 | access-date = 2017-08-03
 | quote = Now, implied multiplication is recognized by the [[Algebraic Operating System|AOS]] and the square root, logarithmic, and trigonometric functions can be followed by their arguments as when working with pencil and paper.}} (NB. The TI-88 only existed as a prototype and was never released to the public.)
</ref>

<ref name = "TI 2011">{{cite web
 | publisher = [[Texas Instruments]]
 | title = Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators
 | year = 2011
 | url = http://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110
 | access-date = 2015-08-24 
 }}</ref>

<ref name = Vanderbeek>{{cite book
 | last = Vanderbeek | first = Greg
 | year = 2007
 | title = Order of Operations and RPN
 | type = Expository paper
 | series = Master of Arts in Teaching (MAT) Exam Expository Papers
 | id = Paper 46
 | place = Lincoln | publisher = University of Nebraska
 | url = https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1045&context=mathmidexppap
 | access-date = 2020-06-14 
 }}</ref>

<ref name = Mathworld>{{cite web
 | last = Weisstein | first = Eric Wolfgang | author-link = Eric Wolfgang Weisstein
 | title = Precedence
 | website = [[MathWorld]]
 | url = https://mathworld.wolfram.com/Precedence.html
 | access-date = 2020-08-22 
 }}</ref>

<ref name = Wu>{{cite web
 | first = Hung-Hsi | last = Wu
 | year = 2007 | orig-year = 2004
 | title = "Order of operations" and other oddities in school mathematics
 | publisher = Dept. of Mathematics, University of California
 | url = https://math.berkeley.edu/~wu/order5.pdf
 | access-date = 2007-07-03
 }}</ref>

}} <!-- END REFLIST -->

== Further reading ==

* {{cite conference
 | editor-first1 = Michael | editor-last1 = Fothe
 | editor-first2 = Thomas | editor-last2 = Wilke
 | year = 2015
 | title = Keller, Stack und automatisches Gedächtnis – eine Struktur mit Potenzial | language = de
 | trans-title = Cellar, stack and automatic memory – a structure with potential 
 | conference = Kolloquium 14 Nov 2014 in Jena, Germany
 | place = Bonn | publisher = Gesellschaft für Informatik
 | isbn = 978-3-88579-426-4
 | url = https://dl.gi.de/bitstream/handle/20.500.12116/4381/lni-t-7.pdf?sequence=1&isAllowed=y
 }}

==  External links ==

* {{cite web
 | last = Bergman | first = George Mark | author-link = George Mark Bergman
 | year = 2013
 | title = Order of arithmetic operations; in particular, the 48/2(9+3) question.
 | publisher = Dept. of Mathematics, University of California
 | url = https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html
 | access-date = 2020-07-22 }}
* Zachary, Joseph L. (1997) "Operator Precedence", supplement to ''Introduction to Scientific Programming''. University of Utah. [http://www.cs.utah.edu/~zachary/isp/worksheets/operprec/operprec.html Maple worksheet], [http://www.cs.utah.edu/~zachary/ispmma/notebooks/operprec/operprec.html Mathematica notebook].


{{DEFAULTSORT:Order Of Operations}}
[[Category:Mnemonics]]
[[Category:Operators (programming)]]
[[Category:Algebra]]