Editing Order of operations (section)

===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}}