Editing Order of operations (section)

==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
 | year = 2017
 | title = Developing Students' Mathematical Skills Involving Order of Operations
 | journal = International Journal of Research in Education and Science
 | volume = 3 | issue = 2
 | doi = 10.21890/ijres.327896
 | 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
 | last = Ameis | first = Jerry A.
 | year = 2011
 | title = The Truth About PEDMAS
 | journal = Mathematics Teaching in the Middle School
 | volume = 16 | number = 7
 | pages = 414–420
 | doi = 10.5951/MTMS.16.7.0414
 | jstor = 41183631
 }}</ref>
 
<ref name = APS>{{cite web
 | publisher = [[American Physical Society]]
 | year = 2012
 | title = Physical Review Style and Notation Guide
 | at = § IV.E.2.e
 | url = https://publish.aps.org/files/styleguide-pr.pdf
 | access-date = 2012-08-05 
 }}</ref>

<ref name = ARGS>{{cite book
 | last1 = Angel | first1 = Allen R.
 | last2 = Runde | first2 = Dennis C.
 | last3 = Gilligan | first3 = Lawrence
 | last4 = Semmler | first4 = Richard
 | year = 2010
 | title = Elementary Algebra for College Students  | edition = 8th
 | publisher = [[Prentice Hall]]
 | isbn = 978-0-321-62093-4
 | at = Ch. 1, §9, Objective 3
 }}</ref>

<ref name = Algol>{{cite report
 | last1 = Backus | first1 = John Warner  | author1-link = John Warner Backus
 | editor-first = Peter | editor-last = Naur | editor-link = Peter Naur
 | year = 1963
 | 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 -->