Order of operations (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Order of operations" in English language version.

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Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010). NIST Handbook of Mathematical Functions. National Institute of Standards and Technology. ISBN 978-0-521-19225-5. MR 2723248.
Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994). Concrete Mathematics (2nd ed.). Reading, Mass: Addison-Wesley. "A Note on Notation", p. xi. ISBN 0-201-55802-5. MR 1397498. An expression of the form a/bc means the same as a/(bc). Moreover, log x/log y = (log x)/(log y) and 2n! = 2(n!).

aps.org

publish.aps.org

"Physical Review Style and Notation Guide" (PDF). American Physical Society. 2012. § IV.E.2.e. Retrieved 2012-08-05.

archive.org

Chrystal, George (1904) [1886]. Algebra. Vol. 1 (5th ed.). "Division", Ch. 1 §§19–26, pp. 14–20.

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.
Cajori, Florian (1928). A History of Mathematical Notations. Vol. 1. La Salle, Illinois: Open Court. §242. "Order of operations in terms containing both ÷ and ×", p. 274.
Ball, John A. (1978). Algorithms for RPN calculators (1st ed.). Cambridge, Mass: Wiley. p. 31. ISBN 0-471-03070-8.

bell-labs.com

Ritchie, Dennis M. (1996). "The Development of the C Language". History of Programming Languages (2 ed.). ACM Press.

berkeley.edu

math.berkeley.edu

Wu, Hung-Hsi (2007) [2004]. ""Order of operations" and other oddities in school mathematics" (PDF). Dept. of Mathematics, University of California. Retrieved 2007-07-03.

people.eecs.berkeley.edu

Fateman, R. J.; Caspi, E. (1999). Parsing TEX into mathematics (PDF). International Symposium on Symbolic and Algebraic Computation, Vancouver, 28–31 July 1999.

books.google.com

Swokowski, Earl William (1978). Fundamentals of Algebra and Trigonometry (4 ed.). Boston: Prindle, Weber & Schmidt. ISBN 0-87150-252-6. p. 1: The language of algebra [...] may be used as shorthand, to abbreviate and simplify long or complicated statements.
Henderson, Harry (2009) [2003]. "Operator Precedence". Henderson's Encyclopedia of Computer Science and Technology (Rev. ed.). New York: Facts on File. p. 355. ISBN 978-0-8160-6382-6. Retrieved 2023-09-17.

bos.nsw.edu.au

syllabus.bos.nsw.edu.au

"Order of operations" (DOC). Syllabus.bos.nsw.edu.au. Retrieved 2019-08-02.

caltech.edu

feynmanlectures.caltech.edu

For example, the third edition of Mechanics by Landau and Lifshitz contains expressions such as hP_z/2 $π$ (p. 22), and the first volume of the Feynman Lectures contains expressions such as 1/2√N (p. 6–7). In both books, these expressions are written with the convention that the solidus is evaluated last.

casio.com

support.casio.com

"Calculation Priority Sequence". support.casio.com. Casio. Retrieved 2019-08-01.

columbia.edu

journals.library.columbia.edu

Lee, Jae Ki; Licwinko, Susan; Taylor-Buckner, Nicole (2013). "Exploring Mathematical Reasoning of the Order of Operations: Rearranging the Procedural Component PEMDAS". Journal of Mathematics Education at Teachers College. 4 (2): 73–78. doi:10.7916/jmetc.v4i2.633. p. 73: [...] students frequently make calculation errors with expressions which have either multiplication and division or addition and subtraction next to each other. [...]

datamath.net

Announcing the TI Programmable 88! (PDF). Texas Instruments. 1982. Retrieved 2017-08-03. Now, implied multiplication is recognized by the 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.)

doi.org

Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) [1987]. An Atlas of Functions: with Equator, the Atlas Function Calculator (2nd ed.). Springer. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48806-6.
Lennes, N. J. (1917). "Discussions: Relating to the Order of Operations in Algebra". The American Mathematical Monthly. 24 (2): 93–95. doi:10.2307/2972726. JSTOR 2972726.
Ali Rahman, Ernna Sukinnah; Shahrill, Masitah; Abbas, Nor Arifahwati; Tan, Abby (2017). "Developing Students' Mathematical Skills Involving Order of Operations" (PDF). International Journal of Research in Education and Science. 3 (2): 373–382. doi:10.21890/ijres.327896 (inactive 2024-12-13). p. 373: 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).{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link)
Knight, I. S. (1997). "Why BODMAS?". The Mathematical Gazette. 81 (492): 426–427. doi:10.2307/3619621. JSTOR 3619621.
Ameis, Jerry A. (2011). "The Truth About PEDMAS". Mathematics Teaching in the Middle School. 16 (7): 414–420. doi:10.5951/MTMS.16.7.0414. JSTOR 41183631.
Lee, Jae Ki; Licwinko, Susan; Taylor-Buckner, Nicole (2013). "Exploring Mathematical Reasoning of the Order of Operations: Rearranging the Procedural Component PEMDAS". Journal of Mathematics Education at Teachers College. 4 (2): 73–78. doi:10.7916/jmetc.v4i2.633. p. 73: [...] students frequently make calculation errors with expressions which have either multiplication and division or addition and subtraction next to each other. [...]
Dupree, Kami M. (2016). "Questioning the Order of Operations". Mathematics Teaching in the Middle School. 22 (3): 152–159. doi:10.5951/mathteacmiddscho.22.3.0152.
Taff, Jason (2017). "Rethinking the Order of Operations (or What Is the Matter with Dear Aunt Sally?)". The Mathematics Teacher. 111 (2): 126–132. doi:10.5951/mathteacher.111.2.0126.

ed.gov

files.eric.ed.gov

Ali Rahman, Ernna Sukinnah; Shahrill, Masitah; Abbas, Nor Arifahwati; Tan, Abby (2017). "Developing Students' Mathematical Skills Involving Order of Operations" (PDF). International Journal of Research in Education and Science. 3 (2): 373–382. doi:10.21890/ijres.327896 (inactive 2024-12-13). p. 373: 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).{{cite journal}}: CS1 maint: DOI inactive as of December 2024 (link)

edweek.org

blogs.edweek.org

Ginsburg, David (2011-01-01). "Please Excuse My Dear Aunt Sally (PEMDAS)--Forever!". Education Week - Coach G's Teaching Tips. Retrieved 2023-09-17.

iso.org

In the ISO 80000 standard, the division symbol '÷' is entirely disallowed in favor of a slash symbol:
ISO 80000-2:2019, "Quantities and units – Part 2: Mathematics". International Standards Organization.

jstor.org

Lennes, N. J. (1917). "Discussions: Relating to the Order of Operations in Algebra". The American Mathematical Monthly. 24 (2): 93–95. doi:10.2307/2972726. JSTOR 2972726.
Davies, Peter (1979). "BODMAS Exposed". Mathematics in School. 8 (4): 27–28. JSTOR 30213488.
Knight, I. S. (1997). "Why BODMAS?". The Mathematical Gazette. 81 (492): 426–427. doi:10.2307/3619621. JSTOR 3619621.
Foster, Colin (2008). "Higher Priorities". Mathematics in School. 37 (3): 17. JSTOR 30216129.
Ameis, Jerry A. (2011). "The Truth About PEDMAS". Mathematics Teaching in the Middle School. 16 (7): 414–420. doi:10.5951/MTMS.16.7.0414. JSTOR 41183631.
Krtolica, Predrag V.; Stanimirović, Predrag S. (1999). "On some properties of reverse Polish Notation". Filomat. 13: 157–172. JSTOR 43998756.

knosof.co.uk

Jones, Derek M. (2008) [2006]. "Developer beliefs about binary operator precedence". CVu. 18 (4): 14–21. Retrieved 2023-09-17.

masswerk.at

Backus, John Warner; et al. (1963). "§ 3.3.1: Arithmetic expressions". In Naur, Peter (ed.). Revised Report on the Algorithmic Language Algol 60 (Report). Retrieved 2023-09-17. (CACM Vol. 6 pp. 1–17; The Computer Journal, Vol. 9, p. 349; Numerische Mathematik, Vol. 4, p. 420.)

mathcentre.ac.uk

"Rules of arithmetic" (PDF). Mathcentre.ac.uk. 2009. Retrieved 2019-08-02.

mathforum.org

Doctor Peterson, The Math Forum (2000-11-22). "History of the Order of Operations". Ask Dr Math. Archived from the original on 2002-06-19. Retrieved 2024-10-04.

microsoft.com

support.microsoft.com

"Calculation operators and precedence: Excel". Microsoft Support. Microsoft. 2023. Retrieved 2023-09-17.
"Formula Returns Unexpected Positive Value". Microsoft. 2005-08-15. Archived from the original on 2015-04-19. Retrieved 2012-03-05.

nyti.ms

Strogatz, Steven (2019-08-02). "The Math Equation That Tried to Stump the Internet". The New York Times. Retrieved 2024-02-12. In this article, Strogatz describes the order of operations as taught in middle school. However, in a comment, he points out,
"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.
"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. [...]"

nytimes.com

Strogatz, Steven (2019-08-02). "The Math Equation That Tried to Stump the Internet". The New York Times. Retrieved 2024-02-12. In this article, Strogatz describes the order of operations as taught in middle school. However, in a comment, he points out,
"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.
"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. [...]"

python.org

docs.python.org

"6. Expressions". Python documentation. Retrieved 2023-12-31.

researchgate.net

Foster, Colin (2008). "Higher Priorities". Mathematics in School. 37 (3): 17. JSTOR 30216129.

ruby-doc.org

"precedence - RDoc Documentation". ruby-doc.org. Retrieved 2023-12-31.

slate.com

Haelle, Tara (2013-03-12). "What Is the Answer to That Stupid Math Problem on Facebook? And why are people so riled up about it?". Slate. Retrieved 2023-09-17.

stanford.edu

plato.stanford.edu

Simons, Peter Murray (2021). "Łukasiewicz's Parenthesis-Free or Polish Notation". Stanford Encyclopedia of Philosophy. Dept. of Philosophy, Stanford University. Retrieved 2022-03-26.

themathdoctors.org

Peterson, Dave (September–October 2019). The Math Doctors (blog). Order of Operations: "Why?"; "Why These Rules?"; "Subtle Distinctions"; "Fractions, Evaluating, and Simplifying"; "Implicit Multiplication?"; "Historical Caveats". Retrieved 2024-02-11.
Peterson, Dave (August–September 2023). The Math Doctors (blog). Implied Multiplication: "Not as Bad as You Think"; "Is There a Standard?"; "You Can't Prove It". Retrieved 2024-02-11.

ti.com

epsstore.ti.com

"Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators". Texas Instruments. 2011. Retrieved 2015-08-24.

uga.edu

esploro.libs.uga.edu

Naddor, Josh (2020). Order of Operations: Please Excuse My Dear Aunt Sally as her rule is deceiving (MA thesis). University of Georgia.

unl.edu

digitalcommons.unl.edu

Vanderbeek, Greg (2007). Order of Operations and RPN (Expository paper). Master of Arts in Teaching (MAT) Exam Expository Papers. Lincoln: University of Nebraska. Paper 46. Retrieved 2020-06-14.

usu.edu

5010.mathed.usu.edu

Jensen, Patricia. "History and Background". 5010.mathed.usu.edu. Utah State University. Retrieved 2024-10-04.

web.archive.org

"Formula Returns Unexpected Positive Value". Microsoft. 2005-08-15. Archived from the original on 2015-04-19. Retrieved 2012-03-05.
Doctor Peterson, The Math Forum (2000-11-22). "History of the Order of Operations". Ask Dr Math. Archived from the original on 2002-06-19. Retrieved 2024-10-04.

Order of operations (English Wikipedia)

ams.org

mathscinet.ams.org

aps.org

publish.aps.org

archive.org

bell-labs.com

berkeley.edu

math.berkeley.edu

people.eecs.berkeley.edu

books.google.com

bos.nsw.edu.au

syllabus.bos.nsw.edu.au

caltech.edu

feynmanlectures.caltech.edu

casio.com

support.casio.com

columbia.edu

journals.library.columbia.edu

datamath.net

doi.org

ed.gov

files.eric.ed.gov

edweek.org

blogs.edweek.org

iso.org

jstor.org

knosof.co.uk

masswerk.at

mathcentre.ac.uk

mathforum.org

microsoft.com

support.microsoft.com

nyti.ms

nytimes.com

python.org

docs.python.org

researchgate.net

ruby-doc.org

slate.com

stanford.edu

plato.stanford.edu

themathdoctors.org

ti.com

epsstore.ti.com

uga.edu

esploro.libs.uga.edu

unl.edu

digitalcommons.unl.edu

usu.edu

5010.mathed.usu.edu

web.archive.org

wolfram.com

mathworld.wolfram.com

youtube.com