# 運算次序

## 基本規則

1. 乘冪方根 ${\displaystyle x^{y}}$ ${\displaystyle {\sqrt {}}}$
2. ${\displaystyle \times }$ ${\displaystyle \div }$
3. ${\displaystyle +}$ ${\displaystyle -}$

${\displaystyle {\sqrt {1+3}}+5={\sqrt {4}}+5=2+5=7.}$

${\displaystyle {\frac {1+2}{3+4}}+5={\frac {3}{7}}+5.}$

${\displaystyle [(1+2)-3]-(4-5)=[3-3]-(-1)=1.}$

### 乘除混合

1 ÷ 2 × x = 1 × 1/2 × x = 1/2 × x

## 記憶術

• 中文地區一般以「先乘除後加減」作為口訣，但是當中沒有涵蓋指數
• 英國使用「BOMDAS」或「BIMDAS」，與前者相比，這兩個縮寫純粹改變了指數的詞彙（Orders/Indices）[9]
• 其他英語地區普遍使用「BOMDAS」

10 - 3 + 2

abc = a(bc)

(10 ÷ 5) ÷ 2

10 ÷ (5 ÷ 2)

## 資料來源

1. Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič. 2.4.1.1.. Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (编). Taschenbuch der Mathematik 1. 由Ziegler, Viktor翻译. Weiß, Jürgen 23. Thun, Switzerland / Frankfurt am Main, Germany: Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig). 1987: 115–120 [1945]. ISBN 3-87144-492-8 （德语）.
2. ^ Ask Dr. Math. Math Forum. 2000-11-22 [2012-03-05]. （原始内容存档于2021-04-21）.
3. ^ Angel, Allen R. Elementary Algebra for College Students 8. . Chapter 1, Section 9, Objective 3.
4. ^ Formula Returns Unexpected Positive Value. Microsoft. 2005-08-15 [2012-03-05]. （原始内容存档于2015-04-19）.
5. Ball, John A. Algorithms for RPN calculators 1. Cambridge, Massachusetts, USA: Wiley-Interscience, John Wiley & Sons, Inc. 1978: 31. ISBN 0-471-03070-8.
6. ^ Rules of arithmetic (PDF). Mathcentre.ac.uk. [2019-08-02]. （原始内容存档 (PDF)于2021-02-24）.
7. ^ Physical Review Style and Notation Guide (PDF). American Physical Society. Section IV–E–2–e. [2012-08-05]. （原始内容存档 (PDF)于2013-04-20）.
8. Vanderbeek, Greg. Order of Operations and RPN (Expository paper). Master of Arts in Teaching (MAT) Exam Expository Papers. Lincoln, Nebraska, USA: University of Nebraska. June 2007 [2020-06-14]. Paper 46. （原始内容存档于2020-06-14）.
9. ^ Order of operations (DOC). Syllabus.bos.nsw.edu.au. [2019-08-02]. （原始内容存档于2021-02-24）.
10. ^ Robinson, Raphael Mitchel. A report on primes of the form k · 2n + 1 and on factors of Fermat numbers (PDF). Proceedings of the American Mathematical Society (University of California, Berkeley, California, USA). October 1958, 9 (5): 673–681 [677] [1958-04-07] [2020-06-28]. doi:10.1090/s0002-9939-1958-0096614-7. （原始内容 (PDF)存档于2020-06-28）.
11. ^ Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W. (编). NIST Handbook of Mathematical Functions. National Institute of Standards and Technology (NIST), U.S. Department of Commerce, Cambridge University Press. 2010. ISBN 978-0-521-19225-5. MR 2723248.[1] Archive.is存檔，存档日期2013-07-03
12. ^ Zeidler, Eberhard; Schwarz, Hans Rudolf; Hackbusch, Wolfgang; Luderer, Bernd; Blath, Jochen; Schied, Alexander; Dempe, Stephan; Wanka, Gert; Hromkovič, Juraj; Gottwald, Siegfried. Zeidler, Eberhard , 编. Springer-Handbuch der Mathematik I I 1. Berlin / Heidelberg, Germany: Springer Spektrum, Springer Fachmedien Wiesbaden. 2013: 590 [2012]. ISBN 978-3-658-00284-8. doi:10.1007/978-3-658-00285-5. ISBN 3-658-00284-0 （德语）. (xii+635 pages)
13. ^ Van Winkle, Lewis. Exponentiation Associativity and Standard Math Notation. Codeplea - Random thoughts on programming. 2016-08-23 [2016-09-20]. （原始内容存档于2020-06-28）.