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according to the order of operations, what is not calculated after exponentiation?

Gild in which mathematical operations are performed

In mathematics and computer programming, the lodge of operations (or operator precedence) is a drove of rules that reflect conventions about which procedures to perform outset in order to evaluate a given mathematical expression.

For case, in mathematics and most computer languages, multiplication is granted a college precedence than addition, and it has been this way since the introduction of modern algebraic notation.[1] [2] Thus, the expression 1 + 2 × 3 is interpreted to have the value i + (two × 3) = 7, and not (1 + 2) × iii = ix. When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication, and could be placed only equally a superscript to the right of their base of operations.[i] Thus 3 + 52 = 28 and iii × 5ii = 75.

These conventions exist to eliminate notational ambiguity, while allowing notation to be every bit brief equally possible. Where information technology is desired to override the precedence conventions, or fifty-fifty simply to emphasize them, parentheses ( ) tin can be used. For example, (ii + 3) × 4 = 20 forces improver to precede multiplication, while (3 + v)two = 64 forces improver 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 brackets or braces to avert confusion, as in [2 × (three + 4)] − 5 = 9.

Definition [edit]

The club of operations, which is used throughout mathematics, science, technology and many computer programming languages, is expressed here:[i] [iii] [4]

  1. exponentiation and root extraction
  2. multiplication and partition
  3. add-on and subtraction

This ways that if, in a mathematical expression, a subexpression appears betwixt ii operators, the operator that is college in the above list should be applied first.

The commutative and associative laws of addition and multiplication allow adding terms in whatever order, and multiplying factors in any lodge—only mixed operations must obey the standard order of operations.

In some contexts, it is helpful to supervene upon a partition with multiplication past the reciprocal (multiplicative inverse) and a subtraction by addition of the contrary (additive changed). For instance, in computer algebra, this allows ane to handle fewer binary operations, and makes it easier to employ commutativity and associativity when simplifying large expressions (for more, encounter Computer algebra § Simplification). Thus 3 ÷ iv = 3 × 1 / 4 ; in other words, the caliber of 3 and 4 equals the product of three and ane / four . Also 3 − iv = 3 + (−iv); in other words the difference of 3 and 4 equals the sum of three and −iv. Thus, ane − 3 + seven tin be idea of equally the sum of 1 + (−iii) + vii, and the 3 summands may be added in any order, in all cases giving five every bit the result.

The root symbol √ is traditionally prolongated by a bar (called vinculum) over the radicand (this avoids the need for parentheses effectually the radicand). Other functions use parentheses around the input to avert ambiguity.[5] [vi] [a] The parentheses can exist omitted if the input is a single numerical variable or constant[1] (as in the case of sin 10 = sin(x) and sin π = sin(π).[a] Another shortcut convention that is sometimes used is when the input is monomial; thus, sin threeten = sin(3x) rather than (sin(3))ten , only sin x + y = sin(ten) + y , because x + y is not a monomial. This, however, is ambiguous and not universally understood exterior of specific contexts.[b] Some calculators and programming languages crave parentheses effectually office inputs, some do non.

Symbols of grouping tin be used to override the usual order of operations.[1] Grouped symbols can exist treated as a single expression.[1] Symbols of grouping can be removed using the associative and distributive laws, likewise they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal.

Examples [edit]

i + three + v = 4 + 5 = 2 + v = 7. {\displaystyle {\sqrt {1+3}}+5={\sqrt {4}}+five=2+5=vii.}

A horizontal partial line also acts equally a symbol of group:

one + ii three + four + 5 = 3 seven + 5. {\displaystyle {\frac {1+2}{3+4}}+5={\frac {3}{7}}+5.}

For ease in reading, other grouping symbols, such as curly braces { } or foursquare brackets [ ], are ofttimes used forth with parentheses ( ). For example:

( [ one + two ] ÷ [ 3 + iv ] ) + 5 = ( 3 ÷ 7 ) + 5 {\displaystyle ([one+2]\div [3+four])+v=(3\div seven)+5}

Mnemonics [edit]

Mnemonics are oftentimes used to help students recollect the rules, involving the kickoff letters of words representing various operations. Dissimilar mnemonics are in use in different countries.[seven] [8] [9]

  • In the The states[10] and in French republic,[xi] the acronym PEMDAS is mutual. It stands for Parentheses, Exponents, Multiplication/Division, Addition/Due southubtraction.[x] PEMDAS is often expanded to the mnemonic "Please Excuse My Dear Aunt Emerge" in schools.[12]
  • Canada and New Zealand use BEDMAS , standing for Brackets, Eastwardxponents, Division/Multiplication, Addition/Subtraction.[10]
  • Most common in the UK, Pakistan, India, Bangladesh and Australia[xiii] and some other English-speaking countries is BODMAS pregnant either Brackets, Order, Division/Multiplication, Addition/Southubtraction or Brackets, Of/Division/Multiplication, Addition/Due southubtraction.[c] [fourteen] Nigeria and another West African countries also use BODMAS. Similarly in the U.k., BIDMAS is also used, continuing for Brackets, Indices, Division/Multiplication, Addition/Subtraction.

These mnemonics may exist misleading when written this mode.[12] For example, misinterpreting any of the higher up rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression[12]

a − b + c = ( a − b ) + c ≠ a − ( b + c ) {\displaystyle a-b+c=(a-b)+c\neq a-(b+c)}

The "Improver/Subtraction" in the mnemonics should be interpreted as that any additions and subtractions should be performed in guild from left to correct. Similarly, the expression a ÷ b × c might be read multiple ways, but the "Multiplication/Division" in the mnemnonic means the multiplications and divisions should be performed from left to right.

a ÷ b × c = ( a ÷ b ) × c ≠ a ÷ ( b × c ) {\displaystyle a\div b\times c=(a\div b)\times c\neq a\div (b\times c)}

Additional ambiguities caused by the apply of multiplication by juxtaposition and using the slash to correspond partition are discussed beneath. In full general, the surest style to avoid ambiguity is to use parentheses.

Special cases [edit]

Serial exponentiation [edit]

If exponentiation is indicated by stacked symbols using superscript notation, the usual rule is to work from the tiptop downward:[fifteen] [1] [6] [16]

a b c = a (b c )

which typically is not equal to (a b ) c . This convention is useful considering there is a holding of exponentiation that (a b ) c = a bc , so it's unnecessary to utilise serial exponentiation for this.

However, when using operator notation with a caret (^) or arrow (↑), there is no common standard.[17] For example, Microsoft Excel and computation programming language MATLAB evaluate a^b^c as (a b ) c , but Google Search and Wolfram Alpha as a (b c ). Thus 4^3^2 is evaluated to 4,096 in the kickoff example and to 262,144 in the second case.

Unary minus sign [edit]

At that place are differing conventions concerning the unary operator − (ordinarily read "minus"). In written or printed mathematics, the expression −32 is interpreted to mean −(32) = −9.[1] [18]

In some applications and programming languages, notably Microsoft Excel, PlanMaker (and other spreadsheet applications) and the programming language bc, unary operators have a higher priority than binary operators, that is, the unary minus has college precedence than exponentiation, so in those languages −iii2 will exist interpreted every bit (−3)2 = ix.[xix] This does not utilize to the binary minus operator −; for example in Microsoft Excel while the formulas =−2^2, =-(2)^2 and =0+−two^ii render iv, the formula =0−2^2 and =−(2^2) return −4.

Mixed division and multiplication [edit]

In some of the bookish literature, multiplication denoted by juxtaposition (too known every bit implied multiplication) is interpreted as having higher precedence than division, and then that 1 ÷ twodue north equals 1 ÷ (2n), not (1 ÷ 2)n .[1] [7] For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division,[20] and this is likewise the convention observed in prominent physics textbooks such every bit the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[d] This ambiguity is often exploited in internet memes such as "viii÷2(2+2)".[21]

Ambiguity can also be caused past the use of the slash symbol, '/', for division. The Physical Review submission instructions propose to avoid expressions of the form a/b/c; ambiguity can exist avoided by instead writing (a/b)/c or a/(b/c).[20]

Calculators [edit]

Unlike calculators follow different orders of operations.[1] Many simple calculators without a stack implement concatenation input working left to right without whatsoever priority given to dissimilar operators, for example typing

1 + 2 × three yields 9,

while more than sophisticated calculators volition utilise a more standard priority, for example typing

1 + 2 × 3 yields 7.

The Microsoft Reckoner program uses the former in its standard view and the latter in its scientific and programmer views.

Concatenation input expects ii operands and an operator. When the next operator is pressed, the expression is immediately evaluated and the answer becomes the left hand of the next operator. Avant-garde calculators permit entry of the whole expression, grouped as necessary, and evaluates only when the user uses the equals sign.

Calculators may acquaintance exponents to the left to the right. For example, the expression a^b^c is interpreted every bit a (b c ) on the TI-92 and the TI-30XS MultiView in "Mathprint mode", whereas it is interpreted as (a b ) c on the TI-30XII and the TI-30XS MultiView in "Classic mode".

An expression like i/iix is interpreted equally 1/(twox) by TI-82, as well as many modern Casio calculators,[22] but as (1/two)x by TI-83 and every other TI calculator released since 1996,[23] likewise as past all Hewlett-Packard calculators with algebraic notation. While the first interpretation may exist expected past some users due to the nature of implied multiplication, the latter is more in line with the standard rule that multiplication and division are of equal precedence,[24] [25] where 1/210 is read ane divided by two and the answer multiplied past x.

When the user is unsure how a calculator will interpret an expression, parentheses tin can be used to remove the ambivalence.

Calculators that use reverse Smooth notation (RPN), also known as postfix notation, utilize a stack to enter expressions in the correct order of precedence without a demand for parentheses or any possibly model-specific lodge of execution.[12] [10]

Programming languages [edit]

Some programming languages use precedence levels that adapt to the order commonly used in mathematics,[17] though others, such as APL, Smalltalk, Occam and Mary, have no operator precedence rules (in APL, evaluation is strictly right to left; in Smalltalk, it is strictly left to right).

Furthermore, because many operators are non associative, the order within whatsoever single level is usually defined by grouping left to right and so that 16/4/4 is interpreted equally (sixteen/4)/4 = ane rather than 16/(4/4) = 16; such operators are perhaps misleadingly referred to every bit "left associative". Exceptions exist; for example, languages with operators corresponding to the cons operation on lists usually make them group right to left ("correct associative"), e.k. in Haskell, one:two:3:4:[] == 1:(2:(three:(iv:[]))) == [1,2,3,4].

The creator of the C linguistic communication has said of the precedence in C (shared by programming languages that infringe those rules from C, for case, C++, Perl and PHP) that it would have been preferable to move the bitwise operators above the comparison operators.[26] Still, many programmers have go accepted to this gild. The relative precedence levels of operators found in many C-fashion languages are as follows:

1 ()   []   ->   .   :: Office call, scope, array/fellow member access
2 !   ~   -   +   *   &   sizeof type cast ++   -- (most) unary operators, sizeof and type casts (right to left)
iii *   /   % Modernistic Multiplication, partitioning, modulo
4 +   - Improver and subtraction
5 <<   >> Bitwise shift left and right
6 <   <=   >   >= Comparisons: less-than and greater-than
seven ==   != Comparisons: equal and not equal
8 & Bitwise AND
nine ^ Bitwise exclusive OR (XOR)
ten | Bitwise inclusive (normal) OR
11 && Logical AND
12 || Logical OR
13 ? : Provisional expression (ternary)
fourteen =   +=   -=   *=   /=   %=   &=   |=   ^=   <<=   >>= Assignment operators (right to left)
15 , Comma operator

Examples: (Note: in the examples below, '≡' is used to mean "is identical to", and non to be interpreted as an bodily consignment operator used every bit part of the instance expression.)

  • !A + !B ≡ (!A) + (!B)
  • ++A + !B ≡ (++A) + (!B)
  • A + B * C ≡ A + (B * C)
  • A || B && C ≡ A || (B && C)
  • A && B == C ≡ A && (B == C)
  • A & B == C ≡ A & (B == C)

Source-to-source compilers that compile to multiple languages need to explicitly deal with the effect of unlike order of operations across languages. Haxe for example standardizes the order and enforces it by inserting brackets where information technology is appropriate.[27]

The accurateness of software developer knowledge well-nigh binary operator precedence has been institute to closely follow their frequency of occurrence in source lawmaking.[28]

Run into too [edit]

  • Common operator notation (for a more formal description)
  • Hyperoperation
  • Operator associativity
  • Operator overloading
  • Operator precedence in C and C++
  • Smooth notation
  • Contrary Smoothen note

Notes [edit]

  1. ^ a b Some authors deliberately avoid whatever omission of parentheses with functions even in the case of single numerical variable or constant arguments (f.e. Oldham in Atlas), whereas other authors (like NIST) apply this notational simplification merely conditionally in conjunction with specific multi-character function names (like sin), but don't utilize it with generic office names (like f ).
  2. ^ To avoid any ambiguity, this notational simplification for monomials is deliberately avoided in works such equally Oldham's Atlas of Functions or the NIST Handbook of Mathematical Functions.
  3. ^ "Of" is equivalent to division or multiplication, and unremarkably used especially at chief school level, as in "Half of fifty".
  4. ^ For instance, the third edition of Mechanics by Landau and Lifshitz contains expressions such as hP z /twoπ (p. 22), and the get-go volume of the Feynman Lectures contains expressions such every bit 1/two√ N (p. 6–7). In both books, these expressions are written with the convention that the solidus is evaluated last. This also implies that an expression like viii/2(four) has solution 1 every bit the omission of the multiplication sign (x * or .) implies that the solidus is evaluated terminal fifty-fifty if positioned more to the left.

References [edit]

  1. ^ a b c d e f k h i j Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. "two.iv.1.1. Definition arithmetischer Ausdrücke" [Definition of arithmetic expressions]. Written at Leipzig, Germany. In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik [Pocketbook of mathematics] (in German). Vol. one. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun, Switzerland / Frankfurt am Master, Frg: Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig). pp. 115–120, 802. ISBN3-87144-492-8. 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 A dafür geschrieben werden. [Darüber hinaus ist noch die Abkürzung F n (A) für (F(A)) n üblich. Dabei kann F sowohl Funktionenkonstante als auch Funktionenvariable sein.]
  2. ^ "Order of Operations: Why?". The Math Doctors. 2019-09-30. Retrieved 2021-10-21 .
  3. ^ Weisstein, Eric Due west. "Precedence". mathworld.wolfram.com . Retrieved 2020-08-22 .
  4. ^ Stapel, Elizabeth. "The Order of Operations: PEMDAS". Purplemath . Retrieved 2020-08-22 .
  5. ^ Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) [1987]. An Atlas of Functions: with Equator, the Atlas Function Calculator (2 ed.). Springer Science+Business Media, LLC. doi:10.1007/978-0-387-48807-3. ISBN978-0-387-48806-6. LCCN 2008937525.
  6. ^ a b Olver, Frank W. J.; Lozier, Daniel West.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010). NIST Handbook of Mathematical Functions. National Institute of Standards and Engineering (NIST), U.S. Department of Commerce, Cambridge University Press. ISBN978-0-521-19225-5. MR 2723248. [1]
  7. ^ a b "Rules of arithmetic" (PDF). Mathcentre.air-conditioning.uk . Retrieved 2019-08-02 .
  8. ^ "Please Excuse My Dear Aunt Sally (PEMDAS)--Forever!". Teaching Week - Autobus G's Teaching Tips. 2011-01-01.
  9. ^ "What is PEMDAS? - Definition, Rule & Examples". Report.com.
  10. ^ a b c d Vanderbeek, Greg (June 2007). Order of Operations and RPN (Expository paper). Master of Arts in Teaching (MAT) Examination Expository Papers. Lincoln, Nebraska, USA: Academy of Nebraska. Paper 46. Archived from the original on 2020-06-fourteen. Retrieved 2020-06-fourteen .
  11. ^ Le calcul qui divise : 6÷2(1+ii) - Micmaths , retrieved 2021-xi-01 Archived at Ghostarchive and the Wayback Automobile
  12. ^ a b c d Ball, John A. (1978). Algorithms for RPN calculators (1 ed.). Cambridge, Massachusetts, USA: Wiley-Interscience, John Wiley & Sons, Inc. p. 31. ISBN0-471-03070-8.
  13. ^ "Order of operations" (DOC). Syllabus.bos.nsw.edu.au . Retrieved 2019-08-02 .
  14. ^ "Bodmas Dominion - What is Bodmas Rule - Order of Operations". vedantu.com . Retrieved 2019-08-21 .
  15. ^ Robinson, Raphael Mitchel (Oct 1958) [1958-04-07]. "A report on primes of the form chiliad · 2due north + 1 and on factors of Fermat numbers" (PDF). Proceedings of the American Mathematical Society. University of California, Berkeley, California, USA. ix (5): 673–681 [677]. doi:10.1090/s0002-9939-1958-0096614-7. Archived (PDF) from the original on 2020-06-28. Retrieved 2020-06-28 .
  16. ^ Zeidler, Eberhard; Schwarz, Hans Rudolf; Hackbusch, Wolfgang; Luderer, Bernd; Blath, Jochen; Schied, Alexander; Dempe, Stephan; Wanka, Gert; Hromkovič, Juraj; Gottwald, Siegfried (2013) [2012]. Zeidler, Eberhard (ed.). Springer-Handbuch der Mathematik I (in German language). Vol. I (1 ed.). Berlin / Heidelberg, Deutschland: Springer Spektrum, Springer Fachmedien Wiesbaden. p. 590. doi:10.1007/978-iii-658-00285-5. ISBN978-three-658-00284-8. (xii+635 pages)
  17. ^ a b Van Winkle, Lewis (2016-08-23). "Exponentiation Associativity and Standard Math Notation". Codeplea - Random thoughts on programming. Archived from the original on 2020-06-28. Retrieved 2016-09-20 .
  18. ^ Affections, Allen R. Uncomplicated Algebra for Higher Students (8 ed.). Chapter 1, Section 9, Objective 3.
  19. ^ "Formula Returns Unexpected Positive Value". Microsoft. 2005-08-15. Archived from the original on 2015-04-19. Retrieved 2012-03-05 .
  20. ^ a b "Concrete Review Style and Notation Guide" (PDF). American Physical Society. Department IV–Due east–2–east. Retrieved 2012-08-05 .
  21. ^ Lakritz, Talia. "This equation has 2 wildly dissimilar answers depending on what you lot learned in school, and it's dividing the internet". Insider . Retrieved 2022-02-18 .
  22. ^ "Calculation Priority Sequence". support.casio.com. Casio. Retrieved 2019-08-01 .
  23. ^ "Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators". Texas Instruments. 2011-01-xvi. 11773. Archived from the original on 2016-04-17. Retrieved 2015-08-24 .
  24. ^ Zachary, Joseph 50. (1997). "Introduction to scientific programming - Computational problem solving using Maple and C - Operator precedence worksheet". Retrieved 2015-08-25 .
  25. ^ Zachary, Joseph 50. (1997). "Introduction to scientific programming - Computational problem solving using Mathematica and C - Operator precedence notebook". Retrieved 2015-08-25 .
  26. ^ Ritchie, Dennis K. (1996). "The Development of the C Language". History of Programming Languages (2 ed.). ACM Press.
  27. ^ Li, Andy (2011-05-02). "6÷2(1+2)=?". Andy Li's Web log . Retrieved 2012-12-31 .
  28. ^ Jones, Derek M. "Developer behavior about binary operator precedence". CVu. xviii (4): 14–21.

Further reading [edit]

  • Bergman, George Mark (2013-02-21). "Order of arithmetic operations; in item, the 48/two(9+3) question". Department of Mathematics, Academy of California. Archived from the original on 2020-05-twenty. Retrieved 2020-07-22 .
  • "The Order of Operations". MathSteps: What is information technology?. Houghton Mifflin Company. 1999. Archived from the original on 2020-07-21. Retrieved 2020-07-22 .

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Source: https://en.wikipedia.org/wiki/Order_of_operations