Urutan operasi: Perbedaan antara revisi

Since the introduction of modern algebraic notation, multiplication has taken precedence over addition.<ref>{{Cite web|url=http://mathforum.org/library/drmath/view/52582.html |title=Ask Dr. Math |publisher=Math Forum |date=22 November 2000 |accessdate=5 March 2012}}</ref> Thus 3&nbsp;+&nbsp;4&nbsp;&times;×&nbsp;5 =&nbsp;4&nbsp;&times;×&nbsp;5&nbsp;+&nbsp;3 = 23. When exponents were first introduced in the 16th and 17th centuries, exponents took precedence over both addition and multiplication and could be placed only as a superscript to the right of their base. Thus 3&nbsp;+&nbsp;5² = 28 and 3&nbsp;&times;×&nbsp;5² =&nbsp;75. To change the order of operations, originally a [[vinculum (symbol)|vinculum]] (an overline or underline) was used. Today, [[bracket (mathematics)|parentheses or brackets]] are used to explicitly denote precedence by grouping parts of an expression that should be evaluated first. Thus, to force addition to precede multiplication, we write (2&nbsp;+&nbsp;3)&nbsp;&times;×&nbsp;4 =&nbsp;20, and to force addition to precede exponentiation, we write (3&nbsp;+&nbsp;5)² = 64.

Urutan operasi yang digunakan dalam [[matematika]], [[sains]], [[teknologi]] dan banyak [[bahasa pemrograman]] [[komputer]] didaftarkan di sini :<ref>{{Cite web|url=http://www.algebrahelp.com/lessons/simplifying/oops/ |title=Order of Operations Lessons |publisher=Algebra.Help |date= |accessdate=5 March 2012}}</ref>

Untuk memudahkan pembacaan, simbol kelompok yang lain, seperti [[tanda kurung]], seringkalisering kali disebut "tanda kurung melengkung" ( ), atau tanda kurung himpunan (''curly braces'') { }, atau tanda kurung siku (''square brackets'') [ ], seringkalisering kali digunakan bergantian. Misalnya,

There exist differing conventions concerning the unary operator − (usually read "minus"). In written or printed mathematics, the expression &minus;3−3² is interpreted to mean &minus;−(3²)&nbsp;=&nbsp;&minus;9−9,<ref>Allen R. Angel, Elementary Algebra for College Students 8/E; Chapter 1, Section 9, Objective 3</ref> but in some applications and programming languages, notably the [[BASIC language|BASIC programming language]], the application [[Microsoft Office Excel]] and [[bc programming language|the programming language bc]], unary operators have a higher priority than binary operators, that is, the unary minus (negation) has higher precedence than exponentiation, so in those languages &minus;3−3² will be interpreted as (&minus;3−3)²&nbsp;=&nbsp;9.<ref>{{Cite web|url=http://support.microsoft.com/kb/q132686/ |title=Formula Returns Unexpected Positive Value |publisher=Support.microsoft.com |date=15 August 2005 |accessdate=5 March 2012}}</ref> Note this does not apply to the binary operator −; for example while the formulas <code>=-2^2</code> and <code>=0+-2^2</code> return 4 in Microsoft Excel, the formula <code>=0-2^2</code> returns −4. In cases where there is the possibility that the notation might be misinterpreted, parentheses are usually used to clarify the intended meaning.

Similarly, there can be ambiguity in the use of the [[Slash (punctuation)#Arithmetic|slash]] ('/') symbol in expressions such as 1/2''x''. If one rewrites this expression as 1 ÷ 2 &times;× ''x'' and then interprets the division symbol as indicating multiplication by the reciprocal, this becomes

Hence, with this interpretation we have that 1/2''x'' is equal to (1/2)''x'', and not 1/(2''x''). However, there are examples, including in published literature, where implied multiplication is interpreted as having higher precedence than division, so that 1/2''x'' equals 1/(2''x''), not (1/2)''x''. For example, the manuscript submission instructions for the ''[[Physical Review]]'' journals state that multiplication is of higher precedence than division with a slash,<ref>{{Cite web|url=https://publish.aps.org/files/styleguide-pr.pdf |title=Physical Review Style and Notation Guide |publisher=[[American Physical Society]]|accessdate=5 August 2012|at=Section IV&ndash;E&ndash;2&ndash;eIV–E–2–e}}</ref> and this is also the convention observed in prominent physics textbooks such as the ''[[Course of Theoretical Physics]]'' by [[Lev Landau|Landau]] and [[Evgeny Lifshitz|Lifshitz]] and the ''[[Feynman Lectures on Physics]]''.<ref>For example, the third edition of ''Mechanics'' by Landau and Lifshitz contains expressions such as ''hP_z''/2π (p. 22), and the first volume of the ''Feynman Lectures'' contains expressions such as 1/2{{sqrt|''N''}} (p. 6&ndash;86–8). In both books these expressions are written with the convention that the solidus is evaluated last.</ref> [[Wolfram Alpha]] changed in early 2013 to treat implied multiplication the same as explicit multiplication (formerly, implied multiplication without parentheses was assumed to bind more strongly than explicit multiplication). 2''x''/2''x'', 2*''x''/2*''x'', and 2(x)/2(x) now all yield x².<ref>{{Cite web|url=http://www.wolframalpha.com/input/?i=2x%2F2x%2C+2*x%2F2*x%2C+2%28x%29%2F2%28x%29+|title=2x&#x2f;/2x, 2&#x2a;*x&#x2f;/2&#x2a;*x, 2&#x28;(x&#x29;&#x2f;)/2&#x28;(x&#x29;) - Wolfram&#124;Alpha|publisher=Wolframalpha.com |date= |accessdate=11 February 2013}}</ref> Newer TI calculators (TI 83 or later) also yield ''x''² in all three cases.

| url = httphttps://archive.istoday/20130204120803/http://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110

|1 || () &nbsp; [] &nbsp; -> &nbsp; . &nbsp; :: || Function call, scope, array/member access

|3 || * &nbsp; / &nbsp; % MOD || Multiplication, division, [[modularAritmetika arithmeticmodular|modulo]]

The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.<ref>"[http://www.knosof.co.uk/cbook/accu06.html Developer beliefs about binary operator precedence]" Derek M. Jones, CVu 18(4):14&ndash;2114–21</ref>

* {{planetmathen}} reference{{PlanetMath|id=3951orderofoperations|title=Orderorder of operations (ID: urutan operasi)}}