Dual number - Revision history

en>Rgdboer: /* Geometry */ ce

2014-12-28T01:26:13Z

Geometry: ce

← Older revision		Revision as of 03:26, 28 December 2014
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en>Quondum: /* Algebraic properties */ Removal of unnecessary qualifier "nonzero" when describing the zero divisors

2014-02-15T22:10:24Z

Algebraic properties: Removal of unnecessary qualifier "nonzero" when describing the zero divisors

← Older revision		Revision as of 00:10, 16 February 2014
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	In [[logic]], [[mathematics]], and [[computer science]], the '''arity''' {{IPAc-en\|audio=en-us-arity.ogg\|ˈ\|æ\|r\|ɨ\|t\|i}} of a [[function (mathematics)\|function]] or [[operation (mathematics)\|operation]] is the number of [[argument of a function\|arguments]] or [[operand]]s the function or operation accepts. The arity of a [[relation (mathematics)\|relation]] (or [[predicate (mathematical logic)\|predicate]]) is the dimension of the [[domain of a function\|domain]] in the corresponding [[Cartesian product]]. (A function of arity ''n'' thus has arity ''n''+1 considered as a relation.) The term springs from words like unary, binary, ternary, etc. Unary functions or predicates may be also called "monadic"; similarly, binary functions may be called "dyadic".		My name is Melisa Cleland. I life in Limoges (France).<br><br>my site; [http://Innotomsk.ru/tag/service%20dog%20training%20las%20vegas/ hollywood dog training school]

	In mathematics arity may also be named ''rank'',<ref name="Hazewinkel2001">{{cite book\|author=[[Michiel Hazewinkel]]\|title=Encyclopaedia of Mathematics, Supplement III\|url=http://books.google.com/books?id=47YC2h295JUC&pg=PA3\|year=2001\|publisher=Springer\|isbn=978-1-4020-0198-7\|page=3}}</ref><ref name="Schechter1997">{{cite book\|author=Eric Schechter\|title=Handbook of Analysis and Its Foundations\|url=http://books.google.~~com/books?id=eqUv3Bcd56EC&pg=PA356\|year=1997\|publisher=Academic Press\|isbn=978-0-12-622760-4\|page=356}}</ref> but this word can have many other meanings~~ in mathematics. In logic and philosophy, arity is also called ''adicity'' and ''degree''.<ref name="DetlefsenBacon1999">{{cite book\|author1=Michael Detlefsen\|author2=David Charles McCarty\|author3=John B. Bacon\|title=Logic from A to Z\|url=http://books.google.com/books?id=pc3E4Xdf2GgC&pg=PP17\|year=1999\|publisher=Routeledge\|isbn=978-0-415-21375-2\|page=7}}</ref><ref name="CocchiarellaFreund2008">{{cite book\|author1=Nino B. Cocchiarella\|author2=Max A. Freund\|title=Modal Logic: An Introduction to its Syntax and Semantics\|url=http://books.google.com/books?id=zLmxqytfLhgC&pg=PA121\|year=2008\|publisher=Oxford University Press\|isbn=978-0-19-536658-7\|page=121}}</ref> In [[linguistics]], arity is usually named ''[[valency (~~linguistics~~)~~\|valency]]''~~.<~~ref name="Crystal2008"~~>~~{{cite book\|author=David Crystal\|title=Dictionary of Linguistics and Phonetics\|year=2008\|publisher=John Wiley & Sons\|isbn=978-1-405-15296-9\|page=507\|edition=6th}}~~<~~/ref~~>

	~~In [[computer programming]], there is often a [[Syntax (programming languages)\|syntactical]] distinction between [[Operator (programming)\|operators]] and [[Function (computer science)\|functions]]~~; ~~syntactical operators usually have arity 0, 1, or 2. Functions vary widely in the number of arguments, though large numbers can become unwieldy. Some programming languages also offer support for~~ [~~[variadic functions]], i.e. functions syntactically accepting a variable number of arguments.~~

	~~==Examples==~~
	~~{{Expand section\|more math and logic examples\|date=March 2012}}~~
	The term "arity" is rarely employed in everyday usage. For example, rather than saying "the arity of the [[addition]] operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation".
	In general, the naming of functions or operators with a given arity follows a convention similar to the one used for ''n''-based [[numeral system]]s such as [[Binary numeral system\|binary]] and [[hexadecimal]]. One combines a [[Latin]] prefix with the -ary ending; for example:
	* A nullary function takes no arguments.
	* A [[unary operation\|unary function]] takes one argument.
	* A [[binary operation\|binary function]] takes two arguments.
	* A [[ternary operation\|ternary function]] takes three arguments.
	* An ''n''-ary function takes ''n'' arguments.

	~~===Nullary===~~
	~~Sometimes it is useful to consider a [[constant (mathematics)\|constant]] to be an operation of arity 0, and hence call it ''nullary''.~~

	Also, in non-[[functional programming]], a function without arguments can be meaningful and not necessarily constant (due to [[side effect (computer science)\|side effect]]s). Often, such functions have in fact some ''hidden input'' which might be [[global variable]]s, including the whole state of the system (time, free memory, ...). The latter are important examples which usually also exist in "purely" functional programming languages.

	~~===Unary===~~
	Examples of [[unary operator]]s in mathematics and in programming include the unary minus and plus, the increment and decrement operators in [[C (programming language)\|C]]-style languages (not in logical languages), and the [[factorial]], [[Multiplicative inverse\|reciprocal]], [[floor function\|floor]], [[ceiling function\|ceiling]], [[fractional part]], [[sign (mathematics)\|sign]], [[absolute value]], [[complex conjugate]], and [[Norm (mathematics)\|norm]] functions in mathematics. The [[two's complement]], [[Reference (computer science)\|address reference]] and the [[logical NOT]] operators are examples of unary operators in math and programming. According to [[Willard Van Orman Quine\|Quine]], a more suitable term is "singulary".<ref>{{Citation
	~~\| last = Quine~~
	~~\| first = W. V. O.~~
	~~\| title = Mathematical logic~~
	~~\| year = 1940~~
	~~\| place = Cambridge, MA~~
	~~\| publisher = Harvard University Press~~
	~~\| page=13~~
	~~}}<~~/~~ref>~~

	All functions in [[lambda calculus]] and in some [[functional programming language]]s (especially those descended from [[ML (programming language)\|ML]]) are technically unary, but see [[#n-ary\|n-ary]] below.

	~~===Binary===~~
	Most operators encountered in programming are of the [[binary operation\|binary]] form. For both programming and mathematics these can be the [[multiplication operator]], the addition operator, the division operator. Logical predicates such as ''OR'', ''[[exclusive or\|XOR]]'', ''AND'', ''IMP'' are typically used as binary operators with two distinct operands.

	~~===Ternary===~~
	~~From [[C (programming language)\|C]], [[C++]], [[C Sharp (programming language)\|C#]], [[Java (programming language)\|Java]], [[Perl]] and variants comes the [[ternary operator]] <code>[[?:]]<~~/~~code>, which is a so-called [[conditional operator]], taking three parameters~~.
	[[Forth (programming language)\|Forth]] also contains a ternary operator, <code>*/</code>, which multiplies the first two (one-cell) numbers, dividing by the third, with the intermediate result being a double cell number. This is used when the intermediate result would overflow a single cell.
	~~[[Python (programming language)\|Python]] has a ternary conditional statement, <code>x if C else y</code>.~~
	The [[dc (computer program)\|dc calculator]] has several ternary operators, such as <tt>\|</tt>, which will pop three values from the stack and efficiently compute <math>x^y \mod z</math> with arbitrary precision.
	~~Additionally, many [[assembly language]] instructions are ternary or higher, such as <tt>MOV ~~%~~AX, (~~%~~BX,~~%~~CX)</tt>, which will load (MOV) into register <tt>AX</tt> the contents of a calculated memory location that is the sum (parenthesis) of the registers <tt>BX<~~/~~tt> and <tt>CX</tt>.~~
	~~<!-- examples section needs complete rewrite, with links and subsection on math, logic and programming -->~~

	~~===''n''-ary===~~
	From a mathematical point of view, a function of ''n'' arguments can always be considered as a function of one single argument which is an element of some [[product space]]. However, it may be convenient for notation to consider ''n''-ary functions, as for example [[multilinear map]]s (which are not linear maps on the product space, if ''n''≠1).

	The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of some [[object composition\|composite type]] such as a [[tuple]], or in languages with [[higher-order function]]s, by [[currying]].

	~~=== Variable arity===~~
	In computer science, a function accepting a variable number of arguments is called ''[[variadic function\|variadic]]''. In logic and philosophy, predicates or relations accepting a variable number of arguments are called ''[[multigrade predicate\|multigrade]]'', ''[[anadic predicate\|anadic]]'', or ''[[variably polyadic predicate\|variably polyadic]]''.<ref>{{Cite journal \| doi = 10.1093/mind/113.452.609 \| last1 = Oliver \| first1 = Alex \| year = 2004 \| title = Multigrade Predicates \| url = \| journal = Mind \| volume = 113 \| issue = \| pages = 609–681 }}</ref>

	~~==Other names==~~
	* ''Nullary'' means 0-ary.
	* ''[[Unary operation\|Unary]]'' means 1-ary.
	* ''[[Binary operation\|Binary]]'' means 2-ary.
	* ''[[Ternary operation\|Ternary]]'' means 3-ary.
	* ''Quaternary'' means 4-ary.
	* ''Quinary'' means 5-ary.
	* ''Senary'' means 6-ary.
	* ''Septenary'' means 7-ary.
	* ''Octary'' means 8-ary.
	* ''Nonary'' means 9-ary.
	* ''Polyadic'', ''multary'' and ''multiary'' mean 2 or more operands (or parameters).
	* ''n''-''ary'' means ''n'' operands (or parameters), but is often used as a synonym of "polyadic".

	An alternative nomenclature is derived in a similar fashion from the corresponding [[Greek language\|Greek]] roots; for example, ''niladic'' (or ''medadic''), ''monadic'', ''dyadic'', ''triadic'', ''polyadic'', and so on. Thence derive the alternative terms ''adicity'' and ''adinity'' for the Latin-derived ''arity''.

	~~These words are often used to describe anything related to that number (e.g., [[undenary chess]] is a chess variant with an 11×11 board, or the [[Millenary Petition]] of 1603).~~

	~~==See also==~~
	~~{{Portal\|Mathematics\|Logic}}~~
	~~<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">~~
	*[[Logic of relatives]]
	*[[Binary relation]]
	*[[Triadic relation]]
	*[[Theory of relations]]
	*[[Signature (logic)]]
	*[[Parameter]]
	*[[Variadic]]
	*[[Valency (linguistics)\|Valency]]
	*[[n-ary code\|''n''-ary code]]
	*[[n-ary group\|''n''-ary group]]
	~~</div>~~

	~~==References==~~
	~~{{Reflist}}~~

	~~==External links==~~
	~~A monograph available free online:~~
	* Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. ISBN 3-540-90578-2. Especially pp. 22–24.

	~~[[Category:Abstract algebra]]~~
	~~[[Category:Universal algebra]]~~

	~~[[cs:Operace (matematika)#Arita operace]~~]

en>Rgdboer: mv ref inline, lk Dual quaternion in lede, lk Exponential mapping

2013-12-23T03:08:07Z

mv ref inline, lk Dual quaternion in lede, lk Exponential mapping

Show changes

en>ZéroBot: r2.7.1) (Robot: Modifying ja:二重数

2012-08-22T00:11:32Z

r2.7.1) (Robot: Modifying ja:二重数

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