Editing Arity (section)

== Examples ==
In general, functions or operators with a given arity follow the naming conventions of ''n''-based [[numeral system]]s, such as [[Binary numeral system|binary]] and [[hexadecimal]]. A [[Latin]] prefix is combined with the -ary suffix. For example:
* A nullary function takes no arguments.
** Example: <math>f()=2</math>
* A [[unary operation|unary function]] takes one argument.
** Example: <math>f(x)=2x</math>
* A [[binary operation|binary function]] takes two arguments.
** Example: <math>f(x,y)=2xy</math>
* A [[ternary operation|ternary function]] takes three arguments.
** Example: <math>f(x,y,z)=2xyz</math>
* An ''n''-ary function takes ''n'' arguments.
** Example: <math display="inline">f(x_1, x_2, \ldots, x_n)=2\prod_{i=1}^n x_i</math>

=== Nullary ===
A [[constant (mathematics)|constant]] can be treated as the output of an operation of arity 0, called a ''nullary operation''.

Also, outside of [[functional programming]], a function without arguments can be meaningful and not necessarily constant (due to [[side effect (computer science)|side effect]]s). Such functions may have some ''hidden input'', such as [[global variable]]s or the whole state of the system (time, free memory, etc.).

=== 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 [[Successor function|successor]], [[factorial]], [[Multiplicative inverse|reciprocal]], [[floor function|floor]], [[ceiling function|ceiling]], [[fractional part]], [[sign function|sign]], [[absolute value]], [[square root]] (the principal square root), [[complex conjugate]] (unary of "one" [[complex number]], that however has two parts at a lower level of abstraction), and [[Norm (mathematics)|norm]] functions in mathematics.  In programming the [[two's complement]], [[Reference (computer science)|address reference]], and the [[logical NOT]] operators are examples of unary operators.

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.

According to [[Willard Van Orman Quine|Quine]], the Latin distributives being ''singuli'', ''bini'', ''terni'', and so forth, the term "singulary" is the correct adjective, rather than "unary".<ref>
{{Citation
  | last = Quine
  | first = W. V. O.
  | title = Mathematical logic
  | year = 1940
  | place = Cambridge, Massachusetts
  | publisher = Harvard University Press
  | page=13
}}</ref> [[Abraham Robinson]] follows Quine's usage.<ref>
{{Citation
  | last = Robinson
  | first = Abraham
  | title = Non-standard Analysis
  | year = 1966
  | place = Amsterdam
  | publisher = North-Holland
  | page=19
}}</ref>

In philosophy, the adjective ''monadic'' is sometimes used to describe a [[monadic predicate calculus|one-place relation]] such as 'is square-shaped' as opposed to a [[binary relation|two-place relation]] such as 'is the sister of'.

=== Binary ===
Most operators encountered in programming and mathematics are of the [[binary operation|binary]] form. For both programming and mathematics, these include the [[multiplication operator]], the radix operator, the often omitted [[exponentiation]] operator, the [[logarithm]] operator, the [[addition]] operator, and the [[Division (mathematics)|division]] operator. Logical predicates such as ''[[Logical disjunction|OR]]'', ''[[exclusive or|XOR]]'', ''[[Logical conjunction|AND]]'', ''IMP'' are typically used as binary operators with two distinct operands. In [[Complex instruction set computing|CISC]] architectures, it is common to have two source operands (and store result in one of them).

=== Ternary ===
The computer programming language [[C (programming language)|C]] and its various descendants (including [[C++]], [[C Sharp (programming language)|C#]], [[Java (programming language)|Java]], [[Julia (programming language)|Julia]], [[Perl]], and others) provide the [[ternary conditional operator]] <code>?:</code>. The first operand (the condition) is evaluated, and if it is true, the result of the entire expression is the value of the second operand, otherwise it is the value of the third operand. This operator has a lazy or 'shortcut' [[evaluation strategy]] that does not evaluate whichever of the second and third arguments is not used. Some functional programming languages, such as [[Agda (programming language)|Agda]], have such an evaluation strategy for all functions and consequently implement {{code|if...then...else}} as an ordinary function; several others, such as [[Haskell]], can do this but for syntactic, performance or historical reasons choose to define keywords instead.

The [[Python (programming language)|Python]] language has a ternary conditional expression, {{code|x if C else y|python}}. In [[Elixir (programming language)|Elixir]] the equivalent would be {{code|if(C, do: x, else: y)|elixir}}.

The [[Forth (programming language)|Forth]] language 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.

The Unix [[dc (computer program)|dc calculator]] has several ternary operators, such as <code>|</code>, which will pop three values from the stack and efficiently compute <math display="inline">x^y \bmod z</math> with [[arbitrary-precision arithmetic|arbitrary precision]].

Many ([[Reduced instruction set computing|RISC]]) [[assembly language]] instructions are ternary (as opposed to only two operands specified in CISC); or higher, such as <syntaxhighlight lang="asm" inline="">MOV %AX, (%BX, %CX)</syntaxhighlight>, which will load ({{mono|MOV}}) into register {{mono|AX}} the contents of a calculated memory location that is the sum (parenthesis) of the registers {{mono|BX}} and {{mono|CX}}.<!-- examples section needs complete rewrite, with links and subsection on math, logic and programming
-->

=== ''n''-ary ===
The [[arithmetic mean]] of ''n'' real numbers is an ''n''-ary  function: <math>\bar{x}=\frac{1}{n}\left (\sum_{i=1}^n{x_i}\right) = \frac{x_1+x_2+\dots+x_n}{n}</math>

Similarly, the [[geometric mean]] of ''n'' [[positive real numbers]] is an ''n''-ary function: <math>\left(\prod_{i=1}^n a_i\right)^\frac{1}{n} = \ \sqrt[n]{a_1 a_2 \cdots a_n} .</math> Note that a [[logarithm]] of the geometric mean is the arithmetic mean of the logarithms of its ''n'' arguments

From a mathematical point of view, a function of ''n'' arguments can always be considered as a function of a single argument that 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 {{nowrap|''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]].

=== Varying arity ===
In computer science, a function that accepts 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, or variably polyadic.<ref>{{Cite journal | doi = 10.1093/mind/113.452.609 | last1 = Oliver | first1 = Alex | year = 2004 | title = Multigrade Predicates | journal = Mind | volume = 113 | issue = 452| pages = 609–681 }}</ref>