In mathematics, an unary operation is an operation with only one operand, i.e. a single input.^[1] This is in contrast to binary operations, which use two operands.^[2] An example is any function f : A → A, where A is a set. The function f is a unary operation on A.

Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial n!), functional notation (e.g. sin x or sin(x)), and superscripts (e.g. transpose A^T). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument.

Examples

Absolute Value

Obtaining the absolute value of a number is a unary operation. This function is defined as ${\displaystyle |n|={\begin{cases}n,&{\mbox{if ))n\geq 0\\-n,&{\mbox{if ))n<0\end{cases))}$^[3] where ${\displaystyle |n|}$ is the absolute value of ${\displaystyle n}$.

Negation

This is used to find the negative value of a single number. This is technically not a unary operation as ${\displaystyle -n}$ is just short form of ${\displaystyle 0-n}$.^[4] Here are some examples:

${\displaystyle -(3)=-3}$

${\displaystyle -(-3)=3}$

Unary negative and positive

As unary operations have only one operand they are evaluated before other operations containing them. Here is an example using negation:

${\displaystyle 3}$${\displaystyle -}$${\displaystyle -}$${\displaystyle 2}$

Here, the first '−' represents the binary subtraction operation, while the second '−' represents the unary negation of the 2 (or '−2' could be taken to mean the integer −2). Therefore, the expression is equal to:

${\displaystyle 3}$${\displaystyle -}$${\displaystyle (-}$${\displaystyle 2)}$${\displaystyle =5}$

Technically, there is also a unary + operation but it is not needed since we assume an unsigned value to be positive:

${\displaystyle +2=2}$

The unary + operation does not change the sign of a negative operation:

${\displaystyle +}$${\displaystyle (-}$${\displaystyle 2)}$${\displaystyle =}$ ${\displaystyle -2}$

In this case, a unary negation is needed to change the sign:

${\displaystyle -(-2)=+2}$

Trigonometry

In trigonometry, the trigonometric functions, such as ${\displaystyle \sin }$, ${\displaystyle \cos }$, and ${\displaystyle \tan }$, can be seen as unary operations. This is because it is possible to provide only one term as input for these functions and retrieve a result. By contrast, binary operations, such as addition, require two different terms to compute a result.

Examples from programming languages

JavaScript

In JavaScript, these operators are unary:^[5]

• Increment: ++x, x++
• Decrement: −−x, x−−
• Positive: +x
• Negative: −x
• Ones' complement: ~x
• Logical negation: !x

C family of languages

In the C family of languages, the following operators are unary:^[6][7]

• Increment: ++x, x++
• Decrement: −−x, x−−
• Address: &x
• Indirection: *x
• Positive: +x
• Negative: −x
• Ones' complement: ~x
• Logical negation: !x
• Sizeof: sizeof x, sizeof(type-name)
• Cast: (type-name) cast-expression

Unix Shell (Bash)

In the Unix/Linux shell (bash/sh), '$' is a unary operator when used for parameter expansion, replacing the name of a variable by its (sometimes modified) value. For example:

• Simple expansion: $x
• Complex expansion: ${#x}

Windows PowerShell

• Increment: ++$x, $x++
• Decrement: −−$x, $x−−
• Positive: +$x
• Negative: −$x
• Logical negation: !$x
• Invoke in current scope: .$x
• Invoke in new scope: &$x
• Cast: [type-name] cast-expression
• Cast: +$x
• Array: ,$array