Unary operation

In mathematics, a 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 ⁠${\displaystyle f:A\rightarrow A}$⁠, where A is a set; the function ⁠${\displaystyle f}$⁠ 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.

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

Negation is used to find the negative value of a single number. Here are some examples:

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

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

For any positive integer n, the product of the integers less than or equal to n is a unary operation called factorial. In the context of complex numbers, the gamma function is a unary operation extension of factorial.

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.

Below is a table summarizing common unary operators along with their symbols, description, and examples:³

In JavaScript, these operators are unary:⁴

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

In the C family of languages, the following operators are unary:⁵⁶

• 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

In the Unix shell (Bash/Bourne Shell), e.g., the following operators are unary:⁷⁸

• Pre and Post-Increment: ++$x, $x++
• Pre and Post-Decrement: --$x, $x--
• Positive: +$x
• Negative: -$x
• Logical negation: !$x
• Simple expansion: $x
• Complex expansion: ${#x}

In the PowerShell, the following operators are unary:⁹

• 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