Definition:Signum Function

Definition

Let $X \subseteq \R$ be a subset of the real numbers.

The signum function $\sgn: X \to \set {-1, 0, 1}$ is defined as:

$\forall x \in X: \map \sgn x := \sqbrk {x > 0} - \sqbrk {x < 0}$

where $\sqbrk {x > 0}$ etc. denotes Iverson's convention.

That is:

$\forall x \in X: \map \sgn x := \begin {cases} -1 & : x < 0 \\ 0 & : x = 0 \\ 1 & : x > 0 \end {cases}$

The graph of the signum function is illustrated below:

The signum function $\sgn: \N \to \set {0, 1}$ is the restriction of the signum function to the natural numbers, defined as:

$\forall n \in \N: \map \sgn n := \begin {cases} 0 & : n = 0 \\ 1 & : n > 0 \end{cases}$

The signum of a number is known, informally and colloquially, as its sign.

Hence it can also be seen denoted as $\map {\mathrm {sign} } x$.

The concept of a number as being signed or unsigned is used in computer science to distinguish between integers and natural numbers.