# Definition:Signum Function

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## 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:

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

### Natural Numbers

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

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

## Also known as

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

It is also seen denoted as $\map {\operatorname{sign} } x$.

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

## Also see

- Results about
**the signum function**can be found here.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Answers to Exercises: $46$