# Definition:Unit (One)

## Definition

A numerical quantity whose cardinality corresponds to the number $1$ (one) is called **a unit**.

In the words of Euclid:

(*The Elements*: Book $\text{VII}$: Definition $1$)

### One of Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Let $S^*$ be the zero complement of $S$.

By Zero Complement is Not Empty, $S^*$ is not empty.

Therefore, by axiom $(NO4)$, it has a smallest element for $\preceq$.

This smallest element is called **one** and denoted $1$.

## Also known as

In older writings, **a unit** is often rendered as **an unit**; the rules of grammar have since evolved.