# Definition:Unit (One)

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## 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 Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements, $\struct {S^*, \circ, \preceq}$ 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.

## Also see

## Sources

- 1938: A. Geary, H.V. Lowry and H.A. Hayden:
*Mathematics for Technical Students, Part One*... (previous) ... (next): Arithmetic: Chapter $\text I$: Decimals - 2004: Richard K. Guy:
*Unsolved Problems in Number Theory*(3rd ed.) ... (next): $\text A$. Prime Numbers