Definition:Unit (One)

This page is about the number $1$. For other uses, see unit.

Definition

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

In the words of Euclid:

An unit is that of which each of the things that exist is called one.

(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

• 1

• Definition:Unity
• Definition:Zero

• Results about $1$ can be found here.

Sources

• 1938: A. Geary, H.V. Lowry and H.A. Hayden: Mathematics for Technical Students, Part One ... (previous) ... (next): Arithmetic: Chapter $\text I$: Decimals
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): unit: 2.
• 2004: Richard K. Guy: Unsolved Problems in Number Theory (3rd ed.) ... (next): $\text A$. Prime Numbers
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): unit: 2.