Definition:Unit (One)
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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:
(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
- 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.