# Definition:Commensurable

## Definition

Let $a, b \in \R_{>0}$ be (strictly) positive real numbers.

$a$ and $b$ are **commensurable** if and only if $\dfrac a b$ is rational.

In the words of Euclid:

*Those magnitudes are said to be***commensurable**which are measured by the same same measure, and those**incommensurable**which cannot have any common measure.

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

## Notation

There appears to be no universally acknowledged symbol to denote commensurability.

Thomas L. Heath in his edition of *Euclid: The Thirteen Books of The Elements: Volume 3, 2nd ed.* makes the following suggestions:

- $(1): \quad$ To denote that $A$ is commensurable or commensurable in length with $B$:
- $A \mathop{\frown} B$

- $(2): \quad$ To denote that $A$ is commensurable in square with $B$:
- $A \mathop{\frown\!\!-} B$

- $(3): \quad$ To denote that $A$ is incommensurable or incommensurable in length with $B$:
- $A \mathop{\smile} B$

- $(4): \quad$ To denote that $A$ is incommensurable in square with $B$:
- $A \mathop{\smile\!\!-} B$

This convention may be used on $\mathsf{Pr} \infty \mathsf{fWiki}$ if accompanied by a note which includes a link to this page.

## Also known as

When used in the context of **linear measure**, the term **commensurable in length** can be used, in order to distinguish explicitly from **commensurability in square**.

## Examples

### $16$ and $12$

The natural numbers $12$ and $16$ are **commensurable**.

### $3$ Feet and $2$ Inches

The lengths $2$ inches and $3$ feet are **commensurable**.

## Also see

- Results about
**commensurability**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**commensurable** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**commensurable** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**commensurable**