# Definition:Incommensurable

## Definition

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

$a$ and $b$ are **incommensurable** if and only if $\dfrac a b$ is irrational.

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 **incommensurable in length** can be used, in order to distinguish explicitly from **incommensurability in square**.

## Examples

### $\sqrt 2$ and $1$

$\sqrt 2$ and $1$ are **incommensurable**.

### $6$ and $\sqrt 3$

$6$ and $\sqrt 3$ are **incommensurable**.

## Also see

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

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**incommensurable** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**incommensurable**