# 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.

## 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.