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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 $1$)


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.


$\sqrt 2$ 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.