Definition:Commensurable in Square Only

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Definition

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


Then $a$ and $b$ are commensurable in square only iff:

$\left ({\dfrac a b}\right)^2$ is rational.

but:

$\dfrac a b$ is irrational.


That is, such that:

$a$ and $b$ are commensurable in square

but:

$a$ and $b$ are incommensurable in length.


In the words of Euclid:

Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.

(The Elements: Book $\text{X}$: Definition $2$)

and:

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 only can also be used for this concept.


Also see