Definition:Incommensurable in Square

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Definition

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


Then $a$ and $b$ are incommensurable in square iff $\left ({\dfrac a b}\right)^2$ is irrational.


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 $2$)


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 see