# 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** if and only if:

- $\paren {\dfrac a b}^2$ 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.