Definition:Commensurable
Definition
Let $a, b \in \R_{>0}$ be (strictly) positive real numbers.
$a$ and $b$ are commensurable if and only if $\dfrac a b$ is rational.
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 $1$)
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 commensurable in length can be used, in order to distinguish explicitly from commensurability in square.
Examples
$16$ and $12$
The natural numbers $12$ and $16$ are commensurable.
$3$ Feet and $2$ Inches
The lengths $2$ inches and $3$ feet are commensurable.
Also see
- Results about commensurability can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): commensurable
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): commensurable
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): commensurable