Commensurability (mathematics)

In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio ⁠a/b⁠ is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory.

For example, the numbers 3 and 2 are commensurable because their ratio, ⁠3/2⁠, is a rational number. The numbers ${\displaystyle {\sqrt {3}}}$ and ${\displaystyle 2{\sqrt {3}}}$ are also commensurable because their ratio, ${\textstyle {\frac {\sqrt {3}}{2{\sqrt {3}}}}={\frac {1}{2}}}$, is a rational number. However, the numbers ${\textstyle {\sqrt {3}}}$ and 2 are incommensurable because their ratio, ${\textstyle {\frac {\sqrt {3}}{2}}}$, is an irrational number.