# 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

*. (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.*

**incommensurable**For example, the numbers 3 and 2 are commensurable because their ratio, 3/2, is a rational number. The numbers and are also commensurable because their ratio, , is a rational number. However, the numbers and 2 are incommensurable because their ratio, , is an irrational number.

More generally, it is immediate from the definition that if *a* and *b* are any two non-zero rational numbers, then *a* and *b* are commensurable; it is also immediate that if *a* is any irrational number and *b* is any non-zero rational number, then *a* and *b* are incommensurable. On the other hand, if both *a* and *b* are irrational numbers, then *a* and *b* may or may not be commensurable.