Commensurable and incommensurable magnitudes

commensurable and incommensurable quantities

Two magnitude of the same kind (such as lengths or surface areas) that do or do not have a so-called common measure (that is, a magnitude of the same kind contained an integral number of times in both of them). Examples of incommensurable magnitudes are the lengths of a diagonal of a square and the sides of that square, or the surface areas of a circle and the square of its radius. If two magnitudes are commensurable, then their ratio is a rational number, whereas the ratio of incommensurable magnitudes is irrational.