Commensurable and Incommensurable Quantities

Two like quantities—for example, lengths or areas—are said to be commensurable if they have a common measure. A common measure is a quantity that is of the same nature as the quantities under consideration and that is contained in each of these quantities an integral number of times. Two like quantities that do not have a common measure are said to be incommensurable. For example, the lengths of a diagonal and a side of a square are incommensurable, as are the areas of a circle and of a square constructed on the radius of the circle.

If two quantities are commensurable, their ratio is a rational number. The ratio of two incommensurable quantities, on the other hand, is an irrational number. Consequently, if one quantity in a set of like quantities is given the value 1, quantities in the set that are commensurable with this quantity can be expressed by rational numbers, and quantities that are incommensurable with this quantity are expressed by irrational numbers. The discovery of incommensurable quantities was one of the most important achievements of ancient Greek mathematics.