Incommensurable

Incommensurable describes two quantities that do not share a common measure. In mathematics, two positive quantities a and b are incommensurable if their ratio a/b is irrational; equivalently, there is no positive real unit such that both a and b are integer multiples of that unit. If a/b is rational, the quantities are commensurable.

A classic example is the side and the diagonal of a square. If the side length is taken as 1, the diagonal length is √2, which is irrational. Since √2 has no rational multiple that yields the same unit as the side, the side and diagonal are incommensurable. This phenomenon was central to early Greek mathematics and is traditionally linked to the discovery of irrational numbers, often attributed (in historical accounts) to the Pythagoreans and Hippasus.

In general usage, incommensurability refers to the absence of a common measuring unit for two quantities. It is possible for both numbers to be irrational and still be commensurable (for example, √2 and 2√2 have a rational ratio 1/2). Conversely, the ratio of two numbers can be irrational even if one or both are rational in other contexts.

Beyond pure mathematics, the term is also used metaphorically to describe ideas, scales, or theories that cannot be directly compared using a shared framework or unit. In such cases, incommensurability signals fundamental differences that resist simple reduction to a common measure.