in mathematics, concept advanced in 1872 by the German mathematician Richard Dedekind (Dedekind, Richard) that combines an arithmetic formulation of the idea of continuity with a rigorous distinction between rational and irrational numbers (irrational number). Dedekind reasoned that the real numbers (real number) form an ordered continuum, so that any two numbers x and y must satisfy one and only one of the conditions x\\<y, x=y, or x\\>y. He postulated a cut that separates the continuum into two subsets, say X and Y, such that if x is any member of X and y is any member of Y, then x\\<y. If the cut is made so that X has a largest rational member or Y a least member, then the cut corresponds to a rational number. If, however, the cut is made so that X has no largest rational member and Y no least rational member, then the cut corresponds to an irrational number.

For example, if X is the set of all real numbers x less than or equal to 22/7 and Y is the set of real numbers y greater than 22/7, then the largest member of X is the rational number 22/7. If, however, X is the set of all real numbers x such that x² is less than or equal to 2 and Y is the set of real numbers y such that y² is greater than 2, then X has no largest rational member and Y has no least rational member: the cut defines the irrational number √2.