Dedekind cut

A Dedekind cut is a way of pinning down a real number by saying precisely which rationals lie below it. You slice the rational line cleanly into a left piece and a right piece, and the position of the cut itself is the number you have named, even when no rational sits exactly there.

Formally, a Dedekind cut is a set A of rationals that is nonempty, is not all of the rationals, is downward closed, meaning if q is in A and p is less than q then p is in A, and has no greatest element. Each such A is declared to be a real number. The rationals embed as the cuts of the form all rationals less than a fixed q, and the order on reals is just set inclusion of the left pieces.

This construction, due to Dedekind, builds the reals out of the rationals and proves rather than assumes completeness. The least upper bound property becomes almost trivial: given a bounded family of cuts, the union of their left sets is itself a cut, and it is their supremum. The gap in the rationals at the square root of 2 is filled by the cut consisting of all rationals whose square is less than 2 together with all negative rationals.

A Dedekind cut is one of two classical constructions of the reals, the other being equivalence classes of Cauchy sequences of rationals. Both yield the same object up to isomorphism, namely the unique complete ordered field, so it does not matter which you adopt; cuts make order and completeness transparent, while Cauchy sequences make arithmetic and metric completeness transparent.