Dedekind cut
In mathematics, a
Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty parts
A and
B, such that all elements of
A are less than all elements of
B, and
A contains no greatest element. If
B has a smallest element among the rationals, the
cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between
A and
B. In other words,
A contains every rational number
less than the cut, and
B contains every rational number
greater than the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals. Whenever, then, we have to do with a cut produced by no rational number, we create a new
irrational number, which we regard as completely defined by this cut .... From now on, therefore, to every definite cut there corresponds a definite rational or irrational number....