﻿

# Dedekind cut

two nonempty subsets of an ordered field, as the rational numbers, such that one subset is the collection of upper bounds of the second and the second is the collection of lower bounds of the first: can be used to define the real numbers in terms of the rational numbers.
[named after J.W.R. DEDEKIND]

* * *

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.

* * *

Universalium. 2010.

### Look at other dictionaries:

• Dedekind cut — Dedekind used his cut to construct the irrational, real numbers. In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non empty parts A and B, such that all elements of A are less than all… …   Wikipedia

• Dedekind cut — (or Dedekind section ) It has been known since the Greeks that there is no ratio of numbers, a/b, that is equal to the square root of 2. But there is no maximum ratio whose square is less than 2, and no minimum ratio whose square is greater than… …   Philosophy dictionary

• Dedekind cut — Math. two nonempty subsets of an ordered field, as the rational numbers, such that one subset is the collection of upper bounds of the second and the second is the collection of lower bounds of the first: can be used to define the real numbers in …   Useful english dictionary

• Cut — may refer to: The act of cutting, the separation of an object into two through acutely directed force Contents 1 Mathematics 2 Computing 3 …   Wikipedia

• Dedekind–MacNeille completion — The Hasse diagram of a partially ordered set (left) and its Dedekind–MacNeille completion (right). In order theoretic mathematics, the Dedekind–MacNeille completion of a partially ordered set (also called the completion by cuts or normal… …   Wikipedia

• Dedekind, Richard — ▪ German mathematician born Oct. 6, 1831, Braunschweig, duchy of Braunschweig [Germany] died Feb. 12, 1916, Braunschweig  German mathematician who developed a major redefinition of irrational numbers (irrational number) in terms of arithmetic… …   Universalium

• Dedekind , (Julius Wilhelm) Richard — (1831–1916) German mathematician The son of an academic lawyer from Braunschweig, Germany, Dedekind was educated at the Caroline College there and at Göttingen, where he gained his doctorate in 1852. After four years spent teaching at Göttingen,… …   Scientists

• Richard Dedekind — Infobox Scientist name = PAGENAME box width = image size =180px caption =Richard Dedekind, c. 1850 birth date = October 6, 1831 birth place = Braunschweig death date = February 12, 1916 death place = Braunschweig residence = citizenship =… …   Wikipedia

• 0.999... — In mathematics, the repeating decimal 0.999... (which may also be written as 0.9, , 0.(9), or as 0. followed by any number of 9s in the repeating decimal) denotes a real number that can be shown to be the number one. In other words, the symbols 0 …   Wikipedia

• Construction of the real numbers — In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that …   Wikipedia