You can't read this page like it's a novel - it's not a fluffy, gentle read, it requires drawing diagrams and trying to figure out what's being said.
A Dedekind cut 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.

Dedekind cuts are one method of construction of the real numbers.

The set $B$ may or may not have a smallest element among the rationals. 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.

$\quad\quad$
 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 .... Richard Dedekind, Continuity and Irrational Numbers, Section IV
$\quad\quad$

More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts $A$ and $B,$ such that $A$ is closed downwards (meaning that for all $a$ in $A,$ $x\le{a}$ implies that $x$ is in $A$ as well) and $B$ is closed upwards, and $A$ contains no greatest element.

It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the $B$ set). In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.

AbsoluteValue
AlgebraicNumber
ArgandDiagram
CauchySequence
ComplexConjugate
ComplexNumber
ContinuedFraction
ContinuumHypothesis
CubeRoot
DifferenceOfTwoSquares
DividingComplexNumbers
DividingRealNumbers
Divisor
DomainOfAFunction
Function
FundamentalTheoremOfAlgebra
GeorgCantor
ImaginaryNumber
Integer
IrrationalNumber
Logarithm
MagnitudeOfAVector
MathematicsTaxonomy
MultiplyingRealNumbers
NewUserIntroduction
NewtonsMethod
PolarRepresentationOfAComplexNumber
Quaternion
RationalNumber
RiemannHypothesis
RootsOfPolynomials
SquareRoot
SubtractingRealNumbers
TranscendentalNumber
TypesOfNumber
UncountableSet
ComplexPlane Axiom
EuclideanAlgorithm
EuclideanGeometry
MatrixTransformation
PerfectNumber
PoincaresDisc
RealNumber Euclid

You are here

DedekindCut
IrrationalNumber
RationalNumber
AlgebraicNumber
ContinuedFraction
CauchySequence
TranscendentalNumber
CountableSet
Denominator
Integer
Numerator
Pythagoras
ReducingFractionsToLowestTerms
RootTwoIsIrrational
SquareRoot