Dedekind Cut

Dedekind invented cuts to construct the real numbers from the rationals. Another method is to use Cauchy sequences.
Split the rationals in two disjoint sets A and B, such that all the elements of A are smaller than all the element of B. This is called a cut. There are four cases: A has a largest element or not, and B has a smallest element or not.
The case where A has a largest element x and B has a smallest element y is impossible. On the one hand, the average of x and y, being a rational, must belong to one of A or B. On the other hand, their average cannot belong to A (because ) nor to B (because ).
If there is a largest element of A or a smallest element of B, then the cut is rational.
In the fourth case, the most interesting one, A does not have a largest element and B does not have a smallest element. In that case the cut is irrational.
This visualization draws circles with rational radii smaller than 1. Examples of rational cuts are selected from these, with a red circle used to indicate that the rational is included in one of the two sets. Examples for irrational cuts are generated as multiples of .


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+