Dedekind Cut

cut

0.5

kind of cut

irrational

rational

mark radii

smaller than cut

larger than cut

range of numerator and denominator

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

x<

x+y

2

) nor to B (because

x+y

2

<y

).

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

2

.