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
, 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
has a smallest element y
is impossible. On the one hand, the average of x
, being a rational, must belong to one of A
. On the other hand, their average cannot belong to A
) nor to B
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