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 .

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