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
.
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