have this useful property. Construction of the real numbers A typical Dedekind cut of the rational numbers is given by the partition with This cut represents the irrational number in Dedekind's construction. The essential idea is that we use a set , which is the set of all rational numbers whose squares are less than 2, to "represent" number , and further, by defining properly arithmetic operators over these sets (addition, subtraction, multiplication, and division), these sets (together with these arithmetic operations) form the familiar real numbers. To establish this, one must show that really is a cut (according to the definition) and the square of , that is (please refer to the link above for the precise definition of how the multiplication of cuts is defined), is (note that rigorously speaking this number 2 is represented by a cut ). To show the first part, we show that for any positive rational with , there is a rational with and . The choice works, thus is indeed a cut. Now armed with the multiplication between cuts, it is easy to check that (essentially, this is because ). Therefore to show that , we show that , and it suffices to show that for any , there exists , . For this we notice that if , then for the constructed above, this means that we have a sequence in whose square can become arbitrarily close to , which finishes the proof. Note that the equality cannot hold since is not rational. Relation to interval arithmetic Given a Dedekind cut representing the real number by splitting the rationals into where rationals in are less than and rationals in are greater than , it can be equivalently represented as the set of pairs with and , with the lower cut and the upper cut being given by projections. This corresponds exactly to the set of intervals approximating . This allows the basic arithmetic operations on the real numbers to be defined in terms of interval arithmetic. This property and its relation with real numbers given only in terms of and is particularly important in weaker foundations such as constructive analysis. Generalizations Arbitrary linearly ordered sets In the general case of an arbitrary linearly ordered set X, a cut is a pair such that and , imply . Some authors add the requirement that both A and B are nonempty. If neither