Construction of real numbers

Real numbers can be constructed from rational numbers in a number of ways.

Construction from Cauchy sequences

If we have a space where Cauchy sequences are meaningful (such as a metric space, i.e., a space where distance is defined, or more generally a uniform space), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completion). By starting with rational numbers and the metric d(x,y) = |x - y|, we can construct the real numbers, as will be detailed below. (If we started with a different metric on the rationals, we'd end up with the p-adic numbers instead.)

Let R be the set of Cauchy sequences of rational numbers. Cauchy sequences (x_n) and (y_n) can be added, multiplied and compared as follows:

(x_n) + (y_n) = (x_n + y_n)

(x_n) × (y_n) = (x_n × y_n)

(x_n) ≥ (y_n) if and only if for every ε > 0, there exists an integer N such that x_n ≥ y_n - ε for all n > N.

Two Cauchy sequences are called equivalent if the sequence (x_n - y_n) has limit 0. This does indeed define an equivalence relation, it is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all the usual axioms of the real numbers. We can embed the rational numbers into the reals by identifying the rational number r with the sequence (r,r,r,...).

A practical and concrete representative for an equivalence class representing a real number is provided by the representation to base b -- in practice, b is usually 2 (binary), 8 (octal), 10 (decimal) or 16 (hexadecimal). For example, the number π = 3.14159... corresponds to the Cauchy sequence (3,3.1,3.14,3.141,3.1415,...). Notice that the sequence (0,0.9,0.99,0.999,0.9999,...) is equivalent to the sequence (1,1.0,1.00,1.000,1.0000,...); this shows that 0.9999... = 1.

Construction by Dedekind cuts

A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards. Real numbers can be constructed as Dedekind cuts of rational numbers. In detail, one can make the following definitions. (These are of value in extending some definitions to combinatorial game theory.)

Certain arithmetic operations and set-theoretic notions which apply to the real numbers can be defined correspondingly for Dedekind cuts as follows:

1. Comparison. Two Dedekind cuts, (A_x, B_x) and (A_y, B_y) are equal:

and (A_x, B_x) is less than, or equal to, (A_y, B_y):

2. Addition. The sum of two Dedekind cuts:

3. Subtraction. The difference between two Dedekind cuts:

4. Multiplication. The product of two Dedekind cuts, in case

5. Division. The quotient of two Dedekind cuts, in case  

6. Completeness. The supremum of a set of Dedekind cuts which is bounded above:

and the infimum of a set of Dedekind cuts which is bounded below:

Based on the above definitions it is perhaps worth noting that the sum of certain pairs of Dedekind cuts is not necessarily itself a Dedekind cut. Considering for instance the sum   of

and  ,

i. e.   ,

is not a Dedekind cut at all; it is not a partition of the set   of rational numbers because the rational number   is neither an element of set  , nor an element of set  . In this regard (at least) the real numbers are apparently not represented as Dedekind cuts of the rational numbers.

Construction by decimal expansions

We can take the infinite decimal expansion to be the definition of a real number, considering expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally.

Construction from ultrafilters

As in the hyperreal numbers, we construct ^*Q from the rational numbers using an ultrafilter. We take then the ring of all elements in ^*Q whose absolute value is less than some nonzero natural number (it doesn't matter which). This ring has a unique maximal ideal, the infinitesimal numbers. Factoring a ring by a maximal ideal gives a field, in this case the field of reals. It turns out that the maximal ideal respects the order on ^*Q, so the field we get is an ordered field. Completeness can be proven in a similar way to the construction from the Cauchy sequences.

Construction from surreal numbers

Every ordered field can be embedded in the surreal numbers. The real numbers form the largest subfield that is Archimedean (meaning that no real number is infinitely large).