1 History

Fractions had been used by the Egyptians around 1000 BC; around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers. Negative numbers began to be generally accepted in the 1600s and were invented by Muslim mathematicians.

The development of the calculus in the 1700s used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871.

2 Definition

2.1 Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers.

2.2 Axiomatic approach

Let R denote the set of all real numbers. Then:

• The set R is a field, i.e., addition, subtraction, multiplication and division are defined and have the usual properties.
• The field R is ordered, i.e., there is a total order ≥ such that, for all real numbers x, y and z:
  • if x ≥ y then x + z ≥ y + z;
  • if x ≥ 0 and y ≥ 0 then xy ≥ 0.
• The order is Dedekind-complete, i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.

The latter property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.

The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R₁ and R₂, there exists a unique field isomorphism from R₁ to R₂, allowing us to think of them as essentially the same mathematical object.

3 Properties

3.1 Completeness

The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following:

A sequence (x_n) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n - x_m| is less than ε provided that n and m are both greater than N. In other words, a sequence is a Cauchy sequence if its elements x_n eventually come and remain arbitrarily close to each other.

A sequence (x_n) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n - x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x.

It is easy to see that every convergent sequence is a Cauchy sequence. Now the important fact about the real numbers is that the converse is true:

Every Cauchy sequence of real numbers is convergent.

That is, the reals are complete.

Note that the rationals are not complete. For example, the sequence (1,1.4,1.41,1.414,1.4142,1.41421,...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.)

The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance.

For example the standard series of the exponential function

converges to a real number because for every x the sums

can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we don't know ahead of time what the limit is.

3.2 "The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

First, an order can be lattice complete. It's easy to see that no ordered field can be lattice complete, because it can have no largest element (given any element z, z + 1 is larger). So this is not the sense that is meant.

Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way.

These two notions of completeness ignore the field structure. However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way.

But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

3.3 Advanced properties

The reals are uncountable, that is, there are strictly more real numbers than natural numbers (even though both sets are infinite). This is proved with Cantor's diagonal argument. In fact, the cardinality of the reals is 2^ω (see cardinal numbers), i.e., the cardinality of the set of subsets of the natural numbers. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The nonexistence of a subset of the reals with cardinality strictly in between that of the integers and the reals is known as the continuum hypothesis. This can neither be proved nor be disproved, but is independent from the axioms of set theory.

The real numbers form a metric space: the distance between x and y is defined to be the absolute value |x - y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical. The reals are a contractible (hence connected and simply connected), locally compact separable metric space, of dimension 1, and are everywhere dense. The real numbers are not compact. There are various properties that uniquely specify them; for instance, all unbounded, continuous, and separable order topologies are necessarily homeomorphic to the reals.

Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra.

The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1.

The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim-Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of hyperreal numbers is much bigger than R but also satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R.

4 Generalizations and extensions

The real numbers can be generalized and extended in several different directions. Perhaps the most natural extension are the complex numbers which contain solutions to all polynomial equations. However, the complex numbers are not an ordered field. Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean. Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers.

Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.

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