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Real number

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In mathematics, real numbers are thought of informally as quantities identified with points on an infinitely long gapless straight line. The number zero is one such point; positive numbers are to its right and negative numbers to its left. The integers (positive or negative "whole numbers") ..., −3, −2, −1, 0, 1, 2, 3, ... are real numbers, and points on the line between integers (e.g. 2.75) are also real numbers.

The reason such numbers are called real numbers is that mathematicians have found it useful to introduce what are called imaginary numbers—a misnomer—such as \scriptstyle{i}, sometimes represented as \scriptstyle{\sqrt{-1}}. The term real number was introduced to distinguish real numbers from complex numbers (so-called because they can be expressed as the sum of a real number and a purely imaginary number, i.e., the product of a real number and i ).

Contents

Basic properties

A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero.

Real numbers measure continuously varying quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823211247…. The three dots indicate that there would still be more digits to come.

More formally, the field of real numbers has the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the field of real numbers up to isomorphism, and thus allow its other properties to be deduced. For instance, we can prove from these properties that every polynomial of odd degree with real numbers as coefficients has a root in the real numbers.

Uses

Measurements in the physical sciences are almost always conceived of as approximations to real numbers. While the numbers used for this purpose are generally decimal fractions representing rational numbers, writing them in decimal terms suggests they are an approximation to a theoretical underlying real number.

A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.

Computers can only represent most real numbers approximately. Most commonly, they can represent a certain subset of the rationals exactly, via either floating point numbers or fixed-point numbers, and these rationals are used as an approximation for other nearby real values. Arbitrary-precision arithmetic is a method to represent arbitrary rational numbers, limited only by available memory, but more commonly one uses a fixed number of bits of precision determined by the size of the processor registers. In addition to these rational values, computer algebra systems are able to treat many (countable) irrational numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their rational approximation.

Mathematicians use the symbol R (or alternatively,  \Bbb{R} ) to represent the set of all real numbers. The notation Rn refers to an n-dimensional space of real numbers; for example, a value from R3 consists of three real numbers and specifies a location in 3-dimensional space.

In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.

History

Vulgar fractions had been used by the Egyptians around 1000 BCE; the Vedic "Sulba Sutras" ("rule of chords" in Sanskrit), ca. 600 BCE, contained the first use of irrational numbers, and an approximation of π at 3.16.

Around 500 BCE, the Greek mathematicians led by Pythagoras realized the need for irrational numbers in particular the irrationality of the square root of two. Negative numbers were invented by Indian mathematicians around 600 CE, and then possibly invented independently in China shortly after. They were not used in Europe until the 17th century, but even in the late 1700s, Leonhard Euler discarded negative solutions to equations as unrealistic.

In the 18th and 19th centuries there was much work on irrational and transcendental numbers. Lambert (1761) gave the first flawed proof that π cannot be rational, Legendre (1794) completed the proof, and showed that π is not the square root of a rational number. Ruffini (1799) and Abel (1842) both construct proofs of Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.

Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals which gave rise to Galois theory. Joseph Liouville (1840) showed that neither e nor e^2 can be a root of an integer quadratic equation, and then established existence of transcendental numbers, the proof being subsequently displaced by Georg Cantor (1873). Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Hurwitz and Paul Albert Gordan.

The development of 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. In 1874 he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his method was not his famous diagonal argument, which he published three years later.

Definition

Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like {3, 3.1, 3.14, 3.141, 3.1415,…} converges to a unique real number. For details and other construction of real numbers, see construction of real numbers.

Axiomatic approach

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

The last 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.

For another axiomatization of R, see Tarski's axiomatization of the reals.

Properties

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 (xn) 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 |xn − xm| 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 xn eventually come and remain arbitrarily close to each other.

A sequence (xn) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − 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. An important fact about the real numbers is that the converse is also 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

\mathrm{e}^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

converges to a real number because for every x the sums

\sum_{n=N}^{M} \frac{x^n}{n!}

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 the limit is not known in advance.

"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 is 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 (in this case, the additive group of the field) 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.

Advanced properties

Continuum hypothesis

The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly bigger than the cardinality of N. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory.

Metric structure

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), connected metric space of dimension 1, and are everywhere dense. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals.

Real closed field

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.

Group-invariant measure

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.

Model theory

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 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.

Generalizations and extensions

The real numbers can be generalized and extended in several directions. Perhaps the most natural extension are the complex numbers, which contain solutions to all polynomial equations, and, unlike the real numbers are, are therefore an algebraically closed field . 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, the two 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.

Further reading




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