Real number

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The real numbers are those numbers that are used to represent a continuous quantity (including 0 and negatives). One may think of a real number as a possibly infinite decimal fraction, such as 324.823211247...; the real numbers also stand in a one-to-one correspondence with the points on a line. Measurements in the physical sciences are almost always expressed as real numbers.

The real numbers contain the integers and the rational numbers as well as irrational numbers such as the square root of 2 and transcendental numbers such as pi.

The term "real number" is a retronym coined in response to "imaginary number".

Computers can only approximate most real numbers; these approximations are known as floating point numbers. Computer algebra systems are able to treat some real numbers exactly by storing their algebraic description (such as sqrt(2)) rather than their decimal approximation.

Mathematicians use the symbol R (or more properly, a double-barred R represented by the unicode character ℝ if your browser supports unicode display) to represent the set of all real numbers.

What follows is a rigorous mathematical description of the concept. The real numbers are the central object of study in the mathematical field of analysis.

Axioms

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

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 least upper bound, because the square root of 2 is not rational.

The real numbers are uniquely specified by the above properties.

Completeness

The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete. 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 depending on ε such that the distance |xn - xm| < ε provided that n, m > N. In other words, a sequence is a Cauchy sequence if its elements xn eventually remain arbitrarily close to each other.

A sequence {xn} converges to the limit x if for any ε > 0 there exists an integer N depending on ε such that the distance |xn - x | < ε provided that n > N. In other words, a sequence has limit x if its elements eventually come arbitrary 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.

The reals are complete. Note that the rationals are not complete.

The existence of limits of Cauchy sequences is what makes analysis and the 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

ex = Σn = 0 xn / n!

converges to a real number because for every x the tail sums

Σn=NM xn / n!

can be made arbitrarily small by choosing N sufficiently large.

Construction from the rational numbers

If we have a space where Cauchy sequences are meaningful (a metric space, i.e. a space where distance is defined), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completing). 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 (xn) and (yn) can be added, multiplied and compared as follows:

(xn) + (yn) = (xn + yn)
(xn) * (yn) = (xn * yn)
(xn) ≥ (yn) if and only if there exists an integer N such that xnyn for all n > N.

Two Cauchy sequences are called equivalent if the sequence (xn - yn) 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 axioms of the real numbers given above. 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 (usually 2 (binary), 10 (decimal) or 16 (hexadecimal)). For example the number π = 3.14159.... would correspond to the Cauchy sequence {3, 3.1, 3.14, 3.141, 3.1415, ...}. A real number can have two representations. For example: 1 = 0.99999.... .

Advanced properties

The reals are uncountable, that is, there are strictly more real numbers than integers. This is proved with Cantor's diagonal argument. In fact, the cardinality of the reals is 2ω (see cardinal numbers), i.e. the cardinality of the set of subsets of the integers. 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 non negative real number has a square root. 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.

The reals are one of the two local fields of characteristic 0 (the other one being the complex numbers).

The reals carry a canonical measure, the Lebesgue measure which is the Haar measure normalised such that the 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 Lowenheim-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.

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 infintely large numbers. Self-adjoint operators on a Hilbert space (including 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.

History

Fractions had been used by the Egyptians a thousand years BC; the Greek mathematicians around 500 BC realized the need for irrational numbers. Negative numbers begun to be generally accepted in the 1600s. The development of the calculus in the 1700s used the real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871.


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