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A more rigorous definition of the real numbers was one of the most important developments of 19th Century mathematics. Popular definitions in use today include Equivalence Class es of Cauchy Sequence s of rational numbers, Dedekind Cut s, a more sophisticated version of "decimal representation", and an axiomatic definition of the real numbers as the unique Complete Archimedean Ordered Field . Real numbers are so called to distinguish them from " Imaginary Number s" (what we now call " Complex Number s"). BASIC PROPERTIES A real number may be either Rational or Irrational ; either Algebraic or Transcendental ; and either Positive , Negative , or Zero . Real numbers measure Continuous quantities. They may in theory be expressed by Decimal Representation s 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 Ellipsis (three dots) indicate that there would still be more digits to come. More formally, real numbers have 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 on a number line 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 real numbers completely, and allow its other properties to be deduced. For instance, we can prove from these properties that every polynomial of odd degree with real coefficients has a real root, and that if you add the square root of −1 to the real numbers, obtaining the Complex Number s, the result is Algebraically Closed . USES Measurements in the Physical Science s are almost always conceived of as approximations to real numbers. While the numbers used for this purpose are generally Decimal Fraction s 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 Number s is broader, but still only countable. Computer s can only approximate most real numbers. 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 Bit s 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, , the letter " R " in Blackboard Bold , Unicode ℝ) to represent the Set of all real numbers. The Notation R''n'' refers to an ''n''- Dimension al space with real coordinates; 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 ''. As a substantive, the term is used almost strictly in reference to the real numbers, themselves (e.g., The "set of all reals"). HISTORY Vulgar Fraction s had been used by the Egyptians around 1000 BC ; the Vedic " Sulba Sutras " ("rule of chords" in Sanskrit ), ca. 600 BC , include what may be the first 'use' of Irrational Numbers . Around 500 BC , the Greek mathematicians led by Pythagoras realized the need for Irrational Number s in particular the irrationality of the Square Root Of Two . In the 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 the field of 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 Number s is Countably Infinite . Contrary to widely held beliefs, his method was not his famous Diagonal Argument , which he published in 1891. DEFINITION See Also: Construction of real numbers 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 constructions 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 Space s or Uniform Space s, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: | ||
| The Real Numbers Form A ''x'' &minus ''y'' By Virtue Of Being A | "http://wwwinformationdelightinfo/information/entry/total_order" class="copylinks">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 ), Separable 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 |
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