Real number

the quantity of the real numbers forms the largest human experience for accessible counting range: Each measurable size can be assigned a real number as yardstick. Thus this number term extends the quantity of the rational numbers, under those forsome lengths (for example for the diagonal of a square with the side length 1) no yardstick is present.

The difference quantity from real and rational numbers, i.e. the quantity of the numbers, which are real numbers, but not rational numbers, is called quantity that surds; their existence was discovered by the Pythagoräern.

Descriptive expressed the quantity of the real numbers corresponds to the quantity of all points of the number lines. One says: The real numbers are biuniquely (bijektiv) assigned to these points.

For thoseQuantity of the real numbers becomes the symbol <math> \ mathbb {R}< /math> (also <math> \ mathbf {R}< /math>) uses. The name „real numbers “ is to point out that by it measurable (thus material or „real “) sizes are described. The Gegenbegriff is imaginary numbers.

The real numbers and Functions of <math> \ mathbb {R}< /math> after <math> \ mathbb {R}< /math> is the investigation article of the real analysis.

Table of contents

organization of the real numbers

the quantity of the real numbers consists of the rational numbers (whole numbers such as −1, 0, 1, 2 and break numbers like 3/4, −2/3 etc.) and the surds. Typical one surds are for example:

  • the circle number π (pi),
  • the roots from whole numbers, which do not have integral values like z. B. <math> \ sqrt {2}< /math>, but sqrt <not> math \ {4} =2< /math>.

Characteristic of surds is that it represented as decimal numbers no finite number of placesafter the comma and the numbers have no periodic consequence after the comma also to form.

The rational numbers a comprehensive subset of the real numbers is the quantity of the real algebraic numbers, i.e. the real solutions of polynomial equations with integral coefficients;this quantity covers all radicals. Their complement is the quantity of the transcendental numbers; it contains for example e and π.


the term of power permits a size comparison of infinite quantities. While the quantities of the natural, wholeor rational numbers countable are, thus essentially equally large, are over countable the quantity of the real numbers, as CAN gate proved; to the proof see CAN gate second diagonal argument. Briefly said the over counting barness means that each list <math> x_1, x_2, x_3, \ ldots< /math> real numbers incompletelyis.

The quantity of the algebraic numbers is countable, the quantity of the irrational and the quantity of the transcendental numbers is alike in each case for the quantity of all real numbers.

The assumption that each over-countable quantity at least as powerfully as the quantity of the realNumbers is, continuum hypothesis is called. It is independently of the usually used axiom systems such as ZFC, i.e. it is not possible to prove or disprove it.

construction of R from Q

the quantity of the real numbersmathematically as completion of the rational numbers one defines. That is, real numbers are equivalence classes of rational Cauchy - consequences. Two Cauchy consequences are equivalent, if their (point for point) difference forms a zero-sequence. As one tests relatively easily, is this relation actually reflexiv, transitiv and symmetrically, thus to the education by equivalence classes suitably.

The addition and multiplication induced by the rational numbers are well-defined, i.e. independent of the selection of the representative. With these well-defined operations the real formPay a body. Likewise by the rational numbers a total order is induced. Altogether the real numbers thereby are an arranged body.

A further construction possibility is the representation of the real numbers than Dedekind cuts of rational numbers. Usesone out the fact that each subset of the real numbers limited upward has a smallest upper barrier and “completes” the rational numbers regarding this characteristic.

With the solution of cubic equations one stated that every now and then a square root outnegative numbers to be pulled must, which leads in further consequence again to real solutions (Casus irreducibilis). At first that was understood only as a kind computing trick, in further consequence led however to the introduction of the complex numbers.

axiomatic introduction of the real numbers

the construction of the real numbers as number range extension of the rational numbers is somewhat toilsome. A further possibility of seizing the real numbers is to be introduced it axiomatically. One essentially needs for it three groups ofAxioms - the body axioms, the axioms of the order structure as well as an axiom, which guarantee the completeness.

  1. The real numbers are a body
  2. the real numbers are totally arranged (see body also arranged), i.e. for all real numbers <math> A, b, C< /math>applies:
    1. it applies exactly one of the relations <math> for A < b, A = b, b < A< /math> (Trichotomie)
    2. out <math> A < b< /math> and <math> b < C< /math> math <A> follows < C< /math> (Transitivity)
    3. out <math> A < b< /math> math <A> follows+ C < b + C< /math> (Compatibility with the addition)
    4. out <math> A < b< /math> and <math> C > 0< /math> math <AC> follows < UC< /math> (Compatibility with the multiplication)
  3. the real numbers are order-completely, i.e. each nonempty, upward limited subset of<math> \ mathbb {R}< /math> a least upper bound possesses

alternative can the body of the real numbers be also charaktisiert as more completely, Archimedeanly arranged body, i.e. as a body fulfills the following axioms:

  • the body axioms and Ordnungsaxiome
  • the Archimedean axiom:
    Are <math> A< /math> and <math> b< /math>, then there are positive real numbers <math> a n \ in \ mathbb {N}< /math>, so that <math> well > b< /math> is.
  • the completeness axiom:
    The real numbers are concerning. the Metrik a complete area induced by the absolute value, i.e. each Cauchy consequence converges

in place of the completeness axiomone can set also the interval nesting axiom:

  • the interval nesting axiom:
    The average of each monotonous falling consequence of final limited intervals is nonempty.

By each of these axiom systems the body of the real numbers (up to isomorphism) is clearly certain.

topology, compactness,if the usual topology

extended real numbers, with which the real numbers will provide, that is, those from the basis of the open balls <math> B_r (p): = \ {x \ in \ R:\|x-p \|<r \}, p \ in \ R, r \ in \ R^+< /math> one produces. Since the rational numbers lie in this topology closely, it is enough itself,on rational <math> p, r< /math> to limit, the topology is sufficient therefore for both Abzählbarkeitsaxiomen. One can do equivalent ones the usual topology of the real numbers also as the topology of <math> \ R< /math> as metric area with the Metrik <math> D (x, y): =|XY|</math> define.

Contrary to thatrational numbers are the real numbers a locally compact area, to each real number <math> x< /math> thus open environment can be indicated, whose conclusion is compact. Such an open environment is simple to find; each limited, open quantity<math> U< /math> with <math> x \ in U< /math>, the wishing carries out: after the sentence of Heine Borel math <\> {U is cash}< /math> compactly.

The real numbers are only locally compact, not however compact. A common Kompaktifizierung are cash the so-called extended <real numbers> math \ {\ R}: = \ R \ cup \ {- \ infty, + \ infty \}< /math>, whereby those Environments of <math> - \ infty< /math> by the environment basis <math> \ mathfrak {B}: = \ {B_r (- \ infty)|r \ in {\ Bbb Q} ^+ \}< /math> with <math> B_r (- \ infty): = \ {x \ in \ R|x< -1/r \}< /math> and the environments of <math> + \ infty< /math> by the environment basis <math> \ mathfrak {B}: = \ {B_r (+ \ infty)|r \ in {\ Bbb Q} ^+ \}< /math> with <math> B_r (\ infty): = \ {x \ in \ R|x> 1/r \}< /math> are defined. This topology is sufficient further for both Abzählbarkeitsaxiomen. <math> \ bar {\ R} \; </math> is homöomorph to the final interval [0,1], a Homöomorphismus is for example the illustration <math> x \ mapsto \ frac {2} {\ pi} \ arctan x< /math>.Bestimmt divergente Folgen sind in der Topologie der erweiterten reellen Zahlen konvergent, beispielsweise ist die Aussage

<math>\lim_{n\to\infty} n^2\to\infty</math>

in this topology a genuine limit value.

With <math> - \ infty< x< \ infty< /math> for all <math> x \ in \ R< /math> are the extended real numbersfurther totally arranged; it is not possible however to transfer the body structure of the real numbers to the extended real numbers for example has the equation <math> \ infty+x= \ infty< for /math> no clear solution.

an approach

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