Real number

The real numbers form a major in mathematics, speed range. They form an extension of the range of rational numbers of fractions, whereby the measured values usual for physical quantities, such as length, temperature, and mass can be considered as real numbers. The real numbers have special topological properties over the rational numbers. These are, inter alia, that exist for every problem for which any good in a sense approximate solutions in the form of real numbers, a real number exists as an exact solution. Therefore, they can be used in many ways in analysis, topology and geometry. For example, lengths and surface of very diverse geometric objects can be usefully defined as real numbers, but not as rational numbers. If mathematical concepts in empirical sciences - such as lengths - used to describe, therefore, the theory of real numbers often plays an important role there as well.

Classification of real numbers

The set of real numbers corresponds to the set of all points of the real line. To their name, the symbol ( also, Unicode U 211 D: ℝ ) is used. The real numbers are divided into:

Rational numbers - integers -. natural numbers - ( without 0 ) or ( 0 ) is (also ).

Irrational algebraic numbers and
Transcendental numbers.

The rational numbers are those numbers that can be represented as a fraction of integers. A number is called irrational if it is real, but not rational. The first evidence that the number line contains irrational numbers were led by the Pythagoreans. Irrational numbers are for example the circle number (Pi), the Euler number or the non-integer roots of integers as or.

A rational numbers rich subset of the real numbers is the set of (real) algebraic numbers, that is, the real solutions of polynomial equations with integer coefficients. This amount includes, among other things, all real - th roots of rational numbers and their finite sums for, but not limited to (eg solutions of appropriate equations of degree 5 ). Your complement is the set of (real) transcendental numbers.

Notation for frequently used subsets of the real numbers

Is then referred to

This notation is particularly common with a = 0 is used to designate the set of positive real numbers, or the amount of the non-negative real numbers. Occasionally found in this special case (a = 0) the names or. Here, however, caution is required because at some authors is included zero, but not in others.

Construction of the real from the rational numbers

The construction of the real numbers as a number field extension of the rational numbers in the 19th century was an important step to provide the analysis on a solid mathematical foundation. The first exact construction probably goes back to Karl Weierstrass, who defined the real numbers on limited series of positive terms.

Today, common constructions of the real numbers:

View as Dedekind cuts of rational numbers: The real numbers as the smallest upper bounds of limited upward subsets of the rational numbers are defined.
View as equivalence classes of Cauchy sequences: This now widespread construction of the real numbers probably goes back to Georg Cantor, who defined the real numbers as equivalence classes of rational Cauchy sequences. Two Cauchy sequences are considered equivalent if their ( pointwise ) differences form a null sequence. How to nachprüft relatively easy, this relation is indeed reflexive, transitive and symmetric, so suitable for the formation of equivalence classes.

View as equivalence classes of intervals of rational intervals.
Completion of the topological group of the rational numbers in the sense that the canonical uniform structure is completed.

The four methods of construction called " complete " ( complete ) all the rational numbers and lead to the ( up to isomorphism ) of the same structure, the field of real numbers. Each of the methods lit another property of the rational and real numbers and their relationship to each other:

The method of Dedekind cuts completes the order on the rational numbers to order complete order. As a result, there are the rational numbers (in the sense of order ) is dense in the real numbers and each upward bounded subset has a supremum.
The method of Cauchy completes the set of rational numbers and a metric space to complete metric space in the topological sense. Thus, the rational numbers in the topological sense are dense in the reals and every Cauchy sequence has a limit. This method the complete ( completion ) is also applicable to many other mathematical structures.
The method of nested intervals reflects the numeric calculation of real numbers: they are approximated by approximate values with a certain accuracy ( approximation error ), ie enclosed in an interval around the estimate. The proof that the approximation can be arbitrarily improved ( by iterative or recursive process ), then a proof of the "existence" of a real limit.
The method of the completion of a uniform structure uses a special general concept which can be applied not only to parent or provided with a notion of distance structures such as the rational numbers.

Construction of the real numbers from the Euclidean geometry

Based on purely geometrical concepts such as points, lines, and planes can define real numbers as ratios of line sizes. The starting point is for example Hilbert's system of axioms of Euclidean geometry. In addition to the geometrical axioms is especially called " axiom of measuring " a variation of the Archimedean axiom of meaning and a " completeness axiom ", which states that one can not take points added without the axioms are violated.

Axiomatic introduction of real numbers

The construction of the real numbers as a number field extension of the rational numbers is often made in the literature in four steps: From the set theory of the natural, the whole, the rational finally to the real numbers as described above. A direct way to measure the real numbers is mathematically, to describe by axioms. This requires three groups of axioms - the field axioms, the axioms of order structure and an axiom that guarantees completeness.

Alternatively, the field of real numbers can also be characterized as a complete, Archimedean ordered field, that is, as a body satisfies the following axioms:

The field axioms and order axioms
The Archimedean axiom: Are and be positive real numbers, then there is a, so that.
The completeness axiom: Converges Every Cauchy sequence in, or in other words the real numbers are, in terms of the induced from the absolute value of a complete metric space.

Instead of the completeness axiom can also set the Intervallschachtelungsaxiom:

The Intervallschachtelungsaxiom: The average of each episode monotonically decreasing completed limited intervals is not empty.

If one introduces the real numbers axiomatically, then the design as a number range extension their " existence proof ", or more precisely: the construction in four steps from the set theory proves that a model for the described by the axioms structure in set theory, from which went out the construction is available.

Widths

The cardinality of is denoted by ( cardinality of the continuum ). It is larger than the cardinality of the set of natural numbers, which is called as the smallest infinite cardinality. The set of real numbers is therefore uncountable. A proof of its uncountable Cantor 's second diagonal argument. Informal means " uncountable " that each list of real numbers is incomplete. Since the set of real numbers is equally powerful to the power set of the natural numbers, we are also their thickness with.

The aforementioned less extensive enhancements to the set of natural numbers, however, are equally powerful with the natural numbers, that is countable. For the rational numbers, this can be proved by Cantor's diagonal argument first. Even the algebraic numbers are countable. So the uncountable obtained only through the addition of the transcendental numbers.

In set theory, the question was examined after Cantor's discoveries: "Is there a thickness between " countable " and the cardinality of the real numbers between and? " - Or, formulated for the real numbers: " Is every uncountable subset of the real numbers the same cardinality as the set of all real numbers? " The assumption that the answer to the first question "No " and to the second question is "Yes" is called the continuum hypothesis (CH ), briefly formulated as = and. It could be shown that the continuum hypothesis is independent of the axiom systems such as the Zermelo -Fraenkel set theory with the axiom of choice commonly used ( ZFC ), that is, it can not be proved or disproved in the context of these systems.

Topology, compactness, extended real numbers

The usual topology which the real numbers are provided, is the one from the base of the open intervals

Is generated. Written in this form, it is the order topology. Open intervals in the real numbers can also using the center and represent "Radius": So as open balls

With respect to the metric defined by the absolute value function, the topology generated by the open intervals is thus simultaneously the topology of this metric space. Since the rational numbers are dense in this topology, it is enough to be confined at the interval boundaries and the centers and radii of the balls that define the topology on rational numbers, so the topology satisfies both Abzählbarkeitsaxiomen.

In contrast to the rational numbers, the real numbers are a locally compact space; to any real number then, to an open environment indicate their completion is compact. So an open environment is easy to find; any bounded, open set with making the Desired: by the theorem of Heine- Borel compact.

The real number field is only locally compact but not compact. A common compactification are the so-called extended real numbers with the environments of the surrounding base

And the environments of the surrounding base

Be defined. This topology is sufficient to continue both Abzählbarkeitsaxiomen. is homeomorphic to the closed interval [0,1], for example, the mapping is a homeomorphism and all compact intervals are homeomorphic by affine- linear functions. Determines divergent sequences are convergent in the topology of the extended real numbers, for example, is the statement

In this topology of a real limit.

With all the extended real numbers are still totally ordered. However, it is not possible to transfer the body structure of the real numbers in the expanded real numbers, for example, the equation has no unique solution.