The real numbers form an important number range in mathematics . It is an extension of the range of rational numbers , the openings , whereby the numerical values of the measured values for conventional physical quantities such as length , temperature or mass can be considered as real numbers. The real numbers include the rational numbers and the irrational numbers .
Compared to the rational numbers, the real numbers have special topological properties . Among other things, these consist in the fact that for every “continuous problem” for which, in a certain sense, arbitrarily good, close approximate solutions exist in the form of real numbers, there is also a real number as an exact solution. Therefore, they can be used in many ways in analysis , topology and geometry . For example, lengths and areas of very diverse geometric objects can be meaningfully defined as real numbers, but not as rational numbers. If mathematical concepts - such as lengths - are used for description in empirical sciences , the theory of real numbers often plays an important role there too.
Classification of the real numbers
- rational numbers :
- irrational numbers : = the set of all elements of that are not in . These can in turn be divided into:
The rational numbers are those numbers that can be represented as fractions of whole numbers. A number is called irrational if it is real but not rational. The first evidence that the number line contains irrational numbers came from the Pythagoreans . For example, irrational numbers are the non-integer roots of integers such as or .
A subset of the real numbers comprising the rational numbers is the set of (real) algebraic numbers, i. H. of real solutions of polynomial equations with integer coefficients. This set includes, among other things, all real -th roots of rational numbers for and their finite sums, but not only these (e.g. solutions of suitable equations of the 5th degree ). Their complement is the set of (real) transcendent numbers. A transcendent number is therefore always irrational. For example, the circle number (Pi) and Euler's number are transcendent . All examples mentioned so far are calculable, in contrast to the limit value of a Specker sequence .
Notation for frequently used subsets of real numbers
Is , then designated
- the set of all real numbers except the number a,
This notation is used particularly often to denote the set of positive real numbers or the set of nonnegative real numbers. Occasionally the terms or are also used for special cases . Caution is advised here, however, as some authors include the zero and others do not.
Construction of the real from the rational numbers
The construction of the real numbers as a range extension of the rational numbers was an important step in the 19th century in order to put analysis on a solid mathematical foundation. The first exact construction probably goes back to Karl Weierstraß , who defined the real numbers by means of limited series with positive terms.
Common constructions of real numbers today:
- Representation as Dedekindian cuts of rational numbers: The real numbers are defined as the smallest upper bound of subsets of the rational numbers with an upper limit .
- Representation as equivalence classes of Cauchy sequences : This construction of real numbers, which is most widespread today, 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 (point-wise) differences form a zero sequence . As is relatively easy to check, this relation is actually reflexive , transitive and symmetrical , i.e. suitable for the formation of equivalence classes.
The addition and multiplication of the equivalence classes induced by the rational numbers are well defined , that is, independent of the choice of the representative, i.e. a Cauchy sequence. With these well-defined operations, the real numbers defined in this way form a field . A total order is also induced by the rational numbers . Overall, the real numbers thus form an ordered body .
- Representation as equivalence classes of interval nesting of rational intervals.
- Completion of the topological group of rational numbers in the sense that the canonical uniform structure is completed.
The four construction methods mentioned “complete” (complete) all the rational numbers and lead to the same structure (except for isomorphism ), the field of real numbers. Each of the methods illuminates a different property of the rational and real numbers and their relationship to one another:
- The method of Dedekindian cuts completes the order on the rational numbers to an order-complete order. As a result, the rational numbers (in the sense of order) lie close to the real numbers and every subset bounded upwards has a supremum.
- The Cauchy sequence method completes the set of rational numbers as metric space to a complete metric space in the topological sense. Thus the rational numbers in the topological sense lie close to the real numbers and every Cauchy sequence has a limit value. This method of completion (completion) can also be used with many other mathematical structures.
- The method of intervals reflects the numerical calculation of real numbers: You are approximations with a certain accuracy (approximation error) approximated , so included in an interval around the estimate. The proof that the approximation can be improved at will (by iterative or recursive procedures) is then a proof for the “existence” of a real limit value.
- The method of completing a uniform structure uses a particularly general concept that can not only be applied to ordered or spaced structures such as rational numbers.
Axiomatic introduction of real numbers
The construction of real numbers as a number range extension of the rational numbers is often carried out in four steps in the literature: From set theory to natural, whole, rational and finally to real numbers as described above. A direct way to grasp the real numbers mathematically is to describe them by axioms . For this one needs three groups of axioms - the body axioms, the axioms of the order structure and an axiom which guarantees completeness.
- The real numbers are a field .
- The real numbers are totally ordered (see also ordered fields ), i.e. i.e., for all real numbers :
- The real numbers are order-complete , i. That is, every non-empty, upwardly bounded subset of has a supremum in .
If one introduces the real numbers axiomatically, then the construction as an extension of the number range is a possibility for the proof of their existence, more precisely: The construction in four steps from set theory proves that a model for the structure described by the axioms in set theory, of the the construction ran out, exists. In addition, it can be shown that the specified axioms clearly determine the field of the real numbers except for isomorphism. This essentially follows from the fact that a model of real numbers does not allow any other automorphism besides identity.
Instead of the axioms mentioned above, there are other possibilities to characterize the real numbers axiomatically. In particular, the axiom of completeness can be formulated in different ways. In particular, there are different ways of expressing completeness for the construction options described above, as the next section shows.
Axioms equivalent to the supremum axiom
As an alternative to the supremum axiom, the following can be requested
- The axiom of Archimedes and the completeness axiom, which states that every Cauchy sequence in converged .
- The axiom of Archimedes and the Intervallschachtelungsaxiom, which states that the average, every monotonically decreasing sequence of closed bounded intervals is not empty.
- The infimum axiom, which says that every nonempty downwardly bounded subset of has an infimum.
- The Heine-Borel axiom, which says that if a closed, bounded interval of is covered by any number of open sets of , there are always only finitely many of these open sets that already cover the interval.
- The Bolzano-Weierstrasse axiom, which says that every infinite bounded subset has at least one accumulation point.
- The axiom of monotony, which says that every monotonic bounded sequence converges in.
- The axiom of connection that states that the real numbers, provided with the usual topology, form a connected topological space.
There is also the possibility of describing completeness in terms of continuous functions by raising certain properties of continuous functions to axioms. About:
- The axiom of intermediate values:
- A continuous real function defined on an interval of always assumes every intermediate value in its value range.
- The axiom of limitation:
- A continuous real function defined on a closed and bounded interval of always has an upwardly bounded range of values.
- The maximum axiom:
- A continuous real function defined on a closed and bounded interval of always has a maximum point.
The thickness of is denoted by ( thickness of the continuum ). It is greater than the power of the set of natural numbers , which is called the smallest infinite power . The set of real numbers is therefore uncountable . A proof of their uncountability is Cantor's second diagonal argument . Informally, "uncountability" means that any list of real numbers is incomplete. Since the set of real numbers is equal to the power set of natural numbers, their cardinality is also specified .
The less comprehensive expansions of the set of natural numbers mentioned at the beginning , on the other hand, are equal to the set of natural numbers, that is, countable . For the set of rational numbers , this can be proven by Cantor's first diagonal argument . Even the set of algebraic numbers and more generally the set of computable numbers can be counted. The uncountability arises only through the addition of the non-calculable transcendent numbers.
In set theory, after Cantor's discoveries, the question was investigated: “Is there a power between“ countable ”and the power of the real numbers, between and ?” - Or, formulated for the real numbers: “Is every uncountable subset of real numbers equal to Set of all real numbers? ”The assumption that the answer to the first question is“ No! ”And to the second question“ Yes ”is called the continuum hypothesis (CH) , in short as = and . It could be shown that the continuum hypothesis is independent of the commonly used axiom systems such as the Zermelo-Fraenkel set theory with axiom of choice (ZFC) d. That is, it can neither be proven nor refuted within the framework of these systems.
Topology, compactness, extended real numbers
is produced. Written in this form, it is the order topology . Open intervals in the real numbers can also be represented by the center point and radius : that is, as open spheres
with regard to the metric defined by the absolute value function . The topology generated by the open intervals is therefore also the topology of this metric space . Since the rational numbers are close in this topology , it is sufficient to limit the interval boundaries or the centers and radii of the balls that define the topology to rational numbers ; the topology therefore satisfies both axioms of countability .
In contrast to the rational numbers, the real numbers are a locally compact space ; For every real number , an open environment can be specified, the closure of which is compact. Such an open environment is easy to find; any bounded, open set with satisfies the requirements: after the set of Heine-Borel is compact.
and the surroundings of by the environment base
To be defined. This topology still satisfies both axioms of countability. is homeomorphic to the closed interval [0,1], for example the mapping is a homeomorphism and all compact intervals are homeomorphic by means of affine-linear functions. Certainly divergent sequences are convergent in the topology of the extended real numbers, for example the statement acts
in this topology of a real limit value.
With for all the extended real numbers are still totally ordered. However, it is not possible to transfer the body structure of the real numbers to the extended real numbers, for example the equation has no unique solution.
- A non-standard model of analysis is derived from model theory.
- An approximate representation of real numbers in the computer is done by floating point numbers .
- The interval arithmetic enables calculations taking into account the approximation errors .
- Numbers are represented in a number system .
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