Numbers are abstract, mathematical objects - objects of thought - that historically developed from ideas of greatness. By measuring a size aspect understood as an observation with a number is associated, such as a census. They therefore play a central role in the empirical sciences.
In mathematics, what numbers and their structure studied formally, the term with a very different concepts. This developed as generalizations of existing intuitive number concepts, so they are also referred to as numbers, even though they have some little relation to the originally associated with measurement concepts. Some of these concepts in mathematics is of fundamental importance and are used in almost all areas.
In prehistoric times back the concept of natural numbers, which can be used for counting and have fundamental importance enough. From about 2000 BC Egyptians and Babylonians reckoned with fractions ( rational numbers ). In India, an understanding of the zero and the negative numbers developed in the 7th century AD. Irrational numbers such as or, the need for which arose from findings from ancient Greece (at least from the 4th century BC), were introduced in the heyday of Islam.
The idea of imaginary numbers, the real numbers were later extended to the major complex numbers, dates back to the European Renaissance. The concept of real number could be clarified only in the 19th century sufficiently. End of the 19th century was the first time infinite sizes, a precise meaning be given as numbers. The natural numbers were first defined axiomatically. With the beginning of the 20th century created the first satisfactory foundations of mathematics, the most important number concepts learned a present state corresponding fully formal definition and meaning.
Delineate the concept of number are digits ( special number sign, used to represent certain numbers letters ), number fonts ( spellings of numbers, for example, with the help of digits using certain rules) number words ( numerals, used to designate specific numbers words) and numbers ( identifiers, but even numbers, or - may be strings - usually containing digits).
- 5.1 Examples
- 7.1 Prehistory
- 7.2 First civilizations
- 7.3 Greece
The German word count is probably on the Proto-Germanic word * TALŌ (calculation, number, speech ) back, which include probably the root of the Old High German words zala ( order, orderly presentation, report, list) and Zalon ( report, calculate, calculate, pay ) is. From zala was in Middle High German zale or zal, to today's word goes back number.
The Proto-Germanic word finds its origin probably in a Primitive Indo-European etymon * del ( aim, calculate, adjust ). Also a connection with the Primitive Indo-European * del- (column ) is possible; the original meaning would then possibly " notched Wish characters ."
Links of numbers
Mathematics examines relations between mathematical objects and demonstrates structural properties in these relationships. Elementary examples of defined between numbers relationships are about the well-known arithmetic operations ( basic arithmetic ) over the rational numbers (fractions), comparisons ( " less ", "greater ", " greater than", etc.) between rational numbers and divisibility between integers ("3 is a divisor of nine "). In addition, certain properties of numbers are defined, for example, the property is defined over the integers being a prime number.
Such links are not to be understood by the concept of number independent arbitrary operations, but certain speed ranges are usually considered inseparable from certain links, as these determine the structure under investigation significantly. When speaking about over the natural numbers, one uses almost always at least their order ("", ""), which largely determines our concept of natural numbers.
In school mathematics, computer science and numerical mathematics is concerned with methods to evaluate such linkages on concrete representations of numbers ( arithmetic). As an example we mention the written Addition: Using the representation of numbers in a place value system, it is here possible to obtain by systematic processing of the digits is a representation of the sum of the two numbers. In computer science and numerical mathematics such procedures are developed and tested for their performance. Some such processes are of fundamental importance for today's computers.
In abstract algebra, is concerned with the structure of generalizations of such numerical ranges, where only the presence of links is provided with certain properties over an arbitrary set of objects, which determine the structure of the links is not unique, but many different concrete structures with these properties (models) allow (see algebraic structure). Results can be applied to specific number ranges, which in turn can serve as motivation and elementary examples in abstract algebra.
The number theory, properties ( in the broad sense ) of numbers, such as existence, frequency and distribution of numbers with certain properties. Transfinite properties ( in certain senses " infinite " numbers), however, are the subject of set theory.
In mathematics are such linkages, relationships and properties as predicates or relations, including functions, construed.
Definition of numbers
The concept of number is not defined mathematically, but is a common linguistic term for various mathematical concepts. Therefore, there is no amount in the mathematical sense of the numbers or the like. The math speaks when it deals with numbers, always on certain well-defined ranges of numbers, that is, only on certain objects of our thinking with specified properties, all of which are casually referred to as figures. Since the end of the 19th century figures are purely defined independently of notions of space and time by means of logic in mathematics. Foundations were laid here by Richard Dedekind and Giuseppe Peano with the axiomatization of the natural numbers (see Peano axioms ). Dedekind writes to this new approach:
" What is provable, is not to be believed without proof in science. As obvious as this claim appears to be, it is but, I believe, even in the grounds of the simplest science, namely, of that part of logic which deals with the theory of numbers, by no means be regarded as satisfied even after the latest presentations. [ ... ] The numbers are free creations of the human mind, they serve as a means to conceive the variety of things easier and sharper. Due to the purely logical structure of the numbers - science and the gained in their steady numbers kingdom we are only set in the state to examine our ideas of time and space exactly by the same referring to such created in our mind figures Empire. "
A distinction must be axiomatic definitions of set-theoretic definitions of numbers: In the former case, the existence of certain objects defined on them links is postulated with certain characteristics in terms of axioms, such as even in the early axiomatizations of the natural and real numbers by Peano and Dedekind. As a result, the development of set theory by Georg Cantor you went over to try to limit themselves to set-theoretic axioms, as is customary in mathematics today about the Zermelo -Fraenkel set theory ( ZFC ). The existence of certain sets of numbers and links about them with certain properties is then deduced from these axioms. Sometimes a speed range is defined as a particular class. The axiomatic set theory tries to be a single, unified formal basis for all of mathematics. Within its can be bypassed on rich way with the speed ranges. Formulated she is usually in the first order predicate logic, which defines the structure of mathematical propositions, as well as the possibilities to the conclusion from the axioms.
An elementary example of a set-theoretic definition of a set of numbers is introduced by John von Neumann definition of the natural numbers as the least inductive set whose existence is postulated by the axiom of infinity within the framework of Zermelo -Fraenkel set theory.
As a set-theoretic concepts ordinal and cardinal numbers usually are set-theoretically defined as the generalization of the surreal numbers.
The Peano axioms about and going back to Dedekind definition of the real numbers are based, in contrast to ZFC on the predicate second-order logic. While the first-order predicate logic provides a clear, universally accepted answer to be made as valid conclusions, which can be calculated systematically, attempts to clarify this for the predicate calculus second stage, mostly to the fact that a complex meta-theory must be introduced, which in turn set-theoretical concepts and introduces metalinguistically of the details depend on the described in the following ways of reasoning in the predicate calculus second stage. ZFC is a candidate for such a theory. These limitations make the second -order predicate logic in a part of the philosophy of mathematics appear to be unsuitable to be used at a basic level. The first-order predicate logic, however, is not sufficient to formulate some important intuitive properties of natural numbers, and ( as seen in a set-theoretic metatheory, as a result of the set of Löwenheim - Skolem the countability of these ) to ensure.
Some important numerical ranges are presented here in its mathematical context. Throughout the history of mathematics more speed ranges were always introduced in order to treat general compared to previous numerical ranges certain problems. In particular, existing number ranges have been extended by adding additional elements to the new number of areas in order to speak more generally about certain operations can, see also the article on the speed range extension.
On the concept of the number range, see the section for the definition.
The natural numbers form the one set of numbers that is used to count (0, 1, 2, 3, 4, 5, ...). Depending on the definition, the zero is included or not. The natural numbers are provided with a policy (" small "). There is a smallest element (depending on the definition of the zero or one), and each element has a successor and is smaller than its successor. By repeatedly forms starting from the smallest element of the successor, finally reaching any natural number and gradually getting more, so that infinitely many there are of them. The natural numbers are also provided with addition and multiplication, two natural numbers can thus be assigned a sum and a product that are natural numbers again. These operations are associative and commutative, and they are also in the sense of distributive compatible with each other. These three properties are also essential for many general number of areas such as the whole, rational, real and complex numbers. The order of the natural numbers is in some respects with the addition and multiplication compatible: it is shift invariant, ie, natural numbers follows also, in addition to the shift-invariance also follows.
The existence of the set of all natural numbers is ensured in set theory by the axiom of infinity.
This quantity is denoted by or.
In the set of natural numbers, there is no natural number for two numbers, so. The integers extend the natural numbers such that for any two elements there exists such a number. This is added to a negative number, add the natural numbers: for each natural number, there is a second integer, so that which is referred to as an additive inverse. The above number called difference, then, briefly, as given. Thereby, the subtraction is defined on integers, but which substantially represents a short form.
The order on the natural numbers is extended to the integers, in this case there is no smallest element more, but each element has a predecessor and a successor ( the predecessor of the, the, the, etc.). The compatibility with the addition, the shift-invariance, however, remains. In addition, the product of two integers greater than zero always turn greater than zero.
The integers form a ring.
The set of integers is denoted by or.
Are expanded as well as the natural numbers to integers, to obtain an additive inverse, and the subtraction, one expands the integers to the rational number in order to obtain a multiplicative inverse, and the division. That is, the rational numbers include the integers and each integer adds to the called number ( unit fraction ) as a multiplicative inverse attributes to the. In addition, the product of any two of rational numbers to be defined, generally obtained rational numbers of the form, called fraction, wherein an integer is identified with the fracture. For integers, the fractures are identified and each other; this identification is also called Enhance and shortening. Is thus obtained which is compatible with the multiplication of integers multiplication and division.
By means of Dezimalbruchdarstellung can be defined to be compatible with the order of the integers order, which also receives the compatibility with addition and multiplication.
The rational numbers form a ( parent ) body. The construction of the rational numbers from the integers is generalized by dividing body formation in a ring.
The set of rational numbers is denoted as or.
With the addition and multiplication of whole or rational numbers can be defined in so-called polynomial functions: Each whole or rational number is in this case a sum of powers multiplied by constant numbers (coefficients) assigned. About an arbitrary number of value as defined. For many such polynomial exists no rational number, so that the value of the polynomial at this point is equal to zero ( zero ). Adds one now zeros of certain polynomial functions rational numbers is added, with multiplication and addition remain well defined, one obtains an algebraic extension. If we expand the rational numbers are those zeros for all non-constant polynomials, we obtain the algebraic numbers. If we extend the integers to find roots for all non-constant polynomials whose coefficients are integers and whose coefficient of the highest potency, we obtain the algebraic integers.
Algebraic extensions in the body of theory, in particular in the Galois theory examined.
Looking at problems such as finding zeros of polynomial functions over the rational numbers, one finds that it is possible in the rational numbers construct arbitrarily good approximations: Some can be found in numerous polynomial at any specified tolerance is a rational number, so that the value of different polynomial at this point more than the tolerance of zero. In addition, you can select the approximate solutions so that they are " close together " are, for polynomial functions are continuous ( " have not, 'jumps' on ' ). This behavior occurs not only at zeros of polynomial functions, but also in many other mathematical problems which have a certain continuity, so moving on to guarantee the existence of a solution as soon as any good approximations exist by close to each location rational numbers. Such a solution is what we call a real number. To show the existence of such solutions, it is sufficient to require that there be any set of rational numbers, which does not contain arbitrarily large numbers, among the real numbers that are greater than or equal to all of these elements of the set, a smallest. Alternatively, the real numbers can be explicitly defined as sequences of rational numbers, the " closer " to each other, define.
The set of real numbers is uncountable. Therefore, it is not possible to describe any real number unique language.
The seclusion of the real numbers in such proximity processes is called completeness. This allows many concepts from calculus, such as the dissipation and the integral to define to limit values . Limits also allow the definition of several key functions such as the trigonometric functions (sine, cosine, tangent, etc.), which is not possible over the rational numbers.
The real numbers keep relevant properties of addition, multiplication and order in the rational numbers and thus also form an ordered field. They can not expand without violating this property or the Archimedean axiom, that is " infinitely small strictly positive numbers " introduce.
The idea of transition from the rational to the real numbers is generalized by various concepts of completion.
The set of real numbers is denoted as or.
Some polynomial functions have no zeros in the real numbers. For example, the function for every real number takes a value greater than zero. It can be shown that by adding a number, called the imaginary unit, which satisfies the equation, where the basic properties of addition and multiplication are to be retained, already the real numbers to the complex numbers be extended, in which all non-constant polynomial have a zero point. The complex numbers thus form the algebraic degree of real numbers. Limit processes are the complex numbers as possible as in the real numbers, but complex numbers are not ordered. They can be understood as (two-dimensional vector space over the real numbers ) level. Each complex number can be uniquely in the form " pose ", where and are real numbers and denote the imaginary unit.
The function theory is that part of the analysis, which deals with the analytical properties of functions over the complex numbers.
The set of complex numbers is denoted by or.
Ordinal and cardinal numbers
The ordinal and cardinal numbers are concepts from set theory. In set theory one defines the cardinality of a quantity as a cardinal number, the cardinality is a generalization of the concept of " number of elements " of a finite set of infinite sets. The cardinalities of finite sets are thus natural numbers, which are also included in the cardinal numbers.
Ordinals generalize the concept of " position in a ( well-ordered ) Quantity " on infinite sets. Ordinals then clearly describe the position of an element in such a well-ordering. The ordinals are themselves well-ordered, so that the sequence of well-ordered objects in the order of their assigned " positions " (ie ordinal ) corresponds. For positions in arrays of finitely many objects to natural numbers can be used, which correspond to the smallest ordinal.
Cardinal numbers are now defined as special ordinals, which you will also get a fine. In addition to the order and addition, multiplication and exponentiation are on cardinal numbers and ordinal numbers defined, limited which correspond to the natural numbers with the usual terms for natural numbers, please refer to cardinal arithmetic and transfinite arithmetic.
Both the ordinal and cardinal numbers form proper classes, which means they are in the sense of modern set theory, no quantities.
Hyper Real Numbers
The hyper real numbers are a generalization of real numbers and object of investigation of non- standard analysis. These allow the definition of terms used in the analysis as the continuity of the derivative or without the use of limit values.
Hyper Complex Numbers
The complex numbers can be used as two-dimensional vector space over the reals interpret (see Gauss number plane ), that is as a two-dimensional plane, in addition to the usual coordinate- wise addition of a multiplication between two points in the plane is defined. There are numerous similar structures, which can be summed up under the term hypercomplex numbers. These structures are usually finite dimensional vector spaces over the real numbers ( thought of as a two - or higher-dimensional space ) with an additional multiplication. Often the real numbers can be embedded even in these structures, the multiplication regions to the real numbers of the conventional multiplication of real numbers corresponding to.