# number Overview of some common payment ranges. means that the elements of the payment range can be understood as elements of the payment range while maintaining essential relationships . Real classes are marked in blue.${\ displaystyle A \ subset B}$ ${\ displaystyle A}$ ${\ displaystyle B}$ Numbers are abstract mathematical objects or objects of thought that historically evolved from ideas of size and number. By means of a measurement , an aspect of an observation which is understood as a quantity is associated with a number, for example in a count . They therefore play a central role in the empirical sciences .

In mathematics , which formally examines numbers and their structure, the term includes various concepts. These developed as generalizations of existing intuitive number concepts, so that they are also called numbers , although they have little relation to the concepts originally associated with measurements. Some of these concepts are of fundamental importance in mathematics and are used in almost all sub-areas .

In the prehistory back the concept of sufficient natural numbers that can be used to count and are of fundamental importance. From around 2000 BC BC Egyptians and Babylonians calculated with fractions (rational numbers). In India , an understanding of zero and negative numbers developed in the 7th century AD . Irrational numbers like or , the necessity of which arose from knowledge from ancient Greece (from the 4th century BC at the latest), were introduced in the heyday of Islam . ${\ displaystyle {\ sqrt {2}}}$ ${\ displaystyle {\ sqrt {5}}}$ The idea of imaginary numbers , through which the real numbers were later expanded into the important complex numbers , dates back to the European Renaissance . The concept of the real number could only be adequately clarified in the 19th century. At the end of the 19th century it was possible for the first time to give infinite sizes a more precise meaning than numbers. The natural numbers were also defined axiomatically for the first time . With the first satisfactory fundamentals of mathematics created at the beginning of the 20th century , even the most important number terms were given a fully formal definition and meaning that corresponds to today's standards.

The concept of number delineate are numbers (special numerals ; to illustrate certain figures used characters ), numbers fonts (such spellings of numbers as with the help of numbers using certain rules.), Numerals ( numerals , used to designate specific numbers words ) and Numbers ( identifiers , which themselves can be numbers or - usually containing digits - strings ).

## etymology

The German word number probably goes back to the old Germanic word * talō ( calculation , number , speech ), which is probably the root of the old high German words zala ( order , orderly presentation , report , enumeration ) and zalōn ( report , calculate , count , calculate , numbers ) is. From Zala was in Middle High German zale or zal to which today's word count back.

The primitive Germanic word probably has its origin in a primitive Indo-European etymon * del- ( aim , calculate , readjust ). A connection with the Urindo-European * del- ( splitting ) is possible; the original meaning might then be “notched mark”.

## history

### prehistory

Because of the lack of evidence, it is difficult to say for sure about how people understood numbers in the period before the first written tradition. The importance of regular arrangements of lines or notches that have survived from this period can usually only be guessed at.

On the other hand, the respective languages, historically documented successor cultures as early as possible, or related languages ​​that still exist today, as well as the known languages ​​of old, similar cultures, can provide information on the representation of numbers in a prehistoric culture . By systematically comparing different languages, similarities and differences between them can be determined, so that the peculiarities of each language and language group can be determined and common or different origins can be found to a certain extent. This also results in structures with the numerals that allow conclusions to be drawn about the understanding of numbers.

The fundamental concept of number, recognizable everywhere in human languages ​​- the idea of ​​numbers - is that of the different number or quantity of certain objects, which corresponds most closely to the concept of cardinal number in today's mathematics . At the beginning there was probably the elementary opposition between singular and plural , which was followed by the further division of the plural. In the language of the Pirahã in Brazil, for example, only three or even only two words (“little” and “much”) are known for relative sizes. Attempts to teach some representatives of this people to count have failed. There are also ethnological reports of a people in South Africa and of many Australian indigenous peoples who only know the numerals “one”, “two” and “much” in their languages. The same can be found in Indo-European languages ​​in the form of the singular , the dual (e.g. in Greek, in Latin and earlier also in Germanic languages) and the plural of nouns .

In order to be able to differentiate “much” further and to be able to say more precise numbers, other peoples formed further numerals. Up to a maximum of ten (for larger numbers the numerals would be too long) this is possible simply by adding "two" as often as it is contained in the corresponding number, and with an odd number an "a" is added. added. Another way of obtaining words for larger numbers has been to languages ​​that have invented their own additional words such as “three”, “four” or “five” for smaller numbers and these in turn additively or multiplicatively, e.g. B. "four-two" for eight, connected to new larger numbers. For the formation of numbers significantly larger than ten, it is necessary to combine large numbers into new, larger units and to find new numerals for these, for example in steps of "ten", "hundred" etc.

In this way, such large numbers can be formed that it is necessary to count a corresponding number of objects for their precise detection . However, there is no need to separate the numbers from the type of objects being counted: in some languages ​​there are so-called counting classes, each of which has its own numeric word for the same number. A different word is used for the same number of living beings than for long objects, a third word for round objects and additional words for other objects.

With the detachment from the type of objects , i.e. when the same numerical word is used for the same number regardless of the objects counted, numbers acquire independence and are perceived as something of their own. In Indo-European languages, this can generally be observed for numbers greater than four. Originally there seems to have been a gradation of four, later the numbers were apparently expanded in several steps (this can be seen, for example, in German by the difference between “thirteen” and “three and twenty”). In addition to summaries of two, three or four, there are also languages ​​with levels of five, ten, twelve or twenty as well as mixed forms of these.

### First high cultures

The after the last ice age (after 10,000 v. Chr.) In the Mesolithic onset of climate change led to the drying up of large areas of the Sahara in the west to the Mongolian steppe to the east. The increasing population of the affected areas migrated to the river oases, where more differentiated urban societies developed over time. With the invention of writing in the early high cultures on the Euphrates and Tigris ( Mesopotamia ), on the Nile ( Ancient Egypt ), on the Indus ( Indus culture ) and on the Yellow River ( ancient China ) began between the end of the 4th and the beginning of the 3rd millennium BC The historical time. From the beginning, numerals were created together with writing, since both apparently were needed to manage the increasingly organized societies.

In ancient Egypt , at the latest from around 3,000 BC. Chr. An additive number system for base 10 use to represent natural numbers. The basic arithmetic operations of addition , subtraction , multiplication and division were already carried out there. There were special characters for the former two. The Moscow Papyrus and the Papyrus Rhind - both written in hieratic script in the period between 2000 BC - are particularly significant evidence of the mathematical abilities of this culture . BC and 1800 BC From this a special notation for stem fractions can be derived in addition to the natural numbers . Other ratios were systematically converted into sums of stem fractions ( but also had its own symbol). The motivation of ancient Egyptian mathematics was mostly construction, land surveying and economics, there is no evidence. However, there are also problems that are intended to be interpreted as humorous or entertaining. ${\ displaystyle {\ tfrac {2} {3}}}$ There is also rich mathematical evidence from the Mesopotamia of antiquity . In Sumerian times, an additive number system developed there, based on bases 10 and 60. From ancient Babylonian times between 1,800 and 1,600 BC. There are numerous finds with further achievements: A sexagesimal place value system was created , but with the restriction that there was no digit zero and the notation was therefore ambiguous. Within this system, more general rational numbers were also represented in a manner corresponding to today's decimal fraction expansion , i.e. that is, about and digits could be used. Fractions that cannot be represented in this way or (in modern parlance) logarithms , as they appeared in the calculation of interest , were represented approximately. Systematic approximations were also made in the form of the Babylonian root extraction . In addition, solutions for quadratic , cubic and biquadratic equations were found. These equations were described in geometric terms (a square appearing in modern parlance in such equations was described as area, from which about a side's length is subtracted, so that quantities designated as areas and lengths could be added, but gives a rather abstract, algebraic understanding Near). These achievements came from the practical needs of business, construction, and astronomy. ${\ displaystyle {\ tfrac {1} {60}}}$ ${\ displaystyle {\ tfrac {1} {3600}}}$ ### Ancient Greece

A large number of mathematical findings have come down to us from ancient Greece . For the first time (as far as is known) there was a pronounced understanding of proofs by which the results were proven in a rigor approaching today's mathematics. From the 6th century B.C. The school of the Pythagoreans , founded by Pythagoras of Samos (approx. 570–510 BC), who was probably influenced by trips to Egypt, Mesopotamia and possibly India. In this religious group, mathematics separated from arithmetic, which arises from everyday necessities, with (natural) numbers playing a central role. The tradition of this period in the history of mathematics, including Thales von Miletus , who presumably lived a little earlier , is still thin, most documents come from a later period, so that it is not possible to say with certainty which concepts were already known there and with which Methodology was followed.

For reasons not fully understood, subsequent Greek mathematics placed great emphasis on geometry , despite the influence of the Pythagoreans, among whom arithmetic had been considered fundamental. Important protagonists here were Eudoxus of Knidos (* between approx. 397 and 390 BC, † between approx. 345 and 338 BC) and Euclid (approx. 360–280 BC).

Regarding the number concept of the Greeks, it must be stated that they did not have a concept of rational numbers as algebraic objects or an extension of the natural numbers. The results, which from a modern point of view are often interpreted as statements about such a length or area, were formulated geometrically as statements about length and area relationships: One length or area could be an integral multiple of another; accordingly, relationships between two such multiples of a length or area can be understood today as (positive - concepts comparable to negative numbers were not available) describe rational numbers, but they were not included in the Greek understanding of numbers. All the more, there were no irrational numbers in Greek mathematics - there were only geometric relationships that did not correspond to a ratio of two whole-number multiples of a quantity; one speaks of incommensurability . Even the one was not counted among the numbers in Euclid.

The existence of the incommensurable relationships was at the latest since Aristotle (384–322 BC), who provided quite general evidence, but possibly even before 400 BC. Known in Greece. This showed the impossibility of the Pythagorean approach to describe the relationships occurring in geometry by means of arithmetic - in today's terminology, an inadequacy of rational numbers. The transition to a geometric foundation that allowed such relationships to be dealt with is largely attributed to Eudoxus, who himself was a student of the important Pythagorean Archytas of Taranto , who saw arithmetic as the only possible basis for evidence.

Eudoxos provided a definition of the equality of two geometric relationships (of lengths or areas): Two relationships are therefore equal if all - in modern interpretation - rational relationships that are smaller or larger than one ratio are also smaller or larger than the other. This definition even applies analogously to today's concept of real numbers. Some voices saw or see here a presence of real numbers in Greek mathematics. However, these statements are problematic: On the one hand, the concept of rational numbers was not even present, on the other hand nothing was said about the existence of certain relationships, so that these are roughly orderly , but rather relationships given by geometry were examined. In any case, this definition enabled a multitude of proofs, the techniques of which, such as the exhaustion method, are regarded as forerunners of today's concepts of analysis , with certain estimates already playing a central role. In addition, Richard Dedekind says he was inspired by Eudoxus when he defined the real numbers.

Archimedes of Syracuse (287–212 BC), who based on Eudoxus provided particularly extensive evidence of certain geometrical relationships and certain approximations, is also considered the first person to introduce infinitesimal quantities: in the Archimedes Palimpsest he applied a principle comparable to that Principle from Cavalieri , in which a surface is divided into an infinite number of infinitesimal lines. Even then, such a procedure did not meet the requirements of a mathematical proof, but Archimedes saw this mechanically motivated procedure as a useful tool to approach a problem and later to find a correct proof more easily. The existence of non-zero infinitesimal quantities contradicts the Eudoxus definition of equality and also the so-called Archimedes' axiom established by Archimedes himself .

## Definition of numbers

The concept of number is not defined mathematically, but a general term for various mathematical concepts. Therefore there is no such thing as a set of all numbers or the like in a mathematical sense . When mathematics is concerned with numbers, it always speaks of certain well-defined number ranges , i.e. H. only about certain objects of our thinking with fixed properties, which are all casually referred to as numbers . Since the end of the 19th century, numbers have been defined in mathematics purely by means of logic, independent of notions of space and time. The foundation stones were laid here by Richard Dedekind and Giuseppe Peano with the axiomatization of natural numbers (see Peano axioms ). Dedekind writes about this new approach:

“What can be proven should not be believed in science without proof. As plausible as this requirement appears, I believe it is by no means to be regarded as fulfilled even according to the most recent representations, even with the foundation of the simplest science, namely that part of logic which deals with the theory of numbers. [...] the numbers are free creations of the human spirit, they serve as a means to grasp the difference of things more easily and sharply. Due to the purely logical structure of the science of numbers and the continuous realm of numbers gained in it, we are only able to investigate our ideas of space and time precisely by relating them to this realm of numbers created in our minds . "

- Richard Dedekind : What are and what are the numbers? Foreword to the first edition.

A distinction must be made between axiomatic definitions and set- theoretical definitions of numbers: In the first case, the existence of certain objects with links defined on them with certain properties is postulated in the form of axioms , as was the case with the early axiomatizations of natural and real numbers by Peano and Dedekind . As a result of the development of set theory by Georg Cantor , the attempt was made to limit oneself to set-theoretical axioms, as is common in mathematics today, for example with the Zermelo-Fraenkel set theory (ZFC). The existence of certain sets of numbers and connections over them with certain properties is then deduced from these axioms. Sometimes a range of numbers is defined as a particular class . The axiomatic set theory tries to be a single, uniform formal basis for all mathematics. Within it you can deal with the number ranges in a rich way. It is usually formulated in the first-level predicate logic , which defines the structure of the mathematical sentences and the possibilities for drawing conclusions from the axioms.

An elementary example of a set-theoretical definition of a set of numbers is the definition of natural numbers introduced by John von Neumann as the smallest inductive set , the existence of which is postulated within the framework of the Zermelo-Fraenkel set theory by the axiom of infinity .

Ordinal and cardinal numbers are usually defined as set-theoretical concepts , as is the generalization of surreal numbers .

The Peano axioms, for example, and the definition of real numbers, which goes back to Dedekind, are based, in contrast to ZFC, on the second-order predicate logic . While first-level predicate logic provides a clear, generally accepted answer to how valid inferences are to be made, whereby these can be calculated systematically , attempts to clarify this for second-level predicate logic usually lead to the introduction of a complex metatheory , which for its part introduces set-theoretic terms in metalinguistic terms , and on the details of which the subsequent possibilities of inference in the second-order predicate logic depend. ZFC is a candidate for such a theory. These limitations make second order predicate logic in some part of the philosophy of mathematics unsuitable for use at the fundamental level. The first-order predicate logic, on the other hand, is not sufficient to formulate certain important intuitive properties of the natural numbers and to ensure countability (when considering these in a set-theoretical metatheory, e.g. based on the Löwenheim-Skolem theorem ).

## Linking numbers

Mathematics studies relationships between mathematical objects and proves structural properties in these relationships. Elementary examples for relationships defined between numbers are the generally known arithmetic operations ( basic arithmetic operations ) over the rational numbers (fractions), comparisons (“smaller”, “larger”, “greater than or equal” etc.) between rational numbers and the divisibility relation between whole numbers Numbers (“3 is a divisor of 9”). In addition, properties are defined using certain numbers, for example the property of being a prime number is defined over the whole numbers .

Such links are not to be understood as arbitrary operations that are independent of the number concept; rather, certain number ranges are usually considered inseparable from certain links, as these determine the structure to be examined. If you talk about natural numbers , you almost always use at least their order (“ ”, “ ”), which significantly determines our concept of natural numbers. ${\ displaystyle 1 <5}$ ${\ displaystyle 12 <19}$ In school mathematics , computer science and numerical mathematics, one deals with methods to evaluate such links on concrete representations of numbers ( arithmetic ). The written addition is an example : Using the representation of numbers in a place value system , it is possible here to obtain a representation for the sum of the two numbers by systematically processing the digits . In computer science and numerical mathematics, such processes are developed and examined for their performance. Some of these techniques are fundamental to today's computers .

In abstract algebra one deals with the structure of generalizations of such number ranges, whereby only the existence of links with certain properties is assumed over an arbitrary set of objects, which do not clearly determine the structure of the links, but many different concrete structures with them Allow properties ( models ) (see algebraic structure ). Your results can be applied to specific number ranges, which in turn can serve as motivation and elementary examples in abstract algebra.

The number theory treated properties (broadly defined) of numbers, about existence, frequency and distribution of numbers with certain properties. Properties transfinite (in certain senses “infinite” numbers) are, however, the subject of set theory .

In mathematics, such links, relationships and properties are understood as predicates or relations , including functions .

## Number ranges

Some important number ranges are presented here in their mathematical context. In the course of the history of mathematics , more and more number ranges were introduced in order to be able to deal with certain problems more generally than previous number ranges. In particular, existing number ranges have been expanded to new number ranges by adding additional elements in order to be able to talk about certain operations in more general terms, see also the article on number range expansion .

For the concept of the number range, see the section on the definition .

### Natural numbers

The natural numbers 1, 2, 3, 4, 5,… or 0, 1, 2, 3, 4, 5,… form the set of numbers that are usually used for counting , whereby the zero is included depending on the definition or not. The natural numbers are given an order (“smaller”). There is a smallest element (zero or one , depending on the definition ), and each element has a successor and is smaller than its successor. By starting from the smallest element over and over again, you finally reach every natural number and successively more and more, so that there are infinitely many of them. The natural numbers are also provided with addition and multiplication , so that every two natural numbers can be assigned a sum and a product, which are again natural numbers. These operations are associative and commutative , and they are also compatible with each other within the meaning of distributive: . These three properties are also fundamental to many more general number domains such as whole, rational, real, and complex numbers. The order of the natural numbers is in a certain way compatible with addition and multiplication : it is shift-invariant , i. i.e., for natural numbers , it also follows , in addition to the shift invariance, also follows . ${\ displaystyle a \ cdot (b + c) = a \ cdot b + a \ cdot c}$ ${\ displaystyle m, n, o}$ ${\ displaystyle m \ leq n}$ ${\ displaystyle m + o \ leq n + o}$ ${\ displaystyle m \ cdot o \ leq n \ cdot o}$ The existence of the set of all natural numbers is ensured in set theory by the axiom of infinity .

This amount is referred to as or . ${\ displaystyle \ mathbb {N}}$ ${\ displaystyle \ mathbf {N}}$ ### Whole numbers

In the set of natural numbers, no natural number exists for two numbers , so . The integers expand the natural numbers in such a way that such a number exists for any two elements . To do this, you add the negative numbers to the natural numbers: For every natural number there is a second whole number , so that which is called the additive inverse . The above number , called the difference , is then given as , in short . This defines the subtraction on the whole numbers, which, however, is essentially an abbreviated form. ${\ displaystyle n ${\ displaystyle d}$ ${\ displaystyle m + d = n}$ ${\ displaystyle d}$ ${\ displaystyle n}$ ${\ displaystyle -n}$ ${\ displaystyle n + (- n) = 0}$ ${\ displaystyle d}$ ${\ displaystyle n + (- m)}$ ${\ displaystyle nm}$ The order above the natural numbers is extended to the whole numbers. There is no longer a smallest element; each element has a predecessor and a successor (the predecessor is the one , the one is the etc.). The compatibility with the addition, the shift invariance, is retained. In addition, the product of two whole numbers greater than zero is always greater than zero. ${\ displaystyle 0}$ ${\ displaystyle -1}$ ${\ displaystyle -1}$ ${\ displaystyle -2}$ The whole numbers form a ring .

The set of whole numbers is denoted by or . ${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle \ mathbf {Z}}$ ### Rational numbers

Just as the natural numbers are expanded to the whole numbers to get an additive inverse and the subtraction, the whole numbers are expanded to the rational numbers to get a multiplicative inverse and the division. That means, the rational numbers contain the whole numbers, and to each whole number one adds the mentioned number ( stem fraction ) as a multiplicative inverse, so that . In addition, the product of any two rational numbers should be defined; in general, one obtains rational numbers of the form , called a fraction , where an integer is identified with the fraction . For integers , the fractions and are identified with each other; this identification is also called expanding and shortening . This results in a multiplication and division compatible with the multiplication of whole numbers. ${\ displaystyle z \ neq 0}$ ${\ displaystyle {\ tfrac {1} {z}}}$ ${\ displaystyle \ textstyle z \ cdot {\ frac {1} {z}} = 1}$ ${\ displaystyle \ textstyle {\ frac {x} {y}} = x \ cdot {\ frac {1} {y}}}$ ${\ displaystyle z}$ ${\ displaystyle \ textstyle {\ frac {z} {1}}}$ ${\ displaystyle t \ neq 0}$ ${\ displaystyle \ textstyle {\ frac {x} {y}}}$ ${\ displaystyle \ textstyle {\ frac {t \ cdot x} {t \ cdot y}}}$ Using the decimal fraction representation, an order that is compatible with the order of the whole numbers can be defined, which is also compatible with addition and multiplication.

The rational numbers form an ( ordered ) body . The construction of the rational numbers from the whole numbers is generalized as a quotient field to form a ring.

The set of rational numbers is denoted by or . In (German) school mathematics, the designation ("set of (positive) fractions") also occurs when the positive fractions are introduced before the negative whole numbers. ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbf {Q}}$ ${\ displaystyle \ mathbb {B} (: = \ mathbb {Q} _ {0} ^ {+})}$ ### Algebraic extensions

With the addition and multiplication of whole or rational numbers, so-called polynomial functions can be defined: Each whole or rational number is assigned a sum of powers multiplied by constant numbers ( coefficients ). Roughly any number the value is defined as . For many such polynomial functions there is no rational number, so that the value of the polynomial function at this point equals zero ( zero point ). If one now adds zeros of certain polynomial functions to the rational numbers, with multiplication and addition remaining well-defined, one obtains an algebraic extension . If one extends the rational numbers by such zeros for all non-constant polynomials, one obtains the algebraic numbers . If one expands the whole numbers by zeros for all non-constant polynomials whose coefficients are integer and whose coefficient is to the highest power , then the whole algebraic numbers are obtained . ${\ displaystyle x}$ ${\ displaystyle \ textstyle 12 \ cdot x ^ {0} +4 \ cdot x ^ {2} + \ left (- {\ frac {1} {2}} \ right) \ cdot x ^ {3}}$ ${\ displaystyle \ textstyle 12 + 4 \ cdot x \ cdot x + \ left (- {\ frac {1} {2}} \ right) \ cdot x \ cdot x \ cdot x}$ ${\ displaystyle 1}$ Algebraic extensions are studied in body theory , especially Galois theory .

### Real numbers

If one considers problems such as finding zeros of polynomial functions over the rational numbers, one finds that arbitrarily good approximations can be constructed in the rational numbers : For example, with numerous polynomial functions there is a rational number for every specified tolerance, so that the value of the polynomial function at this point deviates from zero by no more than the tolerance. In addition, the approximate solutions can be chosen so that they are “close to each other”, because polynomial functions are continuous (“do not have any 'jumps'”). This behavior occurs not only with zeros of polynomial functions, but also with numerous other mathematical problems that have a certain degree of continuity, so that one goes over to guaranteeing the existence of a solution as soon as arbitrarily good approximations exist through rational numbers located close together. Such a solution is called a real number . In order to show the existence of such solutions, it suffices to demand that for every set of rational numbers that do not contain arbitrarily large numbers, there is a smallest among the real numbers that are greater than or equal to all these elements of the set. Alternatively, the real numbers can be explicitly defined as sequences of rational numbers that “ approach ” one another .

The set of real numbers is uncountable . It is therefore not possible to uniquely describe any real number in terms of language.

The closeness of the real numbers under such approximation processes is called completeness . This allows numerous terms from analysis , such as the derivative and the integral , to be defined using limit values . Limit values ​​also allow the definition of numerous important functions , such as the trigonometric functions (sine, cosine, tangent etc.), which is not possible with the rational numbers.

The real numbers retain decisive properties of addition, multiplication and order in the rational numbers and thus also form an ordered body . They cannot be expanded without violating this property or the Archimedean axiom , ie introducing “infinitely small, strictly positive numbers”.

The idea of ​​transition from rational to real numbers is generalized by various concepts of completion .

The set of real numbers is denoted by or . ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbf {R}}$ ### Complex numbers

Some polynomial functions have no zeros in the real numbers. For example, the function takes a value greater than zero for every real number . It can be shown that by adding a number , called an imaginary unit , which satisfies the equation , while maintaining the basic properties of addition and multiplication, the real numbers are expanded to the complex numbers in which all non-constant polynomial functions are one Own zero. The complex numbers thus form the algebraic conclusion of the real numbers. Limit value processes are just as possible in the complex numbers as in the real numbers, but the complex numbers are no longer ordered. They can be understood as a plane (two-dimensional vector space over the real numbers). Every complex number can be clearly “represented” in the form , where and are real numbers and denote the imaginary unit. ${\ displaystyle x \ mapsto x ^ {2} +1}$ ${\ displaystyle x}$ ${\ displaystyle i}$ ${\ displaystyle i ^ {2} + 1 = 0}$ ${\ displaystyle a + b \ cdot i}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle i}$ 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 . ${\ displaystyle \ mathbb {C}}$ ${\ displaystyle \ mathbf {C}}$ ### Ordinal numbers and cardinal numbers

The ordinal and cardinal numbers are concepts from set theory . In set theory, the cardinality of a set is defined as a cardinal number, the cardinality is a generalization of the concept of the “number of elements” of a finite set to infinite sets. The cardinalities of finite sets are therefore natural numbers that are also contained in the cardinal numbers.

Ordinal numbers generalize the concept of "position in a ( well-ordered ) set" to infinite sets. Ordinal numbers then clearly describe the position of an element in such a well-ordered system. The ordinal numbers are themselves well-ordered so that the order of well-ordered objects corresponds to the order of the “positions” (i.e. ordinal numbers) assigned to them. For positions in arrangements of finitely many objects, natural numbers can be used that correspond to the smallest ordinal numbers.

Cardinal numbers are nowadays defined as special ordinal numbers, which also gives them an order. In addition to the order, addition, multiplication and exponentiation are also defined for cardinal numbers and ordinal numbers, which, restricted to the natural numbers, agree with the usual terms for natural numbers, see cardinal number arithmetic and transfinite arithmetic .

Both the ordinal numbers and the cardinal numbers form real classes , that is, they are not sets in the sense of modern set theory.

### Hyperreal numbers

The hyper-real numbers are a generalization of the real numbers and are the subject of nonstandard analysis . They allow the definition of terms from analysis such as continuity or derivation without the use of limit values .

### Hypercomplex numbers

The complex numbers can be understood as a two-dimensional vector space over the real numbers (see Gaussian plane of numbers ), i.e. as a two-dimensional plane in which, in addition to the usual coordinate-wise addition, a multiplication between two points on the plane is defined. There are numerous similar structures that are grouped under the term hypercomplex numbers . These structures are usually finite-dimensional vector spaces over the real numbers (imaginable as two- or higher-dimensional space) with an additional multiplication. The real numbers themselves can often be embedded in these structures , whereby the multiplication corresponds to the usual multiplication of real numbers, limited to the real numbers.

### More groups of numbers

• p-adic number , a generalization of the rational numbers with the inclusion of an infinite number of "pre-decimal places", which is used in number theory .
• Surreal number , a generalization of hyper-real numbers and ordinal numbers with applications in game theory .
• Residual class rings can be understood as restrictions of the integers to the first finitely many elements with appropriately defined arithmetic. Their elements are sometimes referred to as numbers.

## Representation of numbers

In mathematics, the language of logic is used to talk about mathematical objects such as numbers that are defined in it, and it can also be used to describe specific numbers, sometimes using formulas. In addition to the common logical formalisms , there are systematic names for certain numbers, for example in the form of special combinations of characters (sometimes digits used for this purpose ) or using specially constructed words from natural language, such as numerals . Designations for specific numbers are used outside of mathematics to describe concrete observations, for example a number of observed objects ( I see five bananas ) or measured values determined by another measuring method ( the door frame is two meters high ). Furthermore, such systematic number representations sometimes allow simple, systematic calculations with concrete numbers - especially with calculating machines and computers . The calculation methods for calculating certain operations between specific numbers depend on the representation chosen.

In the history of culture and mathematics, numerous number systems have developed into such systematic number representations. Evidence for the representation of numbers goes back to the late Stone Age , whereby there are difficulties in distinguishing numerals from mere counting symbols, i.e. to recognize whether people were aware of numbers as the abstract meaning of those, or only a tool-like use in which there was the physical construction of the ticker, but not a meaning, was relevant to accomplish its task. On this problem, see, for example, the article on the Ishango bone , a find from the late Paleolithic , which allows various interpretations.

Examples of such representations are ticklists ( unary system ) and place value systems using digit sequences , as are common today for the representation of natural numbers and are also used for the representation of numbers in computers in the form of the dual system .

If one looks at linguistic representations of numbers formally, then not every number can be assigned such a representation in a formal sense, i. . h, in a mathematical formal sense there are more numbers as possible representations in a language: Since linguistic formulations are always finite, it may of them only countable be many different, while mathematics and uncountable considered speed ranges. Nevertheless, one speaks of representations of uncountable number ranges if one no longer restricts oneself to linguistic formulations in such formal representations, but their structure can be similar to number systems, for example, the real numbers can be defined as special formal series that correspond to the representation are structurally similar in place value systems.

### Examples

Some examples of how numbers are represented:

• In German, "Vier" means a number as a number.
• This number can be described as a tally |||| represent.
• In the Indo-Arabic numerals it is represented as 4.
• In the Roman numerals it is represented as IV.
• As a formula it can be B. represent, which equates to a mathematical definition , if the one and the addition have been previously defined.${\ displaystyle 1 + 1 + 1 + 1}$ • If one understands the natural numbers as an algebraic structure provided with multiplication and addition, then one can be defined as the only natural number , so that and , the symbol then stands for any natural number that fulfills this condition, and is therefore unique.${\ displaystyle x}$ ${\ displaystyle x \ cdot x = x}$ ${\ displaystyle x + x \ neq x}$ ${\ displaystyle 1}$ • If one defines natural numbers in set theory in the variant of John von Neumann , then the four can be represented as via the usual representation of finite sets .${\ displaystyle \ left \ {\ emptyset, \ left \ {\ emptyset \ right \}, \ left \ {\ emptyset, \ left \ {\ emptyset \ right \} \ right \}, \ left \ {\ emptyset, \ left \ {\ emptyset \ right \}, \ left \ {\ emptyset, \ left \ {\ emptyset \ right \} \ right \} \ right \} \ right \}}$ • Rational numbers can be represented as fractions, e.g. B. .${\ displaystyle {\ tfrac {1} {2}}}$ • Solutions of quadratic equations over the rational numbers can be represented as formulas consisting of addition, multiplication and square root formation of rational numbers. For example, the formula describes a solution to the equation for the variable .${\ displaystyle {\ sqrt {2}}}$ ${\ displaystyle x ^ {2} = 2}$ ${\ displaystyle x}$ • Complex numbers are often represented as the sum of the real part and the imaginary part multiplied by the imaginary unit, for example .${\ displaystyle \ textstyle - {\ frac {4} {3}} + {\ frac {9} {2}} \ cdot i}$ • In the dual system, the natural number nine is represented as, this corresponds to the representation as a formula .${\ displaystyle 1001}$ ${\ displaystyle 1 \ times 2 \ times 2 \ times 2 + 0 \ times 2 \ times 2 + 0 \ times 2 + 1}$ • Every real number can be “represented” as a series with an integer and coefficients , but such representations are generally not finite describable because there are uncountably many possible “assignments” of the coefficients. If zero is always for sufficiently large , they correspond to the decimal part in a representation in the dual system (e.g. for ).${\ displaystyle \ textstyle z + \ sum _ {i = 1} ^ {\ infty} a_ {i} \ cdot 2 ^ {- i}}$ ${\ displaystyle z}$ ${\ displaystyle a_ {i} \ in \ left \ {0,1 \ right \}}$ ${\ displaystyle a_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle a_ {i}}$ ${\ displaystyle 0 {,} 101}$ ${\ displaystyle 0 {,} 625}$ ## Numbers as a designation

Just as numbers are assigned linguistic expressions, character strings or the like, conversely, numbers can be assigned to specific objects, on the one hand for abstract considerations, on the other hand to use representations of numbers specifically for the systematic designation of other objects, for example to encode information using numbers. Such a procedure allows the operations defined on numbers to be applied to these designations. A common example is numbering , in which a (mostly natural) number is assigned to each object of a certain considered ensemble: this allows the objects to be named by their numbers, and secondly, creates an order using the natural numbers (" smaller “) an order of the objects; in the case of natural numbers, for example, this allows all objects to be passed through sequentially. It should be noted that not every number is a number as a mathematical object that is independent of the representation. Some numbers are to be understood as special symbol sequences that serve as identifiers, even if they only consist of digits (e.g. ISB or house numbers ).

Another example is the interpretation of digital information in the data processing : As a binary sequence present data can be in a natural way as a natural number, represented in the binary system, interpreted (edge cases like leading zeros must be observed here). Arithmetic operations over this coding as a number are u. a. used in cryptography and data compression .

This principle can also be found in pure mathematics, whereby numbers are usually assigned to mathematical objects that are not understood as numbers, for example in the form of Gödel numbers that identify logical formulas or algorithms .

Further examples are the representation of game situations by means of surreal numbers in game theory , the representation of rotational stretching in two-dimensional Euclidean space by complex numbers and rotations in three-dimensional by means of quaternions .

## Web links

Commons : Numbers  - album with pictures, videos and audio files
Wiktionary: number  - explanations of meanings, word origins, synonyms, translations

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