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It is called Euclidean division and possesses the following important property: In elementary school teaching, integers are often intuitively defined as the positive natural numbers, zero , and the negations of the natural numbers. However there seems to be a somewhat directed preference: However, there are some cases where a more-or-less clear difference exists.

It is called Euclidean division and possesses the following important property: The integer q is called the quotient and r is called the remainder of the division of a by b. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain.

This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. Z is a totally ordered set without upper or lower bound. The ordering of Z is given by: Zero is defined as neither negative nor positive. It follows that Z together with the above ordering is an ordered ring. The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.

In elementary school teaching, integers are often intuitively defined as the positive natural numbers, zero , and the negations of the natural numbers. However, this style of definition leads to many different cases each arithmetic operation needs to be defined on each combination of types of integer and makes it tedious to prove that these operations obey the laws of arithmetic.

The intuition is that a , b stands for the result of subtracting b from a. Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;  denoting by [ a , b ] the equivalence class having a , b as a member, one has:. It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.

Every equivalence class has a unique member that is of the form n ,0 or 0, n or both at once. If the natural numbers are identified with the corresponding integers using the embedding mentioned above , this convention creates no ambiguity.

In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines. Integers are represented as algebraic terms built using a few basic operations such as zero , succ , pred , etc. There exist at least ten such constructions of signed integers. This operation is not free since the integer 0 can be written pair 0,0 , or pair 1,1 , or pair 2,2 , etc.

This technique of construction is used by the proof assistant Isabelle ; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.

An integer is often a primitive data type in computer languages. However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity.

It is, however, certainly possible for a computer to determine whether an integer value is truly positive. Fixed length integer approximation data types or subsets are denoted int or Integer in several programming languages such as Algol68 , C , Java , Delphi , etc.

Variable-length representations of integers, such as bignums , can store any integer that fits in the computer's memory. This is readily demonstrated by the construction of a bijection , that is, a function that is injective and surjective from Z to N. If the domain is restricted to Z then each and every member of Z has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality.

From Wikipedia, the free encyclopedia. For computer representation, see Integer computer science. For the generalization in algebraic number theory, see Algebraic integer. Subgroup Normal subgroup Quotient group Semi- direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics.

Topological and Lie groups. Linear algebraic group Reductive group Abelian variety Elliptic curve. In Bach, Emmon W. Quantification in Natural Languages. The structure of arithmetic. Lecture Notes in Computer Science. Bicomplex numbers Biquaternions Bioctonions. Cardinal numbers Irrational numbers Fuzzy numbers Hyperreal numbers Levi-Civita field Surreal numbers Transcendental numbers Ordinal numbers p -adic numbers Supernatural numbers Superreal numbers.

Leo shows that both bezahlen and zahlen are translated as "to pay". I noticed that people sometimes use the one, and sometimes the other. In many cases, zahlen and bezahlen mean the same and may be used interchangeably:. Sometimes, there is a difference in register; otherwise, it is often a matter of personal taste when to use which. However, there are some cases where a more-or-less clear difference exists.

Sie macht lieber alles selbst, statt einen Handwerker zu bezahlen. If the person is not the recipient of the payment, but the item that is being paid for, i. Ist die Waschmaschine bezahlt? As far as I know, the prefix be- is used on verbs that are born as intransitive to make them transitive.

So, in the case of zahlen, you would say:. That is a tough one, I find me using it interchangeably. However there seems to be a somewhat directed preference:. I would state it as bezahlen expresses the intent and the process of paying. Whilst zahlen only states the intent. The first sentence inquires the intent. The second sentence signals the intent and the payment immediately after. You can also say, ich bezahle die Miete, I pay up the rent, that is, I pay money to satisfy the rent.