Integer

The integers are an extension of the natural numbers.

The integers cover all the numbers

And thus contain all natural numbers and their additive inverses. The set of integers is abbreviated with the symbol (the " Z " stands for the German word " Numbers"). The alternative symbol has become less common; a drawback of this bold symbol is the difficult handwritten representation.

The above list of integers are also simultaneously in ascending order their natural arrangement again. Number theory is the branch of mathematics that deals with properties of integers.

The representation of integers in the computer is usually done by the Integer data type.

The integers are usually introduced in mathematics education in the fifth to seventh grade.

Properties

Ring

The integers form a ring with respect to addition and multiplication, i.e., they can be added without limitation, subtracted and multiplied. This calculation rules apply as the commutative and associative law for addition and multiplication, the distributive laws also apply.

The existence of subtraction, linear equations of the form

With natural numbers and are always solved. Is limited to the set of natural numbers, then it is not any such equation to solve.

Abstract expressed means that the integers form a commutative unitary ring. The neutral element of addition is 0, the additive inverse element of is the neutral element for multiplication is 1

Arrangement

The set of whole numbers is totally ordered, in the order

That is, one can compare two integers. This is called positive, not negative, negative, and not positive integers. The number 0 itself is neither positive nor negative. This order is compatible with the arithmetic operations, ie

As the set of natural numbers is the set of integers is countable.

The integers do not form a body, for example, the equation is not solvable. The smallest body containing, are rational numbers.

Euclidean ring

An important property of the integers is the existence of a division with remainder Because of this property is available for two whole numbers always a greatest common divisor, which can be determined using the Euclidean algorithm. Mathematicians say is a Euclidean ring. It follows also the theorem on the unique prime factorization in.

Construction of the natural numbers

Is the set of natural numbers given, then let the integers construct it as a number range expansion:

On the set of all pairs of natural numbers the following equivalence relation is defined:

Addition and multiplication to is defined by:

Is now the set of equivalence classes.

The addition and multiplication of couples now induce well-defined links that becomes a ring.

The usual order of integers is defined as

Each equivalence class has in the case of a unique representative of the mold, and in case a unique representative of the form, wherein.

The natural numbers can be embedded in the ring of integers, by the natural number is mapped to the equivalence class represented by. Usually, the natural numbers with their images to be identified and represented by the equivalence class is denoted by.

Is a natural number of different, so is the equivalence class denoted by represented as a positive integer, and the equivalence class represented by all as a negative number.

This design of the whole numbers of the natural numbers also works if no defined. Then the natural number should be identified with the equivalence class represented by.