# Integer

The letter Z with a double line
stands for the set of whole numbers

The integers (also integers , lat. Numeri integri ) are an extension of the natural numbers .

The whole numbers include all numbers

..., −3, −2, −1, 0, 1, 2, 3, ...

and thus contain all natural numbers and their additive inverses . The set of whole numbers is usually denoted by the letter with a double line (the “Z” stands for the German word “numbers”). The alternative symbol is now less common; A disadvantage of this bold face symbol is that it is difficult to display by hand. The Unicode of the character is U + 2124 and has the form ℤ. ${\ displaystyle \ mathbb {N} _ {0}}$ ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbf {Z}}$

The above listing of the whole numbers also shows their natural order in ascending order. The number theory is the branch of mathematics that deals with properties of integers.

The representation of whole numbers in the computer is usually done by the data type integer .

The integers are commonly introduced in maths class in the fifth through seventh grades.

## properties

### ring

The whole numbers form a ring with respect to addition and multiplication , i.e. That is, they can be added, subtracted and multiplied without restriction . Calculation rules such as the commutative law and the associative law for addition and multiplication apply, and the distributive laws also apply .

The existence of subtraction allows linear equations of the form

${\ displaystyle a + x = b}$

with natural numbers and always be solved: . If one restricts to the set of natural numbers, then not every such equation is solvable. ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle x = ba}$${\ displaystyle x}$

In abstract terms, this 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 of multiplication is 1. ${\ displaystyle n}$${\ displaystyle -n}$

### arrangement

The set of integers is totally ordered , in order

${\ displaystyle \ cdots <-2 <-1 <0 <1 <2 <\ cdots.}$

In other words, you can compare two whole numbers. We speak of positive , non-negative , negative and non-positive integers. The number 0 itself is neither positive nor negative. This order is compatible with the arithmetic operations, i. H.: ${\ displaystyle \ {1,2,3, \ ldots \} \ quad [= \ mathbb {N}]}$ ${\ displaystyle \ {0,1,2,3, \ ldots \} \ quad [= \ mathbb {N} _ {0}]}$ ${\ displaystyle \ {\ ldots, -2, -1 \} \ quad [= - \ mathbb {N}]}$ ${\ displaystyle \ {\ ldots, -2, -1.0 \} \ quad [= - \ mathbb {N} _ {0}]}$

Is and then is${\ displaystyle a ${\ displaystyle c \ leq d}$${\ displaystyle a + c
Is and then is${\ displaystyle a ${\ displaystyle 0 ${\ displaystyle ac

With the help of the arrangement, the sign function

${\ displaystyle \ operatorname {sgn} (x): = {\ begin {cases} -1 & {\ text {falls}} \ quad x <0 \\ ~~ \, 0 & {\ text {if}} \ quad x = 0 \\ ~~ \, 1 & {\ text {falls}} \ quad x> 0 \ end {cases}}}$

and the amount function

${\ displaystyle | x | = \ operatorname {abs} (x): = {\ begin {cases} ~~ \, x & {\ text {if}} \ quad x \ geq 0 \\ - x & {\ text {if }} \ quad x <0 \ end {cases}}}$

define. They hang like this

${\ displaystyle x = \ operatorname {sgn} (x) \, | x |}$

together.

### Mightiness

Like the set of natural numbers, the set of whole numbers is also countable .

The whole numbers do not form a field , because z. B. the equation is not solvable in. The smallest body that contains are the rational numbers . ${\ displaystyle 2x = 1}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Q}}$

### Euclidean ring

An important property of integers is the existence of division by a remainder . Because of this property, there is always a greatest common divisor for two whole numbers , which can be determined with the Euclidean algorithm . In mathematics it is called the Euclidean ring . The theorem of the unique prime factorization in . ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$

## Construction from the natural numbers

If the set of natural numbers is given, then the whole numbers can be constructed as a number range extension:

The following equivalence relation is defined on the set of all pairs of natural numbers : ${\ displaystyle \ mathbb {N} _ {0} \ times \ mathbb {N} _ {0}}$

${\ displaystyle (a, b) \ sim (c, d)}$if ${\ displaystyle a + d = c + b}$

The addition and multiplication on is defined by: ${\ displaystyle \ mathbb {N} _ {0} \ times \ mathbb {N} _ {0}}$

{\ displaystyle {\ begin {aligned} (a, b) + (c, d) & = (a + c, b + d) \\ (a, b) \ cdot (c, d) & = (ac + bd, ad + bc) \ end {aligned}}}

${\ displaystyle \ mathbb {Z} = \ mathbb {N} _ {0} \ times \ mathbb {N} _ {0} \, / \! \ sim}$is now the set of all equivalence classes .

The addition and multiplication of couples now induce well-defined links to , which is to form a ring. ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$

The usual order of integers is defined as

${\ displaystyle (a, b) <(c, d)}$if .${\ displaystyle a + d

Each equivalence class has in the case a unique representative of the form , where , and in the case a unique representative of the form , where . ${\ displaystyle (a, b)}$${\ displaystyle a \ geq b}$${\ displaystyle (n, 0)}$${\ displaystyle n = from}$${\ displaystyle a ${\ displaystyle (0, n)}$${\ displaystyle n = ba}$

The natural numbers can be embedded in the ring of whole numbers by mapping the natural number to the equivalence class represented by. Usually the natural numbers are identified with their images and the equivalence class represented by is denoted by. ${\ displaystyle n}$${\ displaystyle (n, 0)}$${\ displaystyle (n, 0)}$${\ displaystyle (0, n)}$${\ displaystyle -n}$

If a natural number is different from , then the equivalence class represented by is designated as a positive whole number and the equivalence class represented by is designated as a negative whole number. ${\ displaystyle n}$${\ displaystyle 0}$${\ displaystyle (n, 0)}$${\ displaystyle (0, n)}$

This construction of the whole numbers from the natural numbers also works if instead of the quantity , i.e. without , the initial quantity is taken. Then the natural number is in the equivalence class of and that of .${\ displaystyle \ mathbb {N} _ {0}}$${\ displaystyle \ mathbb {N}}$${\ displaystyle 0}$${\ displaystyle n}$${\ displaystyle (n + 1,1)}$${\ displaystyle 0}$${\ displaystyle (1,1)}$

## Related topics

• A construction similar to the construction of whole numbers from natural numbers is generally possible for commutative semigroups. In this sense, the Grothendieck Group is from .${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {N}}$
• The Gaussian numbers and the Eisenstein numbers are two different extensions of integers to sets of complex numbers.
• The pro- finite completion of the group of integers is formed as the (projective or) inverse limit of all finite factor groups of and represents the totality of the pro- finite integers . It is known under the symbol .${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle {\ widehat {\ mathbb {Z}}}}$