# Integer

For computer representation, see Integer (computer science). Integer_sentence_0

For the generalization in algebraic number theory, see Algebraic integer. Integer_sentence_1

An integer (from the Latin meaning "whole") is colloquially defined as a number that can be written without a fractional component. Integer_sentence_2

For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and √2 are not. Integer_sentence_3

ℤ is a subset of the set of all rational numbers ℚ, which in turn is a subset of the real numbers ℝ. Integer_sentence_4

Like the natural numbers, ℤ is countably infinite. Integer_sentence_5

The integers form the smallest group and the smallest ring containing the natural numbers. Integer_sentence_6

In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. Integer_sentence_7

In fact, (rational) integers are algebraic integers that are also rational numbers. Integer_sentence_8

## Symbol Integer_section_0

The symbol ℤ can be annotated to denote various sets, with varying usage amongst different authors: ℤ, ℤ+ or ℤ for the positive integers, ℤ or ℤ for non-negative integers, and ℤ for non-zero integers. Integer_sentence_9

Some authors use ℤ for non-zero integers, while others use it for non-negative integers, or for {–1, 1}. Integer_sentence_10

Additionally, ℤp is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p-adic integers. Integer_sentence_11

## Algebraic properties Integer_section_1

Like the natural numbers, ℤ is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. Integer_sentence_12

However, with the inclusion of the negative natural numbers (and importantly, 0), ℤ, unlike the natural numbers, is also closed under subtraction. Integer_sentence_13

The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. Integer_sentence_14

This universal property, namely to be an initial object in the category of rings, characterizes the ring ℤ. Integer_sentence_15

ℤ is not closed under division, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Integer_sentence_16

Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative). Integer_sentence_17

The following table lists some of the basic properties of addition and multiplication for any integers a, b and c: Integer_sentence_18

Integer_table_general_0

Properties of addition and multiplication on integersInteger_table_caption_0
Closure:Integer_header_cell_0_1_0 a + b is an integerInteger_cell_0_1_1 a × b is an integerInteger_cell_0_1_2
Associativity:Integer_header_cell_0_2_0 a + (b + c) = (a + b) + cInteger_cell_0_2_1 a × (b × c) = (a × b) × cInteger_cell_0_2_2
Commutativity:Integer_header_cell_0_3_0 a + b = b + aInteger_cell_0_3_1 a × b = b × aInteger_cell_0_3_2
Existence of an identity element:Integer_header_cell_0_4_0 a + 0 = aInteger_cell_0_4_1 a × 1 = aInteger_cell_0_4_2
Existence of inverse elements:Integer_header_cell_0_5_0 a + (−a) = 0Integer_cell_0_5_1 The only invertible integers (called units) are −1 and 1.Integer_cell_0_5_2
Distributivity:Integer_header_cell_0_6_0 a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)Integer_cell_0_6_1
No zero divisors:Integer_header_cell_0_7_0 Integer_cell_0_7_1 If a × b = 0, then a = 0 or b = 0 (or both)Integer_cell_0_7_2

In the language of abstract algebra, the first five properties listed above for addition say that ℤ, under addition, is an abelian group. Integer_sentence_19

It is also a cyclic group, since every non-zero integer can be written as a finite sum 1 + 1 + … + 1 or (−1) + (−1) + … + (−1). Integer_sentence_20

In fact, ℤ under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to ℤ. Integer_sentence_21

The first four properties listed above for multiplication say that ℤ under multiplication is a commutative monoid. Integer_sentence_22

However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that ℤ under multiplication is not a group. Integer_sentence_23

All the rules from the above property table (except for the last), when taken together, say that ℤ together with addition and multiplication is a commutative ring with unity. Integer_sentence_24

It is the prototype of all objects of such algebraic structure. Integer_sentence_25

Only those equalities of expressions are true in ℤ for all values of variables, which are true in any unital commutative ring. Integer_sentence_26

Certain non-zero integers map to zero in certain rings. Integer_sentence_27

The lack of zero divisors in the integers (last property in the table) means that the commutative ring ℤ is an integral domain. Integer_sentence_28

The lack of multiplicative inverses, which is equivalent to the fact that ℤ is not closed under division, means that ℤ is not a field. Integer_sentence_29

The smallest field containing the integers as a subring is the field of rational numbers. Integer_sentence_30

The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. Integer_sentence_31

And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes ℤ as its subring. Integer_sentence_32

Although ordinary division is not defined on ℤ, the division "with remainder" is defined on them. Integer_sentence_33

It is called Euclidean division, and possesses the following important property: given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < | b |, where | b | denotes the absolute value of b. Integer_sentence_34

The integer q is called the quotient and r is called the remainder of the division of a by b. Integer_sentence_35

The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. Integer_sentence_36

Again, in the language of abstract algebra, the above says that ℤ is a Euclidean domain. Integer_sentence_37

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. Integer_sentence_38

This is the fundamental theorem of arithmetic. Integer_sentence_39

## Order-theoretic properties Integer_section_2

ℤ is a totally ordered set without upper or lower bound. Integer_sentence_40

The ordering of ℤ is given by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... An integer is positive if it is greater than zero, and negative if it is less than zero. Integer_sentence_41

Zero is defined as neither negative nor positive. Integer_sentence_42

The ordering of integers is compatible with the algebraic operations in the following way: Integer_sentence_43

Integer_ordered_list_0

1. if a < b and c < d, then a + c < b + dInteger_item_0_0
2. if a < b and 0 < c, then ac < bc.Integer_item_0_1

Thus it follows that ℤ together with the above ordering is an ordered ring. Integer_sentence_44

The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered. Integer_sentence_45

This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring. Integer_sentence_46

## Construction Integer_section_3

In elementary school teaching, integers are often intuitively defined as the (positive) natural numbers, zero, and the negations of the natural numbers. Integer_sentence_47

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 integers obey the various laws of arithmetic. Integer_sentence_48

Therefore, in modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. Integer_sentence_49

The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b). Integer_sentence_50

The intuition is that (a,b) stands for the result of subtracting b from a. Integer_sentence_51

To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule: Integer_sentence_52

precisely when Integer_sentence_53

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using [(a,b)] to denote the equivalence class having (a,b) as a member, one has: Integer_sentence_54

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair: Integer_sentence_55

Hence subtraction can be defined as the addition of the additive inverse: Integer_sentence_56

The standard ordering on the integers is given by: Integer_sentence_57

It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes. Integer_sentence_58

Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). Integer_sentence_59

The natural number n is identified with the class [(n,0)] (i.e., the natural numbers are embedded into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted −n (this covers all remaining classes, and gives the class [(0,0)] a second time since −0 = 0. Integer_sentence_60

Thus, [(a,b)] is denoted by Integer_sentence_61

If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity. Integer_sentence_62

This notation recovers the familiar representation of the integers as {…, −2, −1, 0, 1, 2, …}. Integer_sentence_63

Some examples are: Integer_sentence_64

In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and term rewrite engines. Integer_sentence_65

Integers are represented as algebraic terms built using a few basic operations (e.g., zero, succ, pred) and, possibly, using natural numbers, which are assumed to be already constructed (using, say, the Peano approach). Integer_sentence_66

There exist at least ten such constructions of signed integers. Integer_sentence_67

These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2) and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms. Integer_sentence_68

## Computer science Integer_section_4

Main article: Integer (computer science) Integer_sentence_69

An integer is often a primitive data type in computer languages. Integer_sentence_70

However, integer data types can only represent a subset of all integers, since practical computers are of finite capacity. Integer_sentence_71

Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". Integer_sentence_72

(It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Integer_sentence_73

Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.). Integer_sentence_74

Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Integer_sentence_75

Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10). Integer_sentence_76

## Cardinality Integer_section_5

The cardinality of the set of integers is equal to ℵ0 (aleph-null). Integer_sentence_77

This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from ℤ to ℕ. Integer_sentence_78

If ℕ₀ ≡ {0, 1, 2, ...} then consider the function: Integer_sentence_79

{… (−4,8) (−3,6) (−2,4) (−1,2) (0,0) (1,1) (2,3) (3,5) ...} Integer_sentence_80

If ℕ ≡ {1, 2, 3, ...} then consider the function: Integer_sentence_81

{... (−4,8) (−3,6) (−2,4) (−1,2) (0,1) (1,3) (2,5) (3,7) ...} Integer_sentence_82

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. Integer_sentence_83