An integer (from the Latin ''integer'' meaning "whole") is colloquially defined as a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not.
The set of integers consists of zero (), the positive natural numbers (, , , ...), also called ''whole numbers'' or ''counting numbers'', and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface () or blackboard bold letter "Z"—standing originally for the German word ''Zahlen'' ("numbers").
is a subset of the set of all rational numbers , which in turn is a subset of the real numbers . Like the natural numbers, is countably infinite.
The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
Symbol
The symbol can be annotated to denote various sets, with varying usage amongst different authors: , or for the positive integers, or for non-negative integers, and for non-zero integers. Some authors use for non-zero integers, while others use it for non-negative integers, or for . Additionally, is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of -adic integers.Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008
Algebraic properties
Like the natural numbers, is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, ), , unlike the natural numbers, is also closed under subtraction.
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. This universal property, namely to be an initial object in the category of rings, characterizes the ring .
is not closed under division, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative).
The following table lists some of the basic properties of addition and multiplication for any integers , and :
In the language of abstract algebra, the first five properties listed above for addition say that , under addition, is an abelian group. It is also a cyclic group, since every non-zero integer can be written as a finite sum or . In fact, under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to .
The first four properties listed above for multiplication say that under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that under multiplication is not a group.
All the rules from the above property table (except for the last), when taken together, say that together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings.
The lack of zero divisors in the integers (last property in the table) means that the commutative ring is an integral domain.
The lack of multiplicative inverses, which is equivalent to the fact that is not closed under division, means that is ''not'' a field. The smallest field containing the integers as a subring is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes as its subring.
Although ordinary division is not defined on , the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers and with , there exist unique integers and such that and , where denotes the absolute value of . The integer is called the ''quotient'' and is called the ''remainder'' of the division of by . 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 is a Euclidean domain. This implies that is a principal ideal domain, and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.
Order-theoretic properties
is a totally ordered set without upper or lower bound. The ordering of is given by: An integer is ''positive'' if it is greater than zero, and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: # if and , then # if and , then . Thus it follows that 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. This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.
Construction
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 integers obey the various laws of arithmetic. Therefore, in modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers .
The intuition is that stands for the result of subtracting from . To confirm our expectation that and denote the same number, we define an equivalence relation on these pairs with the following rule:
:
precisely when
:
Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using to denote the equivalence class having as a member, one has:
:
:
The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
:
Hence subtraction can be defined as the addition of the additive inverse:
:
The standard ordering on the integers is given by:
:
Symbol
The symbol can be annotated to denote various sets, with varying usage amongst different authors: , or for the positive integers, or for non-negative integers, and for non-zero integers. Some authors use for non-zero integers, while others use it for non-negative integers, or for . Additionally, is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of -adic integers.Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008
Algebraic properties
Like the natural numbers, is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, ), , unlike the natural numbers, is also closed under subtraction.
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. This universal property, namely to be an initial object in the category of rings, characterizes the ring .
is not closed under division, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative).
The following table lists some of the basic properties of addition and multiplication for any integers , and :
In the language of abstract algebra, the first five properties listed above for addition say that , under addition, is an abelian group. It is also a cyclic group, since every non-zero integer can be written as a finite sum or . In fact, under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to .
The first four properties listed above for multiplication say that under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that under multiplication is not a group.
All the rules from the above property table (except for the last), when taken together, say that together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings.
The lack of zero divisors in the integers (last property in the table) means that the commutative ring is an integral domain.
The lack of multiplicative inverses, which is equivalent to the fact that is not closed under division, means that is ''not'' a field. The smallest field containing the integers as a subring is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes as its subring.
Although ordinary division is not defined on , the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers and with , there exist unique integers and such that and , where denotes the absolute value of . The integer is called the ''quotient'' and is called the ''remainder'' of the division of by . 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 is a Euclidean domain. This implies that is a principal ideal domain, and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.
Order-theoretic properties
is a totally ordered set without upper or lower bound. The ordering of is given by: An integer is ''positive'' if it is greater than zero, and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: # if and , then # if and , then . Thus it follows that 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. This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.
Construction
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 integers obey the various laws of arithmetic. Therefore, in modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers .
The intuition is that stands for the result of subtracting from . To confirm our expectation that and denote the same number, we define an equivalence relation on these pairs with the following rule:
:
precisely when
:
Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using to denote the equivalence class having as a member, one has:
:
:
The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
:
Hence subtraction can be defined as the addition of the additive inverse:
:
The standard ordering on the integers is given by:
: