Integer

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An integer (from the Latin integer meaning "whole") [lower-alpha 1] 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, 5+1/2, and 2 are not.

Contents

The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, [2] [3] and their additive inverses (the negative integers, i.e., −1, −2, −3, ...). The set of integers is often denoted by the boldface (Z) or blackboard bold ${\displaystyle (\mathbb {Z} )}$ letter "Z"—standing originally for the German word Zahlen ("numbers"). [4] [5] [6]

${\displaystyle \mathbb {Z} }$ is a subset of the set of all rational numbers ${\displaystyle \mathbb {Q} }$, which in turn is a subset of the real numbers ${\displaystyle \mathbb {R} }$. Like the natural numbers, ${\displaystyle \mathbb {Z} }$ 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 ${\displaystyle \mathbb {Z} }$ can be annotated to denote various sets, with varying usage amongst different authors: ${\displaystyle \mathbb {Z} ^{+}}$,${\displaystyle \mathbb {Z} _{+}}$ or ${\displaystyle \mathbb {Z} ^{>}}$ for the positive integers, ${\displaystyle \mathbb {Z} ^{0+}}$ or ${\displaystyle \mathbb {Z} ^{\geq }}$ for non-negative integers, and ${\displaystyle \mathbb {Z} ^{\neq }}$ for non-zero integers. Some authors use ${\displaystyle \mathbb {Z} ^{*}}$ for non-zero integers, while others use it for non-negative integers, or for {–1, 1}. Additionally, ${\displaystyle \mathbb {Z} _{p}}$ 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. [7] [8] [9]

Algebraic properties

Like the natural numbers, ${\displaystyle \mathbb {Z} }$ 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,  0), ${\displaystyle \mathbb {Z} }$, unlike the natural numbers, is also closed under subtraction. [10]

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 ${\displaystyle \mathbb {Z} }$.

${\displaystyle \mathbb {Z} }$ 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 a, b and c:

Properties of addition and multiplication on integers
Closure:a + b is an integera × b is an integer
Associativity:a + (b + c) = (a + b) + ca × (b × c) = (a × b) × c
Commutativity:a + b = b + aa × b = b × a
Existence of an identity element:a + 0 = aa × 1 = a
Existence of inverse elements:a + (−a) = 0The only invertible integers (called units) are −1 and 1.
Distributivity:a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)
No zero divisors:If a × b = 0, then a = 0 or b = 0 (or both)

The first five properties listed above for addition say that ${\displaystyle \mathbb {Z} }$, under addition, is an abelian group. 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). In fact, ${\displaystyle \mathbb {Z} }$ under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to ${\displaystyle \mathbb {Z} }$.

The first four properties listed above for multiplication say that ${\displaystyle \mathbb {Z} }$ 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 ${\displaystyle \mathbb {Z} }$ under multiplication is not a group.

All the rules from the above property table (except for the last), when taken together, say that ${\displaystyle \mathbb {Z} }$ 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 ${\displaystyle \mathbb {Z} }$ 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 ${\displaystyle \mathbb {Z} }$ is an integral domain.

The lack of multiplicative inverses, which is equivalent to the fact that ${\displaystyle \mathbb {Z} }$ is not closed under division, means that ${\displaystyle \mathbb {Z} }$ 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 ${\displaystyle \mathbb {Z} }$ as its subring.

Although ordinary division is not defined on ${\displaystyle \mathbb {Z} }$, the division "with remainder" is defined on them. 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. 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.

The above says that ${\displaystyle \mathbb {Z} }$ is a Euclidean domain. This implies that ${\displaystyle \mathbb {Z} }$ is a principal ideal domain, and any positive integer can be written as the products of primes in an essentially unique way. [11] This is the fundamental theorem of arithmetic.

Order-theoretic properties

${\displaystyle \mathbb {Z} }$ is a totally ordered set without upper or lower bound. The ordering of ${\displaystyle \mathbb {Z} }$ 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. Zero is defined as neither negative nor positive.

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

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

Thus it follows that ${\displaystyle \mathbb {Z} }$ 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. [12] 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. [13] Therefore, in modern set-theoretic mathematics, a more abstract construction [14] allowing one to define arithmetical operations without any case distinction is often used instead. [15] The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b). [16]

The intuition is that (a,b) stands for the result of subtracting b from a. [16] 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:

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

precisely when

${\displaystyle a+d=b+c.}$

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; [16] by using [(a,b)] to denote the equivalence class having (a,b) as a member, one has:

${\displaystyle [(a,b)]+[(c,d)]:=[(a+c,b+d)].}$
${\displaystyle [(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)].}$

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

${\displaystyle -[(a,b)]:=[(b,a)].}$

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

${\displaystyle [(a,b)]-[(c,d)]:=[(a+d,b+c)].}$

The standard ordering on the integers is given by:

${\displaystyle [(a,b)]<[(c,d)]}$ if and only if ${\displaystyle a+d

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). 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.

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

${\displaystyle {\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a

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

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

Some examples are:

{\displaystyle {\begin{aligned}0&=[(0,0)]&=[(1,1)]&=\cdots &&=[(k,k)]\\1&=[(1,0)]&=[(2,1)]&=\cdots &&=[(k+1,k)]\\-1&=[(0,1)]&=[(1,2)]&=\cdots &&=[(k,k+1)]\\2&=[(2,0)]&=[(3,1)]&=\cdots &&=[(k+2,k)]\\-2&=[(0,2)]&=[(1,3)]&=\cdots &&=[(k,k+2)].\end{aligned}}}

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 (e.g., zero, succ, pred) and, possibly, using natural numbers, which are assumed to be already constructed (using, say, the Peano approach).

There exist at least ten such constructions of signed integers. [17] 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.

The technique for the construction of integers presented above in this section corresponds to the particular case where there is a single basic operation pair${\displaystyle (x,y)}$ that takes as arguments two natural numbers ${\displaystyle x}$ and ${\displaystyle y}$, and returns an integer (equal to ${\displaystyle x-y}$). 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.

Computer science

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. 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". (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. 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).

Cardinality

The cardinality of the set of integers is equal to 0 (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from ${\displaystyle \mathbb {Z} }$ to ${\displaystyle \mathbb {N} =\{0,1,2,...\}.}$ Such a function may be defined as

${\displaystyle f(x)={\begin{cases}-2x,&{\mbox{if }}x\leq 0\\2x-1,&{\mbox{if }}x>0,\end{cases}}}$

with graph (set of the pairs ${\displaystyle (x,f(x))}$ is

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

Its inverse function is defined by

${\displaystyle {\begin{cases}g(2x)=-x\\g(2x-1)=x,\end{cases}}}$

with graph

{(0, 0), (1, 1), (2, −1), (3, 2), (4, −2), (5, −3), ...}.

Number systems
Complex ${\displaystyle$ :\;\mathbb {C} }
Real ${\displaystyle$ :\;\mathbb {R} }
Rational ${\displaystyle$ :\;\mathbb {Q} }
Integer ${\displaystyle$ :\;\mathbb {Z} }
Natural ${\displaystyle$ :\;\mathbb {N} }
 Zero: 0 One: 1 Prime numbers Composite numbers
Negative integers
Fraction
 Finite decimal Dyadic (finite binary) Repeating decimal
Irrational
 Algebraic irrational Transcendental
Imaginary

Footnotes

1. Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch"). "Entire" derives from the same origin via the French word entier , which means both entire and integer. [1]

Related Research Articles

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial x2x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of x4 + 4.

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted i, called the imaginary unit, and satisfying the equation i2 = −1. Moreover, every complex number can be expressed in the form a + bi, where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

In mathematics, the natural numbers are those numbers used for counting and ordering.

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements.

In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

Exponentiation is a mathematical operation, written as bn, involving two numbers, the baseb and the exponent or powern, and pronounced as "b raised to the power of n". When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:

In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as , where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a ; thus, division by zero is undefined. Since any number multiplied by zero is zero, the expression is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst.

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free-modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.

In algebra, a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero.

In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. For example, −3/7 is a rational number, as is every integer. The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q ; it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient", and first appeared in Bourbaki's Algèbre.

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as . Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol R or and is sometimes called "the reals".

In mathematics, an algebraic number field is an extension field of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .

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