Aninteger is thenumber zero (0), a positivenatural number (1, 2, 3, . . .), or the negation of a positive natural number (−1, −2, −3, . . .).[1] The negations oradditive inverses of the positive natural numbers are referred to asnegative integers.[2] Theset of all integers is often denoted by theboldfaceZ orblackboard bold.[3][4]
The set of natural numbers is asubset of, which in turn is a subset of the set of allrational numbers, itself a subset of thereal numbers.[a] Like the set of natural numbers, the set of integers iscountably infinite. An integer may be regarded as a real number that can be written without afractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75,5+1/2, 5/4, and√2 are not.[8]
The word integer comes from theLatininteger meaning "whole" or (literally) "untouched", fromin ("not") plustangere ("to touch"). "Entire" derives from the same origin via theFrench wordentier, which means bothentire andinteger.[9] Historically the term was used for anumber that was a multiple of 1,[10][11] or to the whole part of amixed number.[12][13] Only positive integers were considered, making the term synonymous with thenatural numbers. The definition of integer expanded over time to includenegative numbers as their usefulness was recognized.[14] For exampleLeonhard Euler in his 1765Elements of Algebra defined integers to include both positive and negative numbers.[15]
The phrasethe set of the integers was not used before the end of the 19th century, whenGeorg Cantor introduced the concept ofinfinite sets andset theory. The use of the letter Z to denote the set of integers comes from theGerman wordZahlen ("numbers")[3][4] and has been attributed toDavid Hilbert.[16] The earliest known use of the notation in a textbook occurs inAlgèbre written by the collectiveNicolas Bourbaki, dating to 1947.[3][17] The notation was not adopted immediately. For example, another textbook used the letter J,[18] and a 1960 paper used Z to denote the non-negative integers.[19] But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.[20]
The symbol is often 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 {–1,1} (thegroup of units of). Additionally, is used to denote either the set ofintegers modulop (i.e., the set ofcongruence classes of integers), or the set ofp-adic integers.[21][22]
Thewhole numbers were synonymous with the integers up until the early 1950s.[23][24][25] In the late 1950s, as part of theNew Math movement,[26] American elementary school teachers began teaching thatwhole numbers referred to thenatural numbers, excluding negative numbers, whileinteger included the negative numbers.[27][28] Thewhole numbers remain ambiguous to the present day.[29]
Algebraic properties
Integers can be thought of as discrete, equally spaced points on an infinitely longnumber line. In the above, non-negative integers are shown in blue and negative integers in red.
Like thenatural numbers, isclosed under theoperations of addition andmultiplication, 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),, unlike the natural numbers, is also closed undersubtraction.[30]
is not closed underdivision, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed underexponentiation, 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 integersa,b, andc:
Properties of addition and multiplication on integers
The first five properties listed above for addition say that, under addition, is anabelian group. It is also acyclic group, since every non-zero integer can be written as a finite sum1 + 1 + ... + 1 or(−1) + (−1) + ... + (−1). In fact, under addition is theonly infinite cyclic group—in the sense that any infinite cyclic group isisomorphic to.
The first four properties listed above for multiplication say that under multiplication is acommutative 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 acommutative ring withunity. It is the prototype of all objects of suchalgebraic structure. Only thoseequalities ofexpressions are true in for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map tozero in certain rings.
The lack ofzero divisors in the integers (last property in the table) means that the commutative ring is anintegral domain.
The lack of multiplicative inverses, which is equivalent to the fact that is not closed under division, means that isnot afield. The smallest field containing the integers as asubring is the field ofrational numbers. The process of constructing the rationals from the integers can be mimicked to form thefield of fractions of any integral domain. And back, starting from analgebraic number field (an extension of rational numbers), itsring of integers can be extracted, which includes as itssubring.
Although ordinary division is not defined on, the division "with remainder" is defined on them. It is calledEuclidean division, and possesses the following important property: given two integersa andb withb ≠ 0, there exist unique integersq andr such thata =q ×b +r and0 ≤r < |b|, where|b| denotes theabsolute value ofb. The integerq is called thequotient andr is called theremainder of the division ofa byb. TheEuclidean algorithm for computinggreatest common divisors works by a sequence of Euclidean divisions.
is atotally ordered set withoutupper or lower bound. The ordering of is given by::... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ....An integer ispositive if it is greater thanzero, andnegative 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:
Ifa <b andc <d, thena +c <b +d
Ifa <b and0 <c, thenac <bc
Thus it follows that together with the above ordering is anordered ring.
In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers,zero, and the negations of the natural numbers. This can be formalized as follows.[33] First construct the set of natural numbers according to thePeano axioms, call this. Then construct a set which isdisjoint from and in one-to-one correspondence with via a function. For example, take to be theordered pairs with the mapping. Finally let 0 be some object not in or, for example the ordered pair (0,0). Then the integers are defined to be the union.
The traditional arithmetic operations can then be defined on the integers in apiecewise fashion, for each of positive numbers, negative numbers, and zero. For examplenegation is defined as follows:
The traditional 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.[34]
Equivalence classes of ordered pairs
Red points represent ordered pairs ofnatural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line.
In modern set-theoretic mathematics, a more abstract construction[35][36] allowing one to define arithmetical operations without any case distinction is often used instead.[37] The integers can thus be formally constructed as theequivalence classes ofordered pairs ofnatural numbers(a,b).[38]
The intuition is that(a,b) stands for the result of subtractingb froma.[38] To confirm our expectation that1 − 2 and4 − 5 denote the same number, we define anequivalence 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;[38] by using[(a,b)] to denote the equivalence class having(a,b) 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:
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 numbern is identified with the class[(n,0)] (i.e., the natural numbers areembedded into the integers by map sendingn 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
If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.
This notation recovers the familiarrepresentation of the integers as{..., −2, −1, 0, 1, 2, ...}.
There exist at least ten such constructions of signed integers.[39] 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 in the previous section corresponds to the particular case where there is a single basic operationpair that takes as arguments two natural numbers and, and returns an integer (equal to). This operation is not free since the integer 0 can be writtenpair(0,0), orpair(1,1), orpair(2,2), etc.. This technique of construction is used by theproof assistantIsabelle; 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.
An integer is often a primitivedata type incomputer languages. However, integer data types can only represent asubset of all integers, since practical computers are of finite capacity. Also, in the commontwo's complement representation, the inherent definition ofsign 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 denotedint or Integer in several programming languages (such asAlgol68,C,Java,Delphi, etc.).
Variable-length representations of integers, such asbignums, 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 set of integers iscountably infinite, meaning it is possible to pair each integer with a unique natural number. An example of such a pairing is
^More precisely, each system isembedded in the next, isomorphically mapped to a subset.[5] The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the reals.[6] Such a convention is "a matter of choice", yet not.[7]
^Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990).Mathematical Methods in Linguistics. Springer Science & Business Media. pp. 78–82.ISBN978-90-277-2245-4.The natural numbers are not themselves a subset of this set-theoretic representation of the integers. Rather, the set of all integers contains a subset consisting of the positive integers and zero which is isomorphic to the set of natural numbers.
^Pisano, Leonardo; Boncompagni, Baldassarre (transliteration) (1202).Incipit liber Abbaci compositus to Lionardo filio Bonaccii Pisano in year Mccij [The Book of Calculation] (Manuscript) (in Latin). Translated by Sigler, Laurence E. Museo Galileo. p. 30.Nam rupti uel fracti semper ponendi sunt post integra, quamuis prius integra quam rupti pronuntiari debeant. [And the fractions are always put after the whole, thus first the integer is written, and then the fraction]
^Martinez, Alberto (2014).Negative Math. Princeton University Press. pp. 80–109.
^Euler, Leonhard (1771).Vollstandige Anleitung Zur Algebra [Complete Introduction to Algebra] (in German). Vol. 1. p. 10.Alle diese Zahlen, so wohl positive als negative, führen den bekannten Nahmen der gantzen Zahlen, welche also entweder größer oder kleiner sind als nichts. Man nennt dieselbe gantze Zahlen, um sie von den gebrochenen, und noch vielerley andern Zahlen, wovon unten gehandelt werden wird, zu unterscheiden. [All these numbers, both positive and negative, are called whole numbers, which are either greater or lesser than nothing. We call them whole numbers, to distinguish them from fractions, and from several other kinds of numbers of which we shall hereafter speak.]
^The University of Leeds Review. Vol. 31–32. University of Leeds. 1989. p. 46.Incidentally, Z comes from "Zahl": the notation was created by Hilbert.
^Bourbaki, Nicolas (1951).Algèbre, Chapter 1 (in French) (2nd ed.). Paris: Hermann. p. 27.Le symétrisé deN se noteZ; ses éléments sont appelés entiers rationnels. [The group of differences ofN is denoted byZ; its elements are called the rational integers.]
^Birkhoff, Garrett (1948).Lattice Theory (Revised ed.). American Mathematical Society. p. 63.the setJ of all integers
^Society, Canadian Mathematical (1960).Canadian Journal of Mathematics. Canadian Mathematical Society. p. 374.Consider the setZ of non-negative integers
^Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008
^LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.
^Mathews, George Ballard (1892).Theory of Numbers. Deighton, Bell and Company. p. 2.
^Betz, William (1934).Junior Mathematics for Today. Ginn.The whole numbers, or integers, when arranged in their natural order, such as 1, 2, 3, are called consecutive integers.
^Peck, Lyman C. (1950).Elements of Algebra. McGraw-Hill. p. 3.The numbers which so arise are called positive whole numbers, or positive integers.
^Warner, Seth (2012).Modern Algebra. Dover Books on Mathematics. Courier Corporation. Theorem 20.14, p. 185.ISBN978-0-486-13709-4.Archived from the original on 6 September 2015. Retrieved29 April 2015..
