An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer with a minus sign (−1, −2, −3, etc.).^{[1]} The negative numbers are the additive inverses of the corresponding positive numbers.^{[2]} In the language of mathematics, the mix of integers is often denoted by the boldface Z or blackboard bold .^{[3]}^{[4]}
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The mix of natural numbers is a subset of , which in turn is a subset of the mix of all rational numbers , itself a subset of the real numbers .^{[a]} Like the natural numbers, is countably infinite. An integer may be regarded as a real 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.^{[8]}
The integers size the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers lớn distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.
History
The word integer comes from the Latin integer meaning "whole" or (literally) "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.^{[9]} Historically the term was used for a number that was a multiple of 1,^{[10]}^{[11]} or lớn the whole part of a mixed number.^{[12]}^{[13]} Only positive integers were considered, making the term synonymous with the natural numbers. The definition of integer expanded over time lớn include negative numbers as their usefulness was recognized.^{[14]} For example Leonhard Euler in his 1765 Elements of Algebra defined integers lớn include both positive and negative numbers.^{[15]} However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century.^{[14]}
The use of the letter Z lớn denote the mix of integers comes from the German word Zahlen ("numbers")^{[3]}^{[4]} and has been attributed lớn David Hilbert.^{[16]} The earliest known use of the notation in a textbook occurs in Algébre written by the collective Nicolas Bourbaki, dating lớn 1947.^{[3]}^{[17]} The notation was not adopted immediately, for example another textbook used the letter J^{[18]} and a 1960 paper used Z lớn denote the nonnegative integers.^{[19]} But by 1961, Z was generally used by modern algebra texts lớn denote the positive and negative integers.^{[20]}
The symbol is often annotated lớn denote various sets, with varying usage amongst different authors: , or for the positive integers, or for nonnegative integers, and for nonzero integers. Some authors use for nonzero integers, while others use it for nonnegative integers, or for {–1, 1} (the group of units of ). Additionally, is used lớn denote either the mix of integers modulo p (i.e., the mix of congruence classes of integers), or the mix of padic integers.^{[21]}^{[22]}
The whole numbers were synonymous with the integers up until the early 1950s.^{[23]}^{[24]}^{[25]} In the late 1950s, as part of the New Math movement,^{[26]} American elementary school teachers began teaching that "whole numbers" referred lớn the natural numbers, excluding negative numbers, while "integer" included the negative numbers.^{[27]}^{[28]} "Whole number" remains ambiguous lớn the present day.^{[29]}
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, 0), , unlike the natural numbers, is also closed under subtraction.^{[30]}
The integers size 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 lớn 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 a, b and c:
Addition  Multiplication  

Closure:  a + b is an integer  a × b is an integer 
Associativity:  a + (b + c) = (a + b) + c  a × (b × c) = (a × b) × c 
Commutativity:  a + b = b + a  a × b = b × a 
Existence of an identity element:  a + 0 = a  a × 1 = a 
Existence of inverse elements:  a + (−a) = 0  The 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 , under addition, is an abelian group. It is also a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1). In fact, under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic lớn .
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 nonzero integers map lớn 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 name.
The lack of multiplicative inverses, which is equivalent lớn 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 lớn size the field of fractions of any integral domain name. 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 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 is a Euclidean domain name. This implies that is a principal ideal domain name, and any positive integer can be written as the products of primes in an essentially unique way.^{[31]} This is the fundamental theorem of arithmetic.
Ordertheoretic properties
is a totally ordered mix without upper or lower bound. The ordering of is given by: :... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... An integer is positive if it is greater than vãn zero, and negative if it is less than vãn zero. Zero is defined as neither negative nor positive.
The ordering of integers is compatible with the algebraic operations in the following way:
 if a < b and c < d, then a + c < b + d
 if a < b and 0 < c, then ac < bc.
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 wellordered.^{[32]} This is equivalent lớn the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.
Construction
Traditional development
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 mix of natural numbers according lớn the Peano axioms, đường dây nóng this . Then construct a mix which is disjoint from and in onetoone correspondence with via a function . For example, take lớn be the ordered pairs with the mapping . Finally let 0 be some object not in or , for example the ordered pair . Then the integers are defined lớn be the union .
The traditional arithmetic operations can then be defined on the integers in a piecewise fashion, for each of positive numbers, negative numbers, and zero. For example negation is defined as follows:
The traditional style of definition leads lớn many different cases (each arithmetic operation needs lớn be defined on each combination of types of integer) and makes it tedious lớn prove that integers obey the various laws of arithmetic.^{[34]}
Equivalence classes of ordered pairs
In modern settheoretic mathematics, a more abstract construction^{[35]}^{[36]} allowing one lớn define arithmetical operations without any case distinction is often used instead.^{[37]} The integers can thus be formally constructed as the equivalence classes of ordered pairs of natural numbers (a,b).^{[38]}
The intuition is that (a,b) stands for the result of subtracting b from a.^{[38]} 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:
precisely when
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Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;^{[38]} by using [(a,b)] lớn denote the equivalence class having (a,b) as a thành viên, 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:
 if and only if
It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.
Every equivalence class has a unique thành viên that is of the size (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 lớn [(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 familiar representation of the integers as {..., −2, −1, 0, 1, 2, ...} .
Some examples are:
Other approaches
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 lớn be already constructed (using, say, the Peano approach).
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 không lấy phí 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 lớn the particular case where there is a single basic operation pair that takes as arguments two natural numbers and , and returns an integer (equal lớn ). This operation is not không lấy phí 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 không lấy phí 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 "nonnegative" rather than vãn "negative, positive, and 0". (It is, however, certainly possible for a computer lớn 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.).
Variablelength 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 mix of integers is equal lớn ℵ_{0} (alephnull). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from lớn Such a function may be defined as
with graph (set of the pairs is
 {... (−4,8), (−3,6), (−2,4), (−1,2), (0,0), (1,1), (2,3), (3,5), ...}.
Its inverse function is defined by
with graph
 {(0, 0), (1, 1), (2, −1), (3, 2), (4, −2), (5, −3), ...}.
See also
 Canonical factorization of a positive integer
 Hyperinteger
 Integer complexity
 Integer lattice
 Integer part
 Integer sequence
 Integervalued function
 Mathematical symbols
 Parity (mathematics)
 Profinite integer
Complex 

Footnotes
 ^ More precisely, each system is embedded in the next, isomorphically mapped lớn a subset.^{[5]} The commonlyassumed settheoretic 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]}
References
 ^ Science and Technology Encyclopedia. University of Chicago Press. September 2000. p. 280. ISBN 9780226742670.
 ^ "Integers: Introduction lớn the concept, with activities comparing temperatures and money.  Unit 1". OER Commons.
 ^ ^{a} ^{b} ^{c} Miller, Jeff (29 August 2010). "Earliest Uses of Symbols of Number Theory". Archived from the original on 31 January 2010. Retrieved 20 September 2010.
 ^ ^{a} ^{b} Peter Jephson Cameron (1998). Introduction lớn Algebra. Oxford University Press. p. 4. ISBN 9780198501954. Archived from the original on 8 December 2016. Retrieved 15 February 2016.
 ^ Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990). Mathematical Methods in Linguistics. Springer Science & Business Media. pp. 78–82. ISBN 9789027722454.
The natural numbers are not themselves a subset of this settheoretic representation of the integers. Rather, the mix of all integers contains a subset consisting of the positive integers and zero which is isomorphic lớn the mix of natural numbers.
 ^ Wohlgemuth, Andrew (10 June 2014). Introduction lớn Proof in Abstract Mathematics. Courier Corporation. p. 237. ISBN 9780486141688.
 ^ Polkinghorne, John (19 May 2011). Meaning in Mathematics. OUP Oxford. p. 68. ISBN 9780191621895.
 ^ Prep, Kaplan Test (4 June 2019). GMAT Complete 2020: The Ultimate in Comprehensive SelfStudy for GMAT. Simon and Schuster. ISBN 9781506248448.
 ^ Evans, Nick (1995). "AQuantifiers and Scope". In Bach, Emmon W. (ed.). Quantification in Natural Languages. Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262. ISBN 9780792333524.
 ^ Smedley, Edward; Rose, Hugh James; Rose, Henry John (1845). Encyclopædia Metropolitana. B. Fellowes. p. 537.
An integer is a multiple of unity
 ^ Encyclopaedia Britannica 1771, p. 367
 ^ Pisano, Leonardo; Boncompagni, Baldassarre (transliteration) (1202). Incipit liber Abbaci compositus lớn 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]  ^ Encyclopaedia Britannica 1771, p. 83
 ^ ^{a} ^{b} Martinez, Alberto (2014). Negative Math. Princeton University Press. pp. 80–109.
 ^ Euler, Leonhard (1771). Vollstandige Anleitung Zur Algebra [Complete Introduction lớn Algebra] (in German). Vol. 1. p. 10.
Alle diese Zahlen, ví 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 vãn nothing. We đường dây nóng them whole numbers, lớn 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 de N se note Z; ses éléments sont appelés entiers rationnels.
[The group of differences of N is denoted by Z; its elements are called the rational integers.]  ^ Birkhoff, Garrett (1948). Lattice Theory (Revised ed.). American Mathematical Society. p. 63.
the mix J of all integers
 ^ Society, Canadian Mathematical (1960). Canadian Journal of Mathematics. Canadian Mathematical Society. p. 374.
Consider the mix Z of nonnegative integers
 ^ Bezuszka, Stanley (1961). Contemporary Progress in Mathematics: Teacher Supplement [to] Part 1 and Part 2. Boston College. p. 69.
Modern Algebra texts generally designate the mix of integers by the capital letter Z.
 ^ Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Vi xử lý 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. McGrawHill. p. 3.
The numbers which ví arise are called positive whole numbers, or positive integers.
 ^ Hayden, Robert (1981). A history of the "new math" movement in the United States (PhD). Iowa State University. p. 145. doi:10.31274/rtd1808135631.
A much more influential force in bringing news of the "new math" lớn high school teachers and administrators was the National Council of Teachers of Mathematics (NCTM).
 ^ The Growth of Mathematical Ideas, Grades K12: 24th Yearbook. National Council of Teachers of Mathematics. 1959. p. 14. ISBN 9780608166186.
 ^ Deans, Edwina (1963). Elementary School Mathematics: New Directions. U.S. Department of Health, Education, and Welfare, Office of Education. p. 42.
 ^ "entry: whole number". The American Heritage Dictionary. HarperCollins.
 ^ "Integer  mathematics". Encyclopedia Britannica. Retrieved 11 August 2020.
 ^ Lang, Serge (1993). Algebra (3rd ed.). AddisonWesley. pp. 86–87. ISBN 9780201555400.
 ^ Warner, Seth (2012). Modern Algebra. Dover Books on Mathematics. Courier Corporation. Theorem trăng tròn.14, p. 185. ISBN 9780486137094. Archived from the original on 6 September 2015. Retrieved 29 April 2015..
 ^ Mendelson, Elliott (1985). Number systems and the foundations of analysis. Malabar, Fla. : R.E. Krieger Pub. Co. p. 153. ISBN 9780898748185.
 ^ Mendelson, Elliott (2008). Number Systems and the Foundations of Analysis. Dover Books on Mathematics. Courier Dover Publications. p. 86. ISBN 9780486457925. Archived from the original on 8 December 2016. Retrieved 15 February 2016..
 ^ Ivorra Castillo: Álgebra
 ^ Kramer, Jürg; von Pippich, AnnaMaria (2017). From Natural Numbers lớn Quaternions (1st ed.). Switzerland: Springer Cham. pp. 78–81. doi:10.1007/9783319694290. ISBN 9783319694276.
 ^ Frobisher, Len (1999). Learning lớn Teach Number: A Handbook for Students and Teachers in the Primary School. The Stanley Thornes Teaching Primary Maths Series. Nelson Thornes. p. 126. ISBN 9780748735150. Archived from the original on 8 December 2016. Retrieved 15 February 2016..
 ^ ^{a} ^{b} ^{c} Campbell, Howard E. (1970). The structure of arithmetic. AppletonCenturyCrofts. p. 83. ISBN 9780390168955.
 ^ Garavel, Hubert (2017). On the Most Suitable Axiomatization of Signed Integers. Postproceedings of the 23rd International Workshop on Algebraic Development Techniques (WADT'2016). Lecture Notes in Computer Science. Vol. 10644. Springer. pp. 120–134. doi:10.1007/9783319720449_9. ISBN 9783319720432. Archived from the original on 26 January 2018. Retrieved 25 January 2018.
Sources
External links
Look up integer in Wiktionary, the không lấy phí dictionary.
 "Integer", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 The Positive Integers – divisor tables and numeral representation tools
 OnLine Encyclopedia of Integer Sequences cf OEIS
 Weisstein, Eric W. "Integer". MathWorld.
This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
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