Integer

Anintegeris thenumberzero (0), a positivenatural number(1, 2, 3,...), or the negation of a positive natural number (−1,−2, −3,...).^[1]The negations oradditive inversesof the positive natural numbers are referred to asnegative integers.^[2]Thesetof all integers is often denoted by theboldface $Z$ orblackboard bold $\mathbb {Z}$ .^[3]^[4]

The set of natural numbers $\mathbb {N}$ is asubsetof $\mathbb {Z}$ ,which in turn is a subset of the set of allrational numbers $\mathbb {Q}$ ,itself a subset of thereal numbers $\mathbb {R}$ .^[a]Like the set of natural numbers, the set of integers $\mathbb {Z}$ 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 $\sqrt 2$ are not.^[8]

The integers form the smallestgroupand the smallestringcontaining thenatural numbers.Inalgebraic number theory,the integers are sometimes qualified asrational integersto distinguish them from the more generalalgebraic integers.In fact, (rational) integers are algebraic integers that are alsorational numbers.

History

The word integer comes from theLatin integermeaning "whole" or (literally) "untouched", fromin( "not" ) plustangere( "to touch" ). "Entire"derives from the same origin via theFrenchwordentier,which means bothentireandinteger.^[9]Historically the term was used for anumberthat 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 numbersas their usefulness was recognized.^[14]For exampleLeonhard Eulerin his 1765Elements of Algebradefined integers to include both positive and negative numbers.^[15]

The phrasethe set of the integerswas not used before the end of the 19th century, whenGeorg Cantorintroduced the concept ofinfinite setsandset theory.The use of the letter Z to denote the set of integers comes from theGermanwordZahlen( "numbers" )^[3]^[4]and has been attributed toDavid Hilbert.^[16]The earliest known use of the notation in a textbook occurs inAlgèbrewritten 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 $\mathbb {Z}$ is often annotated to denote various sets, with varying usage amongst different authors: $\mathbb {Z} ^{+}$ , $\mathbb {Z} _{+}$ or $\mathbb {Z} ^{>}$ for the positive integers, $\mathbb {Z} ^{0+}$ or $\mathbb {Z} ^{\geq }$ for non-negative integers, and $\mathbb {Z} ^{\neq }$ for non-zero integers. Some authors use $\mathbb {Z} ^{*}$ for non-zero integers, while others use it for non-negative integers, or for ${-1, 1}$ (thegroup of unitsof $\mathbb {Z}$ ). Additionally, $\mathbb {Z} _{p}$ is used to denote either the set ofintegers modulo $p$ (i.e., the set ofcongruence classesof integers), or the set of $p$ -adic integers.^[21]^[22]

Thewhole numberswere synonymous with the integers up until the early 1950s.^[23]^[24]^[25]In the late 1950s, as part of theNew Mathmovement,^[26]American elementary school teachers began teaching thatwhole numbersreferred to thenatural numbers,excluding negative numbers, whileintegerincluded the negative numbers.^[27]^[28]Thewhole numbersremain 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-negativeintegers are shown in blue and negative integers in red.

Like thenatural numbers, $\mathbb {Z}$ isclosedunder theoperationsof 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), $\mathbb {Z}$ ,unlike the natural numbers, is also closed undersubtraction.^[30]

The integers form aunital ringwhich is the most basic one, in the following sense: for any unital ring, there is a uniquering homomorphismfrom the integers into this ring. Thisuniversal property,namely to be aninitial objectin thecategory of rings,characterizes the ring $\mathbb {Z}$ .

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

Properties of addition and multiplication on integers
	Addition	Multiplication
Closure:	$a + b$ is an integer	$a \times b$ is an integer
Associativity:	$a + (b + c) = (a + b) + c$	$a \times (b \times c) = (a \times b) \times c$
Commutativity:	$a + b = b + a$	$a \times b = b \times a$
Existence of anidentity element:	$a + 0 = a$	$a \times 1 = a$
Existence ofinverse elements:	$a + (- a) = 0$	The only invertible integers (calledunits) are $-1$ and $1$ .
Distributivity:	$a \times (b + c) = (a \times b) + (a \times c)$ and $(a + b) \times c = (a \times c) + (b \times c)$
Nozero divisors:		If $a \times b = 0$ ,then $a = 0$ or $b = 0$ (or both)

The first five properties listed above for addition say that $\mathbb {Z}$ ,under addition, is anabelian group.It is also acyclic group,since every non-zero integer can be written as a finite sum1 + 1 +... + 1or(−1) + (−1) +... + (−1).In fact, $\mathbb {Z}$ under addition is theonlyinfinite cyclic group—in the sense that any infinite cyclic group isisomorphicto $\mathbb {Z}$ .

The first four properties listed above for multiplication say that $\mathbb {Z}$ under multiplication is acommutative monoid.However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that $\mathbb {Z}$ under multiplication is not a group.

All the rules from the above property table (except for the last), when taken together, say that $\mathbb {Z}$ together with addition and multiplication is acommutative ringwithunity.It is the prototype of all objects of suchalgebraic structure.Only thoseequalitiesofexpressionsare true in $\mathbb {Z}$ for allvalues of variables, which are true in any unital commutative ring. Certain non-zero integers map tozeroin certain rings.

The lack ofzero divisorsin the integers (last property in the table) means that the commutative ring $\mathbb {Z}$ is anintegral domain.

The lack of multiplicative inverses, which is equivalent to the fact that $\mathbb {Z}$ is not closed under division, means that $\mathbb {Z}$ isnotafield.The smallest field containing the integers as asubringis the field ofrational numbers.The process of constructing the rationals from the integers can be mimicked to form thefield of fractionsof any integral domain. And back, starting from analgebraic number field(an extension of rational numbers), itsring of integerscan be extracted, which includes $\mathbb {Z}$ as itssubring.

Although ordinary division is not defined on $\mathbb {Z}$ ,the division "with remainder" is defined on them. It is calledEuclidean division,and possesses the following important property: given two integers $a$ and $b$ with $b \neq 0$ ,there exist unique integers $q$ and $r$ such that $a = q \times b + r$ and $0 \leq r < | b |$ ,where $| b |$ denotes theabsolute valueof $b$ .The integer $q$ is called thequotientand $r$ is called theremainderof the division of $a$ by $b$ .TheEuclidean algorithmfor computinggreatest common divisorsworks by a sequence of Euclidean divisions.

The above says that $\mathbb {Z}$ is aEuclidean domain.This implies that $\mathbb {Z}$ is aprincipal ideal domain,and any positive integer can be written as the products ofprimesin anessentially uniqueway.^[31]This is thefundamental theorem of arithmetic.

Order-theoretic properties

$\mathbb {Z}$ is atotally ordered setwithoutupper or lower bound.The ordering of $\mathbb {Z}$ is given by: $:... -3 < -2 < -1 < 0 < 1 < 2 < 3 <...$ An integer ispositiveif it is greater thanzero,andnegativeif 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 $a < b$ and $c < d$ ,then $a + c < b + d$
if $a < b$ and $0 < c$ ,then $ac < bc$ .

Thus it follows that $\mathbb {Z}$ together with the above ordering is anordered ring.

The integers are the only nontrivialtotally ordered abelian groupwhose positive elements arewell-ordered.^[32]This is equivalent to the statement that anyNoetherian valuation ringis either afield—or adiscrete 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 set of natural numbers according to thePeano axioms,call this $P$ .Then construct a set $P^{-}$ which isdisjointfrom $P$ and in one-to-one correspondence with $P$ via a function $\psi$ .For example, take $P^{-}$ to be theordered pairs $(1,n)$ with the mapping $\psi =n\mapsto (1,n)$ .Finally let 0 be some object not in $P$ or $P^{-}$ ,for example the ordered pair $(0,0)$ .Then the integers are defined to be the union $P\cup P^{-}\cup \{0\}$ .

The traditional arithmetic operations can then be defined on the integers in apiecewisefashion, for each of positive numbers, negative numbers, and zero. For examplenegationis defined as follows: $-x={\begin{cases}\psi (x),&{\text{if }}x\in P\\\psi ^{-1}(x),&{\text{if }}x\in P^{-}\\0,&{\text{if }}x=0\end{cases}}$

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

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 classesofordered pairsofnatural numbers $(a, b)$ .^[38]

The intuition is that $(a, b)$ stands for the result of subtracting $b$ from $a$ .^[38]To confirm our expectation that1 − 2and4 − 5denote the same number, we define anequivalence relation $~$ on these pairs with the following rule:

(a,b)\sim (c,d)

precisely when

a+d=b+c.

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:

[(a,b)]+[(c,d)]:=[(a+c,b+d)].

[(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:

-[(a,b)]:=[(b,a)].

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

[(a,b)]-[(c,d)]:=[(a+d,b+c)].

The standard ordering on the integers is given by:

[(a,b)]<[(c,d)]

if and only if

a+d<b+c.

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 areembeddedinto 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

{\begin{cases}a-b,&{\mbox{if }}a\geq b\\-(b-a),&{\mbox{if }}a<b.\end{cases}}

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

This notation recovers the familiarrepresentationof the integers as ${..., -2, -1, 0, 1, 2,...}$ .

Some examples are:

{\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}}

Other approaches

In theoretical computer science, other approaches for the construction of integers are used byautomated theorem proversandterm rewrite engines. Integers are represented asalgebraic termsbuilt using a few basic operations (e.g.,zero,succ,pred) and, possibly, usingnatural numbers,which are assumed to be already constructed (using, say, thePeano 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 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 $(x,y)$ that takes as arguments two natural numbers $x$ and $y$ ,and returns an integer (equal to $x-y$ ). 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 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 primitivedata typeincomputer languages.However, integer data types can only represent asubsetof all integers, since practical computers are of finite capacity. Also, in the commontwo's complementrepresentation, the inherent definition ofsigndistinguishes 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 denotedintor 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

(0, 1), (1, 2), (-1, 3), (2, 4), (-2, 5), (3, 6),. . . , (1 - k, 2 k - 1), (k, 2 k),. . .

More technically, thecardinalityof $\mathbb {Z}$ is said to equal $ℵ 0$ (aleph-null). The pairing between elements of $\mathbb {Z}$ and $\mathbb {N}$ is called abijection.

Footnotes

^More precisely, each system isembeddedin 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]

References

^Science and Technology Encyclopedia.University of Chicago Press. September 2000. p. 280.ISBN 978-0-226-74267-0.
^Hillman, Abraham P.; Alexanderson, Gerald L. (1963).Algebra and trigonometry;.Boston: Allyn and Bacon.
^^a ^b ^cMiller, Jeff (29 August 2010)."Earliest Uses of Symbols of Number Theory".Archived fromthe originalon 31 January 2010.Retrieved20 September2010.
^^a ^bPeter Jephson Cameron (1998).Introduction to Algebra.Oxford University Press. p. 4.ISBN 978-0-19-850195-4.Archivedfrom the original on 8 December 2016.Retrieved15 February2016.
^Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990).Mathematical Methods in Linguistics.Springer Science & Business Media. pp. 78–82.ISBN 978-90-277-2245-4.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.
^Wohlgemuth, Andrew (10 June 2014).Introduction to Proof in Abstract Mathematics.Courier Corporation. p. 237.ISBN 978-0-486-14168-8.
^Polkinghorne, John (19 May 2011).Meaning in Mathematics.OUP Oxford. p. 68.ISBN 978-0-19-162189-5.
^Prep, Kaplan Test (4 June 2019).GMAT Complete 2020: The Ultimate in Comprehensive Self-Study for GMAT.Simon and Schuster.ISBN 978-1-5062-4844-8.
^Evans, Nick (1995). "A-Quantifiers and Scope". In Bach, Emmon W. (ed.).Quantification in Natural Languages.Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262.ISBN 978-0-7923-3352-4.
^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 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]
^Encyclopaedia Britannica 1771,p.83
^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 deNse noteZ;ses éléments sont appelés entiers rationnels.[The group of differences ofNis denoted byZ;its elements are called the rational integers.]
^Birkhoff, Garrett (1948).Lattice Theory(Revised ed.). American Mathematical Society. p. 63.the setJof all integers
^Society, Canadian Mathematical (1960).Canadian Journal of Mathematics.Canadian Mathematical Society. p. 374.Consider the setZof non-negative 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 set of integers by the capital letter Z.
^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.
^Hayden, Robert (1981).A history of the "new math" movement in the United States(PhD). Iowa State University. p. 145.doi:10.31274/rtd-180813-5631.A much more influential force in bringing news of the "new math" to high school teachers and administrators was the National Council of Teachers of Mathematics (NCTM).
^The Growth of Mathematical Ideas, Grades K-12: 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.Retrieved11 August2020.
^Lang, Serge(1993).Algebra(3rd ed.). Addison-Wesley. pp. 86–87.ISBN 978-0-201-55540-0.
^Warner, Seth (2012).Modern Algebra.Dover Books on Mathematics. Courier Corporation. Theorem 20.14, p. 185.ISBN 978-0-486-13709-4.Archivedfrom the original on 6 September 2015.Retrieved29 April2015..
^Mendelson, Elliott (1985).Number systems and the foundations of analysis.Malabar, Fla.: R.E. Krieger Pub. Co. p. 153.ISBN 978-0-89874-818-5.
^Mendelson, Elliott (2008).Number Systems and the Foundations of Analysis.Dover Books on Mathematics. Courier Dover Publications. p. 86.ISBN 978-0-486-45792-5.Archivedfrom the original on 8 December 2016.Retrieved15 February2016..
^Ivorra Castillo:Álgebra
^Kramer, Jürg; von Pippich, Anna-Maria (2017).From Natural Numbers to Quaternions(1st ed.). Switzerland: Springer Cham. pp. 78–81.doi:10.1007/978-3-319-69429-0.ISBN 978-3-319-69427-6.
^Frobisher, Len (1999).Learning to Teach Number: A Handbook for Students and Teachers in the Primary School.The Stanley Thornes Teaching Primary Maths Series. Nelson Thornes. p. 126.ISBN 978-0-7487-3515-0.Archivedfrom the original on 8 December 2016.Retrieved15 February2016..
^^a ^b ^cCampbell, Howard E. (1970).The structure of arithmetic.Appleton-Century-Crofts. p.83.ISBN 978-0-390-16895-5.
^Garavel, Hubert (2017).On the Most Suitable Axiomatization of Signed Integers.Post-proceedings of the 23rd International Workshop on Algebraic Development Techniques (WADT'2016). Lecture Notes in Computer Science. Vol. 10644. Springer. pp. 120–134.doi:10.1007/978-3-319-72044-9_9.ISBN 978-3-319-72043-2.Archivedfrom the original on 26 January 2018.Retrieved25 January2018.

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A Society of Gentlemen in Scotland (1771).Encyclopaedia Britannica.Edinburgh.

External links

This article incorporates material from Integer onPlanetMath,which is licensed under theCreative Commons Attribution/Share-Alike License.

[8] More precisely, each system isembeddedin 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]

[1] Science and Technology Encyclopedia.University of Chicago Press. September 2000. p. 280.ISBN 978-0-226-74267-0.

[2] Hillman, Abraham P.; Alexanderson, Gerald L. (1963).Algebra and trigonometry;.Boston: Allyn and Bacon.

[earliest-3] Miller, Jeff (29 August 2010)."Earliest Uses of Symbols of Number Theory".Archived fromthe originalon 31 January 2010.Retrieved20 September2010.

[Cameron1998-4] Peter Jephson Cameron (1998).Introduction to Algebra.Oxford University Press. p. 4.ISBN 978-0-19-850195-4.Archivedfrom the original on 8 December 2016.Retrieved15 February2016.

[5] Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990).Mathematical Methods in Linguistics.Springer Science & Business Media. pp. 78–82.ISBN 978-90-277-2245-4.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.

[6] Wohlgemuth, Andrew (10 June 2014).Introduction to Proof in Abstract Mathematics.Courier Corporation. p. 237.ISBN 978-0-486-14168-8.

[7] Polkinghorne, John (19 May 2011).Meaning in Mathematics.OUP Oxford. p. 68.ISBN 978-0-19-162189-5.

[9] Prep, Kaplan Test (4 June 2019).GMAT Complete 2020: The Ultimate in Comprehensive Self-Study for GMAT.Simon and Schuster.ISBN 978-1-5062-4844-8.

[10] Evans, Nick (1995). "A-Quantifiers and Scope". In Bach, Emmon W. (ed.).Quantification in Natural Languages.Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. p. 262.ISBN 978-0-7923-3352-4.

[11] Smedley, Edward; Rose, Hugh James; Rose, Henry John (1845).Encyclopædia Metropolitana.B. Fellowes. p. 537.An integer is a multiple of unity

[12] Encyclopaedia Britannica 1771,p.367

[13] 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]

[14] Encyclopaedia Britannica 1771,p.83

[negmath-15] Martinez, Alberto (2014).Negative Math.Princeton University Press. pp. 80–109.

[16] 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.]

[17] 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.

[18] Bourbaki, Nicolas (1951).Algèbre, Chapter 1(in French) (2nd ed.). Paris: Hermann. p. 27.Le symétrisé deNse noteZ;ses éléments sont appelés entiers rationnels.[The group of differences ofNis denoted byZ;its elements are called the rational integers.]

[19] Birkhoff, Garrett (1948).Lattice Theory(Revised ed.). American Mathematical Society. p. 63.the setJof all integers

[20] Society, Canadian Mathematical (1960).Canadian Journal of Mathematics.Canadian Mathematical Society. p. 374.Consider the setZof non-negative integers

[21] 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 set of integers by the capital letter Z.

[edexcelc1-22] Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008

[23] LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.

[24] Mathews, George Ballard (1892).Theory of Numbers.Deighton, Bell and Company. p. 2.

[25] 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.

[26] Peck, Lyman C. (1950).Elements of Algebra.McGraw-Hill. p. 3.The numbers which so arise are called positive whole numbers, or positive integers.

[27] Hayden, Robert (1981).A history of the "new math" movement in the United States(PhD). Iowa State University. p. 145.doi:10.31274/rtd-180813-5631.A much more influential force in bringing news of the "new math" to high school teachers and administrators was the National Council of Teachers of Mathematics (NCTM).

[28] The Growth of Mathematical Ideas, Grades K-12: 24th Yearbook.National Council of Teachers of Mathematics. 1959. p. 14.ISBN 9780608166186.

[29] Deans, Edwina (1963).Elementary School Mathematics: New Directions.U.S. Department of Health, Education, and Welfare, Office of Education. p. 42.

[30] "entry: whole number".The American Heritage Dictionary.HarperCollins.

[31] "Integer | mathematics".Encyclopedia Britannica.Retrieved11 August2020.

[32] Lang, Serge(1993).Algebra(3rd ed.). Addison-Wesley. pp. 86–87.ISBN 978-0-201-55540-0.

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v t e Numbersystems
Sets ofdefinable numbers	Natural numbers( $\mathbb {N}$ ) Integers( $\mathbb {Z}$ ) Rational numbers( $\mathbb {Q}$ ) Constructible numbers Algebraic numbers( $\mathbb {A}$ ) Closed-form numbers Periods Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers
Composition algebras	Division algebras:Real numbers( $\mathbb {R}$ ) Complex numbers( $\mathbb {C}$ ) Quaternions( $\mathbb {H}$ ) Octonions( $\mathbb {O}$ )
Split types	Over $\mathbb {R}$ : Split-complex numbers Split-quaternions Split-octonions Over $\mathbb {C}$ : Bicomplex numbers Biquaternions Bioctonions
Otherhypercomplex	Dual numbers Dual quaternions Dual-complex numbers Hyperbolic quaternions Sedenions( $\mathbb {S}$ ) Split-biquaternions Multicomplex numbers Geometric algebra/Clifford algebra Algebra of physical space Spacetime algebra Plane-based geometric algebra
Infinitiesandinfinitesimals	Cardinal numbers Extended natural numbers Extended real numbers Projective Hyperreal numbers Levi-Civita field Ordinal numbers Supernatural numbers Surreal numbers Superreal numbers
Other types	Irrational numbers Fuzzy numbers Transcendental numbers p-adicnumbers(p-adicsolenoids) Profinite integers Normal numbers
Classification List