The Integers

An integer 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, $1/2$, and ${\sqrt {2}}$ are not.

The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), 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 (Z) or blackboard bold $(\mathbb {Z} )$ letter "Z".

$\mathbb {Z}$ is a subset of the set of all rational numbers $\mathbb {Q}$, which in turn is a subset of the real numbers $\mathbb {R}$. Like the natural numbers, $\mathbb {Z}$ 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 $\mathbb {Z}$ can be 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}. Additionally, $\mathbb {Z} _{p}$ 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.

Algebraic properties

Integers can be thought of as discrete, equally spaced points on an infinitely long number line. In the above, non-negative integers are shown in blue and negative integers in red.

Like the natural numbers, $\mathbb {Z}$ 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), $\mathbb {Z}$, 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 $\mathbb {Z}$.

$\mathbb {Z}$ 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

	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 $\mathbb {Z}$, 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, $\mathbb {Z}$ under addition is the only infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to $\mathbb {Z}$.

The first four properties listed above for multiplication say that $\mathbb {Z}$ 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 $\mathbb {Z}$ 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 a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in $\mathbb {Z}$ 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 $\mathbb {Z}$ is an integral domain.

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

Although ordinary division is not defined on $\mathbb {Z}$, 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 $\mathbb {Z}$ is a Euclidean domain. This implies that $\mathbb {Z}$ 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

$\mathbb {Z}$ is a totally ordered set without upper or lower bound. The ordering of $\mathbb {Z}$ 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 $\mathbb {Z}$ 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. This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.

Construction

Red points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line.

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 (a,b).

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

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

precisely when

$a+d=b+c.$

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; 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 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

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

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

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

Some examples are:

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

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. 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$(x,y)$ 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 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.

Integer Divisibility

Definition: Let $a,b\in \mathbb {Z}$. Then $b$ is said to be Divisible by $a$, or, $a$ is said to Divide $b$ written $a\mid b$ if there exists a $q\in \mathbb {Z}$ such that $aq=b$. The number $a$ is said to be a Divisor of $b$ and $b$ is said to be a Multiple of $a$. If there does not exist an $q\in \mathbb {Z}$ such that $aq=b$ then we say that $a$ does not divide $b$, or, $b$ is not divisible by $a$ denoted $a\not \mid b$.

For example, the number $a=2$ divides $b=6$ since for $q=3$ we have that $aq=b$, so we write $a\mid b$. However, the number $a=2$ does not divide any odd number $b=2n+1$ for $n\in \mathbb {Z}$, so in this case we have that $a\not \mid b$, i.e., $2\not \mid (2n+1)$ for all integers $n$.

We will now look at some rather elementary proofs regarding the divisibility of integers.

Theorem 1: Let $a,b,c\in \mathbb {Z}$.

a) If $a\mid b$ and $b\mid c$ then $a\mid c$.
b) If $a\mid b$ and $a\mid c$ then for all $x,y\in \mathbb {Z}$, $a\mid (bx+cy)$.

c) If $a\mid b$ and $a\mid c$ then $a\mid (b+c)$ and $a\mid (b-c)$.

For the following proofs we utilize the fact that the set of integers is closed under standard addition and standard multiplication - that is, if $x,y\in \mathbb {Z}$ then $(x+y)\in \mathbb {Z}$ and $x\cdot y=xy\in \mathbb {Z}$.

• Proof of a) If $a\mid b$ and $b\mid c$ then there exists $q_{1},q_{2}\in \mathbb {Z}$ such that:

${\begin{aligned}\quad aq_{1}=b\quad {\text{and}}\quad bq_{2}=c.\end{aligned}}$

• Substituting the first equation into the second gives us:

${\begin{aligned}\quad bq_{2}=c\\\quad a(q_{1}q_{2})=c\end{aligned}}$

• Since $q_{1},q_{2}\in \mathbb {Z}$ we have that their product $q_{1}q_{2}\in \mathbb {Z}$. Therefore $a\mid c$. $\blacksquare$

• Proof of b) If $a\mid b$ and $a\mid c$ then there exists $q_{1},q_{2}\in \mathbb {Z}$ such that:

(3)

${\begin{aligned}\quad aq_{1}=b\quad \mathrm {and} \quad aq_{2}=c.\end{aligned}}$

• Multiply the first equation by $x\in \mathbb {Z}$ and the second equation by $y\in \mathbb {Z}$ to get:

${\begin{aligned}\quad axq_{1}=bx\quad \mathrm {and} \quad ayq_{2}=cy.\end{aligned}}$

• We now add both of these equations:

${\begin{aligned}\quad axq_{1}+ayq_{2}=bx+cy\\\quad a(xq_{1}+yq_{2})=bx+cy\end{aligned}}$

• Since $q_{1},q_{2},x,y\in \mathbb {Z}$ we have that $(xq_{1}+yq_{2})\in \mathbb {Z}$ and so $a\mid (bx+cy)$. $\blacksquare$

• Proof of c) From (b) if we set $x=1$ and $y=1$ we get that $a\mid (b+c)$ and if we set $x=1$ and $y=-1$ we get that $a\mid (b-c)$ as desired. $\blacksquare$

The Division Algorithm

One rather important aspect of the divisibility of integers is that if $a,b\in \mathbb {Z}$ then $a$ can be written as the product of some quotient $q$ with $b$ plus a remainder $r$. For example, if $a=11$ and $b=3$, then $a=3(b)+2$ where $q=3$ and $r=2$.

The following theorem known as the Division Algorithm shows us that for any pair of integers $a,b\in \mathbb {Z}$ where $b>0$ that the form $a=bq+r$ exists and is unique.

Theorem 1 (The Division Algorithm): Let $a,b\in \mathbb {Z}$ with $b>0$. Then there exists unique $q,r\in \mathbb {Z}$ where $0\leq r<b$ and such that $a=bq+r$.

The value of $r$ is often called the Remainder when $a$ is divided by $b$. That is, $a$ is equal to some multiple $q$, known as the Quotient of $b$, plus a remainder $r$ for which $0\leq r<b$.

• Proof: Let $a,b\in \mathbb {Z}$ with $b>0$ and consider the following set $A$:

${\begin{aligned}\quad A=\{a-bq:q\in \mathbb {N} \}=\{a,a-b,a-2b,...\}.\end{aligned}}$

• Consider $B\subset A$ of nonnegative elements in $A$. If $a>0$ then $a\in B$ and if $a<0$ then $a-ba=a(1-b)\in B$, so $B$ is a nonempty set. Moreover, $B$ is a nonempty subset of integers and so by the Well-Ordering principle there exists a least element of $B$, say $r\in B$ is this least element. Since $r\in B$ we have that $r=a-bn$ for some $n\in \mathbb {N}$ and so:

${\begin{aligned}\quad a=bq+r\end{aligned}}$

• So $a$ if of the form we desire. Now since $r\in B$ and every element in $B$ is nonnegative then we have that $0\leq r$. Suppose that $r\geq b$. Then $r=b+p$ where $p\geq 0$ and so:

${\begin{aligned}\quad a=bq+r=bq+(b+p)=b(q+1)+p\end{aligned}}$

• Therefore $p=a-b(q+1)\geq 0$ and so $p\in B$. Since $b>0$ and $q+1>q$ we have that $0\leq p<r$. But this implies that $r$ is not the least element of $B$ which is a contradiction, so our assumption that $r\geq b$ is false, and so $0\leq r<b$.

• We lastly show that $q$ and $r$ are unique for $a=bq+r$ and $0\leq r<b$. Suppose that $a=bq_{1}+r_{1}$ and $a=bq_{2}+r_{2}$ where $0\leq r_{1},r_{2}<b$. Then by subtracting these two equations we get:

${\begin{aligned}\quad 0=bq_{1}-bq_{2}+r_{1}-r_{2}\\\quad 0=b(q_{1}-q_{2})+r_{1}-r_{2}\\\quad r_{2}-r_{1}=b(q_{1}-q_{2})\end{aligned}}$

• Therefore $r_{2}-r_{1}$ is a multiple of $b$. Since $0\leq r_{1}<b$ and $0\leq r_{2}<b$ we must have that $-b<r_{2}-r_{1}<b$. The only multiple of $b$ such that $-b<r_{2}-r_{1}<b$ is $0$, so:

${\begin{aligned}\quad r_{2}-r_{1}=0\\\quad r_{1}=r_{2}\end{aligned}}$

• Since $r_{1}=r_{2}=r$, we have that $0=b(q_{1}-q_{2})$. Since $b>0$ this implies that $q_{1}-q_{2}=0$ and so $q_{1}=q_{2}$. Therefore $a$ can be uniquely expressed in the form $a=bq+r$ where $0\leq r<b$. $\blacksquare$

Licensing

Content obtained and/or adapted from:

• Integer, Wikipedia under a CC BY-SA license