The integers are the arithmetic universe obtained when counting numbers are made stable under subtraction. In the [natural numbers](/page/Natural%20Number), the expression $3 - 5$ has no value inside the same system; in the integers it becomes $-2$. This single enlargement changes the algebraic character of arithmetic: addition becomes reversible, equations such as $x + a = b$ always have integer solutions, and divisibility becomes a structural relation rather than only a computational test. The set of integers is usually denoted by $\mathbb{Z}$, from the German word Zahlen. It is the central example of an ordered commutative [ring](/page/Ring), the base object for divisibility, [congruence](/page/Congruence), prime numbers, and the construction of the rational numbers. Much of elementary and algebraic number theory begins by studying which properties of $\mathbb{Z}$ survive in more general rings. ## Definition The first need is closure under subtraction. Counting measures size, but arithmetic equations require solving for unknowns, and solving $x+a=b$ forces the presence of negative values whenever $b<a$. [definition: Integer] An integer is an element of the set \begin{align*} \mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}. \end{align*} The set $\mathbb{Z}$ is equipped with addition, multiplication, and additive inverse maps \begin{align*} + &: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}. \end{align*} \begin{align*} \cdot &: \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}. \end{align*} \begin{align*} - &: \mathbb{Z} \to \mathbb{Z}. \end{align*} It is also equipped with the order relation $\le$ as a subset of $\mathbb{Z} \times \mathbb{Z}$. [/definition] The definition records the object, but the point is algebraic: $\mathbb{Z}$ is the smallest familiar number system where every addition equation can be solved. To use the order on this enlarged system, we need vocabulary for the direction above $0$, because counting, induction, and prime factorization all single out the positive side of the number line. [definition: Positive Integer] A positive integer is an integer $n \in \mathbb{Z}$ satisfying $n > 0$. [/definition] The positive side alone does not describe the new part of $\mathbb{Z}$. Subtraction creates integers below $0$, and these values are needed to solve equations whose right-hand side is smaller than the quantity being subtracted. We therefore name the integers on the lower side of $0$ before discussing arithmetic across signs. [definition: Negative Integer] A negative integer is an integer $n \in \mathbb{Z}$ satisfying $n < 0$. [/definition] Together, the positive integers, the negative integers, and $0$ partition $\mathbb{Z}$. Positive integers are exactly the natural numbers under the convention used on Androma, where $\mathbb{N}$ starts at $1$. Once the additive structure is in place, the next question is which multiplicative divisions remain inside the integer system, because ordinary fractions are usually not integers. [definition: Divisibility in the Integers] Let $a,b \in \mathbb{Z}$. The integer $a$ divides $b$, written $a \mid b$, if there exists $k \in \mathbb{Z}$ such that \begin{align*} b = ak. \end{align*} [/definition] The integer $0$ requires care in this definition: $0 \mid b$ holds only when $b=0$, while every integer divides $0$. Thus $0 \mid 0$ under the convention used here; some elementary texts reserve divisibility notation for nonzero divisors, so statements involving $0$ should always be read against the stated definition. A quick computation shows both the strength and the limitation of exact divisibility. [example: Divisibility and Non-Divisibility] Let $a=6$ and $b=42$. Since $7 \in \mathbb{Z}$ and \begin{align*} 42 = 6 \cdot 7, \end{align*} the definition of divisibility gives $6 \mid 42$. For $44$, suppose that $44=6k$ for some $k \in \mathbb{Z}$. If $k \le 7$, then multiplying the inequality by the positive integer $6$ gives \begin{align*} 6k \le 6 \cdot 7 = 42 < 44. \end{align*} So $6k$ cannot equal $44$. If $k \ge 8$, then \begin{align*} 6k \ge 6 \cdot 8 = 48 > 44. \end{align*} So $6k$ cannot equal $44$ in this case either. Every integer $k$ satisfies either $k \le 7$ or $k \ge 8$, so no integer $k$ satisfies $44=6k$. Hence $6 \nmid 44$. The closest surrounding multiples are \begin{align*} 6 \cdot 7 = 42 \quad \text{and} \quad 6 \cdot 8 = 48. \end{align*} Since \begin{align*} 44 = 42 + 2 = 6 \cdot 7 + 2, \end{align*} the failure of exact divisibility is measured by remainder $2$ after division by $6$. [/example] Exact divisibility is too rigid for many arithmetic problems: a number may fail to be a multiple of $6$ while still carrying a stable remainder after division by $6$. To study clocks, parity, residue classes, and modular equations, we need a relation that treats two integers as the same whenever their difference is a multiple of a chosen modulus. Congruence is that relation; it keeps precisely the remainder information that survives addition and multiplication. [definition: Congruence Modulo an Integer] Let $a,b,n \in \mathbb{Z}$ with $n \ne 0$. The integers $a$ and $b$ are congruent modulo $n$, written \begin{align*} a \equiv b \pmod{n}, \end{align*} if $n \mid (a-b)$. [/definition] Congruence turns arithmetic in $\mathbb{Z}$ into arithmetic on residue classes, retaining exactly the remainder information that ordinary divisibility discards. [example: Congruence Computation] We compute congruence modulo $6$ by checking whether the difference is divisible by $6$. For $44$ and $2$, subtracting gives \begin{align*} 44-2=42. \end{align*} Since $7 \in \mathbb{Z}$ and \begin{align*} 42=6\cdot 7, \end{align*} there exists an integer $7$ such that $44-2=6\cdot 7$. Therefore $6 \mid (44-2)$, so by the definition of congruence modulo an integer, \begin{align*} 44 \equiv 2 \pmod{6}. \end{align*} The integer $-4$ gives another representative of the same congruence class. First, \begin{align*} 44-(-4)=44+4. \end{align*} Then \begin{align*} 44+4=48. \end{align*} Since $8 \in \mathbb{Z}$ and \begin{align*} 48=6\cdot 8, \end{align*} we have $6 \mid (44-(-4))$. Hence \begin{align*} 44 \equiv -4 \pmod{6}. \end{align*} Thus $2$ and $-4$ are two integer representatives of the same remainder information modulo $6$. [/example] Remainders describe arithmetic relative to a chosen modulus, but factorization asks a different question: which integers cannot be broken into smaller positive multiplicative pieces? Without naming those atoms, divisibility gives many isolated facts but no systematic decomposition theory. Prime integers are the positive integers whose only positive divisors are forced by the definition of divisibility itself, and they are the pieces from which the multiplicative structure of $\mathbb{Z}$ is built. [definition: Prime Integer] A prime integer is an integer $p \in \mathbb{Z}$ such that $p > 1$ and the only positive divisors of $p$ are $1$ and $p$. [/definition] This page uses the positive convention for primes: $2,3,5,7,\ldots$ are prime, while $-2$ is not called prime. Since signs can be moved from one factor to another without changing divisibility in an essential way, we need to identify the factors that act as invertible sign changes. [definition: Unit in the Integers] A unit in $\mathbb{Z}$ is an integer $u \in \mathbb{Z}$ for which there exists $v \in \mathbb{Z}$ satisfying \begin{align*} uv = 1. \end{align*} [/definition] The definition is short, but in $\mathbb{Z}$ it has a very rigid consequence: the only invertible integers are the two possible signs. This classification matters because multiplying by a unit should not change the essential content of a factorization; it only changes the sign of the product. [quotetheorem:9699] This fact explains why factorization in $\mathbb{Z}$ is unique only up to order and multiplication by units. It also marks the precise obstruction to treating $\mathbb{Z}$ as a field. [example: Failure of Multiplicative Inverses] In a [field](/page/Field), every nonzero element has a multiplicative inverse. We show that $2 \in \mathbb{Z}$ has no multiplicative inverse in $\mathbb{Z}$. Suppose, toward a contradiction, that there exists $m \in \mathbb{Z}$ such that \begin{align*} 2m=1. \end{align*} If $m \le 0$, then multiplying by the positive integer $2$ preserves the inequality, so \begin{align*} 2m \le 2 \cdot 0 = 0. \end{align*} Thus $2m \le 0 < 1$, so $2m \ne 1$. If $m \ge 1$, then multiplying by the positive integer $2$ preserves the inequality, so \begin{align*} 2m \ge 2 \cdot 1 = 2. \end{align*} Thus $2m \ge 2 > 1$, so $2m \ne 1$. Every integer $m$ satisfies either $m \le 0$ or $m \ge 1$, so no integer $m$ satisfies $2m=1$. Therefore $2$ is a nonzero element of $\mathbb{Z}$ without a multiplicative inverse in $\mathbb{Z}$, and hence $\mathbb{Z}$ is not a field even though it is a commutative ring with identity. [/example] To translate divisibility into the language of rings, we need to collect all multiples of a fixed integer at once. This collection is closed under addition and under multiplication by arbitrary integers, so it becomes the algebraic object that records exactly what divisibility by that integer means inside the ring $\mathbb{Z}$. [definition: Principal Ideal Generated by an Integer] Let $a \in \mathbb{Z}$. The principal ideal generated by $a$ is the subset \begin{align*} (a) = a\mathbb{Z} = \{ak : k \in \mathbb{Z}\}. \end{align*} [/definition] The ideal $(a)$ contains exactly the integer multiples of $a$. This is the simplest instance of the connection between number theory and commutative algebra. ## Equivalent Characterisations This section first records elementary ways to recognize the integers inside arithmetic: they are the signed whole numbers, ordered discretely, and closed under addition, subtraction, and multiplication. A few later remarks mention the ring-theoretic language that abstracts these same facts. Those remarks are included only as orientation for readers who already know rings, ideals, and quotient structures; the core integer facts below do not require that vocabulary. The list notation for $\mathbb{Z}$ is concrete, but several equivalent viewpoints are useful in different parts of mathematics. For constructing the number system from scratch, the formal-difference viewpoint is the most useful: it explains exactly why negative integers exist and how subtraction is made internal without appealing to the number line. The construction uses two standard pieces of quotient notation. If $S$ is a set and $\sim$ is an [equivalence relation](/page/Equivalence%20Relation) on $S$, then $S/{\sim}$ means the set of equivalence classes: each element is a whole collection of objects of $S$ that are declared equivalent to one another. The notation $[s]$ means the equivalence class containing $s$. Thus $(\mathbb{N}_0 \times \mathbb{N}_0)/{\sim}$ means that ordered pairs of nonnegative integers are grouped into classes, and one class is treated as one formal integer. [quotetheorem:9700] This construction explains why negative integers are not added by fiat. They are forced by the desire to turn the additive monoid of natural numbers into an additive group. In algebra, however, the important question is not only how to build $\mathbb{Z}$, but how to recognize it inside every unital ring. Repeated addition of the identity element in a ring should produce integer arithmetic without any extra choices, and the universal property is the precise statement that this process is canonical. [quotetheorem:9701] The theorem says that $\mathbb{Z}$ is the initial unital ring. Every unital ring receives a canonical homomorphism from $\mathbb{Z}$, whose image may fail to be an embedded copy in positive characteristic. To explain why induction, minimum counterexample arguments, and division with remainder work in $\mathbb{Z}$, we also need the order-theoretic fact that the integers are discrete rather than dense. [quotetheorem:9702] This well-ordering behavior for bounded subsets is not shared by the rational numbers $\mathbb{Q}$ or the [real numbers](/page/Real%20Numbers) $\mathbb{R}$. It is a major reason that integer arithmetic admits algorithms. ## Standard Examples The first example shows why integers are introduced at all: subtraction needs a closed home. [example: Subtraction Forces Negative Integers] Inside $\mathbb{N}$, where $\mathbb{N}$ starts at $1$, the equation \begin{align*} x+5=3 \end{align*} has no solution. Indeed, if $x \in \mathbb{N}$, then $x \ge 1$, so order compatibility with addition gives $x+5 \ge 1+5$, and \begin{align*} 1+5=6. \end{align*} Since $6>3$, every natural-number value of $x$ gives $x+5>3$, so no $x \in \mathbb{N}$ satisfies $x+5=3$. Inside $\mathbb{Z}$, we can add the additive inverse of $5$ to both sides: \begin{align*} (x+5)+(-5)=3+(-5). \end{align*} By associativity of addition, \begin{align*} x+(5+(-5))=3+(-5). \end{align*} Since $5+(-5)=0$, this becomes \begin{align*} x+0=3+(-5). \end{align*} The additive identity gives $x+0=x$, and $3+(-5)=-2$, so \begin{align*} x=-2. \end{align*} Substituting this value back into the original equation, \begin{align*} (-2)+5=3. \end{align*} Thus passing from $\mathbb{N}$ to $\mathbb{Z}$ supplies the negative integer needed to solve this subtraction equation. [/example] The examples above show the main boundary phenomena: subtraction forces negative integers, exact divisibility can fail, congruence records remainders, and nonzero integers need not be invertible. This failure of unrestricted division is productive rather than defective. Because division is restricted, the concepts of primes, greatest common divisors, and Euclidean algorithms become meaningful. ## Properties The basic properties of $\mathbb{Z}$ should be read in two layers. The elementary layer concerns order, divisibility, congruences, and the [division algorithm](/theorems/725). When terms such as ideals, prime ideals, integral domains, or principal ideal domains appear, they name the later algebraic framework in which these same integer facts are generalized; they are not prerequisites for using the integer arithmetic statements in this page. The most important algorithmic property of the integers is division with remainder. It is the foundation for base expansions, greatest common divisors, and the Euclidean algorithm. Before stating it, we fix the quotient notation that appears in modular arithmetic. For a nonzero integer $n$, the expression $\mathbb{Z}/n\mathbb{Z}$ denotes the set of congruence classes modulo $n$: two integers determine the same class exactly when they are congruent modulo $n$. The symbol $n\mathbb{Z}$ is the ideal of all multiples of $n$, so $\mathbb{Z}/n\mathbb{Z}$ is read as "$\mathbb{Z}$ modulo $n\mathbb{Z}$." Addition and multiplication in this quotient are performed by choosing integer representatives and then passing back to congruence classes. [quotetheorem:9333] The integer $q$ is the quotient and $r$ is the remainder. This theorem is the formal statement behind the ordinary long division algorithm. Once division with remainder is available, repeated division lets us decompose an integer into prime factors. [quotetheorem:730] This theorem is the reason prime numbers are central to number theory. It turns multiplicative questions about arbitrary integers into questions about prime powers. The same divisibility structure has an algebraic expression through ideals, where multiples of a number become subobjects of $\mathbb{Z}$. Here an ideal of $\mathbb{Z}$ means a subset closed under addition, additive inverses, and multiplication by arbitrary integers. The notation $I \trianglelefteq \mathbb{Z}$ means that $I$ is an ideal of the ring $\mathbb{Z}$; the symbol $\trianglelefteq$ marks this special subobject relation and is not an order comparison. The next classification asks whether every ideal of $\mathbb{Z}$ is forced to come from the multiples of a single integer. [quotetheorem:9703] The integer $d$ is the nonnegative generator of the ideal. This theorem is the bridge from arithmetic to ideal theory: divisibility statements become containment statements, but the direction reverses because a larger generator has fewer multiples. [quotetheorem:3257] Factoring and ideal containment both rely on the idea that multiplication by a nonzero integer does not erase information. If two nonzero integers could multiply to $0$, cancellation would fail and the usual construction of fractions from pairs of integers would identify too many pairs. The next property isolates the exact algebraic safeguard: $\mathbb{Z}$ has no zero divisors. [quotetheorem:9704] The [integral domain](/page/Integral%20Domain) property explains why fractions can be built from integers without collapsing distinct denominators. A different structural need comes from inequalities: integer arithmetic supports comparison, and comparison is useful only when the arithmetic operations preserve order in controlled ways. [quotetheorem:9705] This compatibility is what permits inequalities to be manipulated algebraically. It also distinguishes ordered arithmetic from modular arithmetic, where no compatible total order exists on $\mathbb{Z}/n\mathbb{Z}$ for $n>1$. ## Relationship to Other Concepts The integers contain the natural numbers and sit inside the rational numbers: \begin{align*} \mathbb{N} \subsetneq \mathbb{Z} \subsetneq \mathbb{Q}. \end{align*} The first inclusion adds additive inverses. The second inclusion adds multiplicative inverses for nonzero integers. As a ring, $\mathbb{Z}$ is the prototype for integral domains, [principal ideal domains](/page/Principal%20Ideal%20Domain), and [Euclidean domains](/page/Euclidean%20Domain). Many algebraic structures are measured by how closely their arithmetic resembles the arithmetic of the integers. The quotient rings $\mathbb{Z}/n\mathbb{Z}$ are built from congruence modulo $n$. They provide the standard setting for modular arithmetic, finite rings, and elementary cryptographic constructions. Prime ideals are the commutative-algebra version of prime numbers. In $\mathbb{Z}$, they are especially transparent, and the quotient by a prime ideal gives the familiar arithmetic of residues modulo a prime. [quotetheorem:9706] This result connects integer arithmetic to finite fields. It is also the first example of a general pattern: prime ideals are detected by quotient rings that have no zero divisors. The integers also serve as coefficients for polynomials. The ring $\mathbb{Z}[x]$, the polynomial ring in one variable $x$ with integer coefficients, links number theory with the arithmetic of [polynomial rings](/page/Polynomial%20Ring), while algebraic number theory studies rings that extend $\mathbb{Z}$ in number fields. When this page uses the quotient notation $\mathbb{Z}/n\mathbb{Z}$, it means the set of congruence classes modulo $n$, with addition and multiplication performed on representatives and then reduced modulo $n$. When it uses ideal notation such as $I \trianglelefteq \mathbb{Z}$, it means that $I$ is an ideal of the ring $\mathbb{Z}$: a subset closed under addition and under multiplication by arbitrary integers. These algebraic terms package familiar divisibility and congruence behavior into ring language; they are not needed for the first pass through the elementary arithmetic material. ## Beyond and Connections The integers sit at the meeting point of several Androma topics. For foundations, compare their construction and order with [Natural Number](/page/Natural%20Number). For arithmetic structure, the main local tools are divisibility arguments, [Congruence](/page/Congruence), and [Division Algorithm for Integers](/theorems/9333). For the abstract-algebra viewpoint, the relevant next notes are [Ring](/page/Ring), [Ideal](/page/Ideal), [Quotient Ring](/page/Quotient%20Ring), and [Polynomial Ring](/page/Polynomial%20Ring). Reading in that order keeps the elementary integer facts separate from the ring-theoretic language that later generalizes them. ## References [Natural Number](/page/Natural%20Number). [Congruence](/page/Congruence). [Division Algorithm for Integers](/theorems/9333). [Ring](/page/Ring). [Ideal](/page/Ideal). [Quotient Ring](/page/Quotient%20Ring). [Polynomial Ring](/page/Polynomial%20Ring). Hardy and Wright, *An Introduction to the Theory of Numbers* (1979). Ireland and Rosen, *A Classical Introduction to Modern Number Theory* (1990). Dummit and Foote, *Abstract Algebra* (2004).