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.