AlegsaOnline.com

Integer: definition, properties, notation and basic arithmetic

An integer is a whole number with no fractional part. This article explains definitions, notation, arithmetic properties, algebraic structure, ordering, examples, history and common uses.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: June 8, 2026

An integer is a whole number that can be positive, negative, or zero; it does not contain a fractional or decimal part. In elementary mathematics integers are the simplest discrete quantities and include the natural numbers, their negatives, and zero. Integers are commonly represented along an infinite number line, where every point with an integer coordinate corresponds to a distinct integer. A simple depiction of such a line is shown here: ${...-4,-3,-2,-1,0,+1,+2,+3,+4,...}\,\!$

Basic definition and notation

Formally, an integer can be described as a rational number with denominator 1; equivalently, as a decimal with only zeros after the decimal separator (for example, 17, 17.0, and 17.0000 represent the same integer). The set of all integers is usually denoted by the letter Z (from the German Zahlen). Texts often write this as ℤ and refer to subsets such as the positive integers (ℤ+) and the non-negative integers (including zero). For introductory references see natural numbers and rational numbers. A common paired concept is the decimal point or decimal separator; integers have no nonzero digits after that separator decimal notation.

Consecutive integers are those that differ by 1: for example, …, −2, −1, 0, 1, 2, 3, …; a short list of consecutive terms is illustrated here: ${3,4,5,6}$ . The ordering of integers is total: given any two integers a and b, exactly one of these holds — a < b, a = b, or a > b — which makes comparison and ranking straightforward.

Arithmetic properties and algebraic structure

Integers are closed under addition, subtraction and multiplication: the sum, difference, or product of two integers is an integer. Division by a nonzero integer need not yield an integer (for example, 1 divided by 2 equals 1/2, which is not an integer). The integers form an infinite countable set with algebraic structure: (ℤ, +, 0) is an abelian group under addition, and with multiplication it becomes a commutative ring with unity 1. For more about sets and groups see consecutive integers and set, and a basic discussion of groups at abelian group. $\mathbf {Z}$

Important notions that arise in the study of integers include divisibility, greatest common divisors, prime numbers and factorization. Many algorithms in computer science and number theory—such as the Euclidean algorithm for the greatest common divisor—operate in the domain of integers. Modular arithmetic, which studies integers with respect to an equivalence relation determined by a modulus, underpins cryptography, hashing and calendar computations.

Size, order types and historical notes

The set of integers is infinite and countable; its cardinality is the smallest infinite cardinal, often denoted aleph-null. References to cardinality and order are standard in set theory; see entries on cardinal number and ordinal number. Historically the concept of integers developed from counting and ledger-keeping: natural numbers for counting, zero and negative numbers for debts and balances. Symbolic notation such as ℤ and operations on integers were formalized over centuries as algebra and arithmetic matured. $\mathbb {Z}$ $\mathbb {Z} _{+}\!$

Uses, examples and distinctions

Integers serve as indexes, counters, measures of signed quantity (temperature, elevation relative to a datum), and discrete labels. Examples include sequence positions (1, 2, 3), temperature values (−5 °C), financial balances (+100, −250) and algebraic coefficients. Distinguish integers from rational numbers that may have nontrivial fractional parts, and from real numbers that include irrational values. Visual or tabular representations often include special markers for zero and for positive and negative subsets: $\mathbf {Z}$ $\mathbf {Z}$ $\aleph_0$ $\mathbf {Z}$ $^{*}\omega +\omega$

Summary: Integers are whole numbers without fractions, closed under addition, subtraction and multiplication, and fundamental in discrete mathematics and computation.
See also: basic arithmetic, modular arithmetic, prime factorization, number theory and combinatorics.

The integers (ℤ) are part of the rational numbers (ℚ), which in turn are part of the real numbers (ℝ). They themselves contain the natural numbers (ℕ).

Properties

Ring

The integers form a ring with respect to addition and multiplication, i.e. they can be added, subtracted and multiplied without restriction. Calculation rules such as the commutative law and the associative law for addition and multiplication apply, and the distributive laws also apply.

The existence of subtraction allows linear equations of the form

a + x = b

with natural numbers and can always be solved: x = b - a . If one restricts to the set of natural numbers, then not every such equation is solvable.

In abstract terms, this means the integers form a commutative unitary ring. The neutral element of addition is 0, the additive inverse element of is , the neutral element of multiplication is 1.

Arrangement

The set of integers is totally ordered, in the order

. $\cdots < -2 < -1 < 0 < 1 < 2 < \cdots$

That is, you can compare two integers each. One speaks of

positive $\{1,2,3,\ldots \}$	$[=\mathbb {N} ]$ ,		non-negative $\{0,1,2,3,\ldots \}$	$[=\mathbb {N} _{0}]$ ,
negative $\{\ldots ,-2,-1\}$	$[=-\mathbb {N} ]$ and		non-positive $\{\ldots ,-2,-1,0\}$	$[=-\mathbb {N} _{0}]$

integers. The number 0 itself is neither positive nor negative. This order is compatible with the arithmetic operations, ie:

If a<b and $c \leq d$ , then a + c < b + d .

If a<b and 0 < c , then ac < bc .

With the help of the arrangement, the sign function

$\operatorname {sgn}(x):={\begin{cases}-1&{\text{falls }}\quad x<0\\~~\,0&{\text{falls }}\quad x=0\\~~\,1&{\text{falls }}\quad x>0\end{cases}}$

and the magnitude function

$|x|=\operatorname {abs} (x):={\begin{cases}~~\,x&{\text{falls }}\quad x\geq 0\\-x&{\text{falls }}\quad x<0\end{cases}}$

define. They hang as follows

$x=\operatorname {sgn}(x)\,|x|$

together.

Thickness

Like the set of natural numbers, the set of integers is also countable.

The integers do not form a body because, for example, the equation is 2x = 1 not solvable in $\mathbb {Z}$ . The smallest body containing $\mathbb {Z}$ is the rational numbers $\mathbb {Q}$ .

Euclidean ring

An important property of the integers is the existence of division with remainder. Because of this property, two integers always have a greatest common divisor, which can be determined using the Euclidean algorithm. In mathematics, is called $\mathbb {Z}$ a Euclidean ring. From this also follows the theorem of unique prime factorization in $\mathbb {Z}$ .

Construction from the natural numbers

If the set of natural numbers is given, then the integers can be constructed from it as an extension of the number range:

On the set $\N_0 \times \N_0$ of all pairs of natural numbers, the following equivalence relation is defined:

$(a, b) \sim (c, d)$ , if a + d = c + b

Addition and multiplication on $\mathbb {N} _{0}\times \mathbb {N} _{0}$ is defined by:

$\begin{align} (a, b) + (c, d) &= (a + c, b + d)\\ (a, b) \cdot (c, d) &= (ac + bd, ad + bc) \end{align}$

$\mathbb {Z} =\mathbb {N} _{0}\times \mathbb {N} _{0}\,/\!\sim$ is now the set of all equivalence classes.

The addition and multiplication of the pairs now induce well-defined links on $\mathbb {Z}$ , with which $\mathbb {Z}$ becomes a ring.

The usual order of the integers is defined as

(a, b) < (c, d) if a + d < c + b .

Each equivalence class (a, b) has, in the case $a\geq b$ , a unique representative of the form (n,0) , where n=a-b , and in the case a unique a<b representative of the form (0,n) , where n=b-a .

The natural numbers can be embedded in the ring of integers by mapping the natural number to the equivalence class represented by (n,0) Usually, the natural numbers are (n,0) identified with their images and the equivalence (0,n) class represented by denoted by

If is a natural number different $0$ from , the equivalence class represented by (n,0) called a positive integer and the equivalence class represented by (0,n) called a negative integer.

This construction of the integers from the natural numbers works even if instead of $\mathbb {N} _{0}$ the set $\mathbb {N}$ , that is, without $0$ , is taken as the initial set. Then the natural number is in the equivalence class of (n+1,1) and the is $0$ in that of (1,1) .