The integers ℤ are the whole numbers together with their negatives and zero: …, −3, −2, −1, 0, 1, 2, 3, …. They extend the natural numbers by adding negative numbers, making subtraction always possible. The integers form a ring — one of the most fundamental algebraic structures — and are the setting for most of classical number theory.
The natural numbers (1, 2, 3, …) are great for counting things — but they run into a problem the moment you try to subtract a bigger number from a smaller one. What is 3 − 5? With only natural numbers, there's no answer. The integers solve this by adding negative numbers and zero to the mix: …, −3, −2, −1, 0, 1, 2, 3, ….
Mathematicians write the set of integers as ℤ — from the German word Zahlen, meaning "numbers." This notation was introduced by the Bourbaki group of mathematicians in the 20th century and is now used worldwide.
The integers form a commutative ring under addition and multiplication: (ℤ, +, ·). Specifically, ℤ is:
ℤ is not a field — most integers have no multiplicative inverse within ℤ (e.g. there's no integer x with 2x = 1).
Formally, integers can be constructed from the natural numbers ℕ as equivalence classes of ordered pairs (a, b) ∈ ℕ × ℕ, where (a, b) represents "a − b." Two pairs (a, b) and (c, d) are equivalent if a + d = b + c (avoiding subtraction, which isn't yet defined). This construction, due to the formalisation of arithmetic in the 19th–20th centuries, shows that the integers can be built rigorously from simpler foundations, ultimately reducible to set theory.
ℤ is a totally ordered set: for any two integers a, b, exactly one of a < b, a = b, a > b holds. This order is compatible with the ring structure (if a < b then a + c < b + c for any c). ℤ is not well-ordered in the way ℕ is — there is no smallest integer, since you can always subtract 1 to get a smaller one.
The Fundamental Theorem of Arithmetic — every integer greater than 1 factors uniquely into primes — is the central structural fact about ℤ. This makes ℤ the natural setting for elementary number theory: questions about divisibility, GCDs, congruences, and Diophantine equations are all questions about the structure of ℤ.
Negative numbers were treated with suspicion for most of mathematical history. Diophantus (3rd century CE) rejected negative solutions to equations as absurd. European mathematicians as late as the 17th–18th centuries — including prominent figures like Descartes (calling them "false roots") — were uneasy with negatives. Brahmagupta's Brāhmasphuṭasiddhānta (628 CE) gives the first known systematic arithmetic rules treating negative numbers ("debts") on equal footing with positives ("fortunes"). Full acceptance in European mathematics came only in the 19th century, with rigorous algebraic foundations (Hankel, 1867; Peano, 1889) removing any remaining conceptual unease by defining integers purely in terms of formal operations rather than intuitive "quantity."
The equivalence-class construction of ℤ from ℕ (pairs (a,b) representing a−b) is a special case of a general algebraic technique called the Grothendieck group construction, which produces a group from any commutative monoid by formally adjoining inverses. Applied to (ℕ, +), this construction yields exactly (ℤ, +). The same technique, applied to other monoids, produces objects central to K-theory in algebraic topology and algebraic geometry.
Beyond the familiar integers, there is for each prime p a ring of p-adic integers ℤₚ, constructed as the inverse limit of ℤ/pⁿℤ as n → ∞. Every ordinary integer embeds in ℤₚ, but ℤₚ also contains new elements representing "infinite p-adic expansions." The field of fractions of ℤₚ is the p-adic numbers ℚₚ (see the Field entry). p-adic integers are fundamental to modern algebraic number theory and were essential to Wiles's proof of Fermat's Last Theorem.
ℤ generalises to rings of "algebraic integers" in extension fields. The Gaussian integers ℤ[i] = {a + bi : a, b ∈ ℤ} form a Euclidean domain, used to prove results about which integers can be written as a sum of two squares (Fermat's theorem on sums of two squares, 1640). More generally, the ring of integers of any number field is studied in algebraic number theory — and unlike ℤ, these rings do not always have unique factorisation, motivating Dedekind's theory of ideals.