Integer
An integer is any whole number, its negative counterpart, or zero, with no fractional or decimal part.
Definition
An integer is any positive whole number, any negative whole number, or zero, with no fractional or decimal part, forming the set $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$, extending infinitely in both directions and including the natural numbers, their negatives, and zero. Integers are closed under addition, subtraction, and multiplication, but not division. Algebraically, the integers $\mathbb{Z}$ form a commutative ring: an integral domain with no zero divisors (if $ab = 0$ then $a = 0$ or $b = 0$), and a Euclidean domain, which guarantees the division algorithm: for any $a, b$ in $\mathbb{Z}$ with $b \neq 0$, there exist unique $q, r$ such that $a = bq + r$ with $0 \le r < |b|$.
Example
The numbers $-5$, $-2$, $0$, $3$, and $100$ are all integers, while $1.5$ or $3/4$ are not. Integers stay closed under addition and multiplication, such as $(-4) + 7 = 3$ and $(-4) \times 3 = -12$, but not under division: $7 / 2 = 3.5$ is not an integer. The division algorithm applied to $a = 17$, $b = 5$ yields $q = 3$, $r = 2$, since $17 = 5(3) + 2$, the same algorithm that underpins the Euclidean GCD procedure.
Key Insight
Think of integers as the numbers you would find on a thermometer: they go up and down by whole steps, never landing between the marks. The word "integer" comes from Latin for "whole" or "untouched." Every integer is a rational number, but not every rational number is an integer: $\mathbb{Z}$ is a proper subset of $\mathbb{Q}$, which in turn sits inside the reals $\mathbb{R}$. The integers are the natural setting for number theory, where primes, divisibility, and congruences all live.