Integer
ArithmeticAn 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. Integers have no fractions or decimals.
Example
The numbers $-5$, $-2$, $0$, $3$, and $100$ are all integers. Numbers like $1.5$ or $3/4$ are NOT integers.
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.
Definition
The integers are the set $\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$, extending infinitely in both directions. Integers include the natural numbers, their negatives, and zero. They are closed under addition, subtraction, and multiplication, but not division.
Example
$(-4) + 7 = 3$ (integer). $(-4) \times 3 = -12$ (integer). But $7 / 2 = 3.5$, which is NOT an integer, showing integers are not closed under division.
Key Insight
The word "integer" comes from Latin for "whole" or "untouched." Every integer is a rational number, but not every rational number is an integer.
Definition
The integers $\mathbb{Z}$ form a commutative ring under addition and multiplication. $\mathbb{Z}$ is an integral domain: it has no zero divisors, meaning if $ab = 0$ then $a = 0$ or $b = 0$. $\mathbb{Z}$ is also a Euclidean domain, which guarantees the existence of 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 division algorithm applied to $a = 17$, $b = 5$ yields $q = 3$, $r = 2$, since $17 = 5(3) + 2$. This algorithm underpins the Euclidean GCD procedure.
Key Insight
$\mathbb{Z}$ is a proper subset of the rationals $\mathbb{Q}$, which sits inside the reals $\mathbb{R}$. The integers are the natural setting for number theory: primes, divisibility, and congruences all live here.