Integers modulo n

Definition

Let $n$ be a positive integer. The integers modulo $n$ form the set $\mathbb{Z}/n\mathbb{Z} = \{ \overline{0}, \overline{1}, \dots, \overline{n - 1} \}$ of the residue classes modulo $n$. Addition and multiplication on $\mathbb{Z}/n\mathbb{Z}$ are $\overline{a} + \overline{b} = \overline{a + b}$ and $\overline{a} \cdot \overline{b} = \overline{a b}$.

Remarks

The operations are well defined: the result does not depend on the choice of $a$ and $b$ in their classes. This follows from compatibility with addition and compatibility with multiplication. With these operations, $\mathbb{Z}/n\mathbb{Z}$ is a commutative ring with $n$ elements, with zero $\overline{0}$ and identity $\overline{1}$. Other notations are $\mathbb{Z}_n$ and $\mathbb{Z}/(n)$. The Lean library Mathlib writes ZMod n, and it sets ZMod 0 to be $\mathbb{Z}$. The ring is a field if and only if $n$ is prime. See integers modulo a prime form a field.

Explanations

L1 human

In $\mathbb{Z}/12\mathbb{Z}$ you compute as on a clock: $\overline{9} + \overline{5} = \overline{14} = \overline{2}$. Products work the same way: $\overline{5} \cdot \overline{7} = \overline{35} = \overline{11}$.