Real Modular Arithmetic

Why

We extend modular arithmetic to the real numbers.

Definition

Two real numbers $x, y \in \R $ are congruent modulo $\alpha \in \R $ if their difference is a multiple of $\alpha $

Notation

For $x, y \in \R $, if $x$ and $y$ are congruent modulo $\alpha \in \R $ we write

\[ x \equiv y (\mod \alpha ) \]