Congruence (Number Theory)/Examples/Modulo 1

Example of Congruence Modulo an Integer
Let $x \equiv y \pmod 1$ be defined as congruence on the real numbers modulo $1$:


 * $\forall x, y \in \R: x \equiv y \pmod 1 \iff \exists k \in \Z: x - y = k$

That is, if their difference $x - y$ is an integer.

The equivalence classes of this equivalence relation are of the form:


 * $\left[\!\left[{x}\right]\!\right] = \left\{ {\dotsc, x - 2, x - 1, x, x + 1, x + 2, \dotsc}\right\}$

Each equivalence class has exactly one representative in the half-open real interval:
 * $\left[{0 \,.\,.\, 1}\right) = \left\{ {x\in \R: 0 \le x < 1}\right\}$

Also see

 * Definition:Modulo 1