\begin{definition}[Definition:Modulo Operation/Modulo One]
Let $x, y \in \R$ be real numbers.
Let the modulo operation $\bmod$ be defined as:
:$x \bmod y := \begin{cases}
x - y \floor {\dfrac x y} & : y \ne 0 \\
x & : y = 0
\end{cases}$
Then:
:$x \bmod 1 = x - \floor x$
from which it follows directly that:
:$x = \floor x + \paren {x \bmod 1}$
\end{definition}