Definition:Modulo Operation/Modulo One

Definition

Let $x, y \in \R$ be real numbers.

Let the modulo operation $\bmod$ be defined as:

$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$

Then:

$x \bmod 1 = x - \left \lfloor {x}\right \rfloor$

from which it follows directly that:

$x = \left \lfloor {x}\right \rfloor + \left({x \bmod 1}\right)$

Also known as

The value $x \bmod 1$ can be referred to as the fractional part of $x$, and sometimes denoted $\left\{{x}\right\}$.

Also see

From Real Number minus Floor we confirm that $0 \le x \bmod 1 < 1$.