Definition:Modulo Operation/Modulo One

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

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

$x \bmod y := \begin{cases} x - y \floor {\dfrac x y} & : y \ne 0 \\ x & : y = 0 \end{cases}$


$x \bmod 1 = x - \floor x$

from which it follows directly that:

$x = \floor x + \paren {x \bmod 1}$

Also known as

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

Also see

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