Real Number minus Floor

Theorem
Let $x \in \R$ be any real number.

Then:
 * $x - \floor x \in \hointr 0 1$

where $\floor x$ is the floor of $x$.

That is:
 * $0 \le x - \floor x < 1$

Also denoted as
The expression $x - \floor x$ is sometimes denoted $\set x$ and called the fractional part of $x$.

Also see

 * Definition:Fractional Part
 * Definition:Modulo 1: $x \bmod 1 = x - \floor x$