Definition:Fractional Part

Definition

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

Let $\left \lfloor {x}\right \rfloor$ be the floor function of $x$.

Then the quantity $x - \left \lfloor {x}\right \rfloor$ is called the fractional part of $x$, and is frequently denoted:

$\left\{{x}\right\} := x - \left \lfloor {x}\right \rfloor$

Beware, of course, not to get $\left\{{x}\right\}$ confused with the singleton set containing $x$.

Note that from Real Number Minus Floor, we have that $0 \le \left\{{x}\right\} < 1$, or $\left\{{x}\right\} \in \left[{0 . . 1}\right)$.

Compare with the definition of modulo 1:

$x \bmod 1 = \left\{{x}\right\}$

Sources

• Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (1968): $\S 1.2.4$