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 $x$.

Also known as
Some sources, particularly those aimed for the grade-school and muggle market, refer to this as the decimal part, or (even worse) just the decimal.

This misnomer arises from the fact that it is the part of the number after the decimal point.

Burn it with fire.

Also see
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:
 * $x \bmod 1 = \left\{{x}\right\}$