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:
 * $$\left\{{x}\right\} \ \stackrel {\mathbf {def}} {=\!=} \ x - \left \lfloor {x}\right \rfloor$$.

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