Integer Parts

Definition

Suppose $x \in \R $ is a real number. The lower integer part (or floor) of $x$ is the largest integer $a \in \Z $ satisfiying $a \leq x$. Similarly the upper integer part of $x$ is the smallest integer $b \in \Z $ such that $x \leq b$

Notation

Suppose $x \in \R $. We denote the lower and upper integer parts of $x$ by

\[ \floor{x} \quad \text{ and } \quad \ceil{x} \]