Integer Parts

Definition

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

Notation

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

$$\text{floor}x\text{ and }\lceil x\rceil$$