Floors & Ceilings

Overview

The floor of $x$ is the greatest integer less than $x$ . The ceiling of $x$ is the least integer greater than $x$ . These values are denoted $⌊ x ⌋$ and $⌈ x ⌉$ respectively.

Identities

For integers $x$ and $y > 0$ ,

\begin{aligned} ⌊ \frac{x}{y} ⌋ & = ⌈ \frac{x}{y} - \frac{y - 1}{y} ⌉ \\ ⌈ \frac{x}{y} ⌉ & = ⌊ \frac{x}{y} + \frac{y - 1}{y} ⌋ \end{aligned}