The Ceiling and Floor Functions

The Ceiling and Floor Functions

We are now going to briefly look at two very simple yet important functions known as the ceiling and floor functions.

 Definition: The Ceiling Function takes every real number $x \in \mathbb{R}$ and maps it to the smallest integer $z \in \mathbb{Z}$ such that $x \leq z$.

The ceiling of a real number $x$ is often denoted as $\left \lceil x \right \rceil$. For example, let $x = 1.4$. Then the smallest integer $z$ such that $x \leq z$ is $z = 2$ and so we have that $\left \lceil 1.4 \right \rceil = 2$. The graph of the ceiling function is given below: Definition: The Floor Function takes every real number $x \in \mathbb{R}$ and maps it to the largest integer $z \in \mathbb{Z}$ such that $z \leq x$.

The floor of a real number $x$ is often denoted as $\left \lfloor x \right \rfloor$. For example, let $x = 3.14159...$. Then the largest integer $z$ such that $z \leq x$ is $z = 3$ and so we have that $\left \lfloor 3.14159... \right \rfloor = 3$. The graph of the floor function is given below: 