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: