Relation between Floor and Ceiling

Theorem

Let $x \in \R$ be a real number.

Let $\left \lfloor {x}\right \rfloor$ be the floor of $x$, and $\left \lceil {x}\right \rceil$ be the ceiling of $x$.

Then the following results apply:

Floor of Ceiling is Ceiling

$\left \lfloor {\left \lceil {x}\right \rceil}\right \rfloor = \left \lceil {x}\right \rceil$

Ceiling of Floor is Floor

$\left \lceil {\left \lfloor {x}\right \rfloor}\right \rceil = \left \lfloor {x}\right \rfloor$

Floor equals Ceiling iff Integer

$\left \lfloor {x}\right \rfloor = \begin{cases} \left \lceil {x}\right \rceil & : x \in \Z \\ \left \lceil {x}\right \rceil - 1 & : x \notin \Z \\ \end{cases}$

or equivalently:

$\left \lceil {x}\right \rceil = \begin{cases} \left \lfloor {x}\right \rfloor & : x \in \Z \\ \left \lfloor {x}\right \rfloor + 1 & : x \notin \Z \\ \end{cases}$

where $\Z$ is the set of integers.

Floor of Negative equals Negative of Ceiling

$\left \lfloor {-x}\right \rfloor = - \left \lceil {x}\right \rceil$

Ceiling of Negative equals Negative of Floor

$\left \lceil {-x}\right \rceil = - \left \lfloor {x}\right \rfloor$