Relation between Floor and Ceiling

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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

$\ceiling {\floor x} = \floor x$


Floor equals Ceiling iff Integer

$\floor x = \begin {cases} \ceiling x & : x \in \Z \\ \ceiling x - 1 & : x \notin \Z \\ \end {cases}$

or equivalently:

$\ceiling x = \begin {cases} \floor x & : x \in \Z \\ \floor x + 1 & : x \notin \Z \\ \end {cases}$

where $\Z$ is the set of integers.


Floor of Negative equals Negative of Ceiling

$\floor {-x} = -\ceiling x$


Ceiling of Negative equals Negative of Floor

$\ceiling {-x} = -\floor x$