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