Editing Floor and ceiling functions (section)

===Relations among the functions===
It is clear from the definitions that
:<math>\lfloor x \rfloor \le \lceil x \rceil,</math>  with equality if and only if ''x'' is an integer, i.e.
:<math>\lceil x \rceil - \lfloor x \rfloor = \begin{cases}
0&\mbox{ if } x\in \mathbb{Z}\\
1&\mbox{ if } x\not\in \mathbb{Z}
\end{cases}</math>

In fact, for integers ''n'', both floor and ceiling functions are the [[identity function|identity]]:
:<math>\lfloor n \rfloor = \lceil n \rceil = n.</math>

Negating the argument switches floor and ceiling and changes the sign:
:<math> \begin{align}
\lfloor x \rfloor +\lceil -x \rceil &= 0 \\
-\lfloor x \rfloor &= \lceil  -x \rceil \\
-\lceil  x \rceil  &= \lfloor -x \rfloor
\end{align}
</math>

and:
:<math>\lfloor x \rfloor + \lfloor -x \rfloor = \begin{cases}
0&\text{if } x\in \mathbb{Z}\\
-1&\text{if } x\not\in \mathbb{Z},
\end{cases}</math>

:<math>\lceil x \rceil + \lceil -x \rceil = \begin{cases}
0&\text{if } x\in \mathbb{Z}\\
1&\text{if } x\not\in \mathbb{Z}.
\end{cases}</math>

Negating the argument complements the fractional part:

:<math>\{ x \} +  \{ -x \} = \begin{cases}
0&\text{if } x\in \mathbb{Z}\\
1&\text{if } x\not\in \mathbb{Z}.
\end{cases}</math>

The floor, ceiling, and fractional part functions are [[Idempotence|idempotent]]:

:<math>
\begin{align}
\big\lfloor \lfloor x \rfloor \big\rfloor &= \lfloor x \rfloor, \\
\big\lceil \lceil x \rceil \big\rceil &= \lceil x \rceil, \\
\big\{ \{ x \} \big\} &= \{ x \}.
\end{align}
</math>

The result of nested floor or ceiling functions is the innermost function:
:<math>
\begin{align}
\big\lfloor \lceil x \rceil \big\rfloor &= \lceil x \rceil, \\
\big\lceil \lfloor x \rfloor \big\rceil &= \lfloor x \rfloor
\end{align}
</math>
due to the identity property for integers.