Editing Floor and ceiling functions (section)

===Equivalences===
These formulas can be used to simplify expressions involving floors and ceilings.<ref>Graham, Knuth, & Patashink, Ch. 3</ref>

:<math>
\begin{alignat}{3}
\lfloor x \rfloor &= m \ \ &&\mbox{ if and only if } &m &\le x < m+1,\\
\lceil x \rceil &= n     &&\mbox{ if and only if } &\ \ n -1 &< x \le n,\\

\lfloor x \rfloor &= m   &&\mbox{ if and only if } &x-1 &< m \le x,\\
\lceil x \rceil &= n    &&\mbox{ if and only if }  &x &\le n < x+1.
\end{alignat}
</math>

In the language of [[order theory]], the floor function is a [[residuated mapping]], that is, part of a [[Galois connection]]: it is the upper adjoint of the function that embeds the integers into the reals.

:<math>
\begin{align}
x<n &\;\;\mbox{ if and only if } &\lfloor x \rfloor &< n, \\
n<x &\;\;\mbox{ if and only if } &n &< \lceil x \rceil, \\
x\le n &\;\;\mbox{ if and only if } &\lceil  x \rceil &\le n, \\
n\le x &\;\;\mbox{ if and only if } &n &\le \lfloor x \rfloor.
\end{align}
</math>

These formulas show how adding an integer {{mvar|n}} to the arguments affects the functions:

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

The above are never true if {{mvar|n}} is not an integer; however, for every {{mvar|x}} and {{mvar|y}}, the following inequalities hold:

:<math>\begin{align}
\lfloor x \rfloor + \lfloor y \rfloor &\leq \lfloor x + y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor + 1,\\[3mu]
\lceil x \rceil + \lceil y \rceil -1 &\leq \lceil x + y \rceil \leq \lceil x \rceil + \lceil y \rceil.
\end{align}</math>