floorFloor2001-10-18 23:22:572008-11-17 16:02:39Definition% this is the default PlanetMath preamble. as your knowledge
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The \emph{floor} of a real number is the greatest integer less than or equal to the number. The floor of $x$ is usually denoted by $\lfloor x\rfloor$.
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The real function \,$x \mapsto \lfloor{x}\rfloor$\, is monotonically nondecreasing and satisfies
$$x-1 < \lfloor{x}\rfloor \leqq x$$
for all $x$.\, The function is continuous everywhere except in the integer points \,$0,\,\pm1,\,\pm2,\,\ldots$\, where it is only continuous from the right.\, One has
$$\lfloor\lfloor{x}\rfloor\rfloor \;=\; \lfloor{x}\rfloor,$$
i.e. the function is idempotent.\\
\textbf{Some examples}:
\begin{itemize}
\item $\lfloor 6.2\rfloor=6$,
\item $\lfloor 0.4\rfloor=0$,
\item $\lfloor 7\rfloor=7$,
\item $\lfloor -5.1\rfloor=-6$,
\item $\lfloor \pi\rfloor=3$,
\item $\lfloor -4\rfloor=-4$.
\end{itemize}
Note that this function is not the integer part ($[x]$), since
$\lfloor -3.5\rfloor = -4$ and $[ -3.5]=-3$.
However, both functions agree for non-negative numbers.
The notation for floor and ceiling was introduced by Iverson in 1962\cite{Higham}.
In some texts however, the bracket notation is used to denote the floor function (although they actually work with integer part) so it is sometimes also called the \emph{bracket function}.
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\bibitem{Higham} N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.
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