The existence of integers exceeding real numbers motivates the following definitions:

• The greatest integer \(n\) less than or equal to \(x\) denoted by \(\lfloor x \rfloor\).
• The least integer \(n\) greater than or equal to \(x\) denoted by \(\lceil x \rceil\).

Definition: Floor and Ceiling Functions 

The floor $\lfloor \cdot \rfloor:\mathbb R\to\mathbb Z$ and the ceiling function $\lceil \cdot \rceil:\mathbb R\to\mathbb Z$ are functions from the set $\mathbb R$ of real numbers to the set $\mathbb Z$ of integers, defined by:

\[\begin{array}{rcl}
\lfloor x \rfloor=n&\Longleftrightarrow& n\le x < n+1,\\
\lfloor x \rfloor=n&\Longleftrightarrow& x-1 < n \le x,\\
\lceil x \rceil=n&\Longleftrightarrow& x\le n < x+1,\\
\lceil x \rceil=n&\Longleftrightarrow& n-1 < x \le n.\\
\end{array}\]

| | | | | created: 2014-02-21 20:06:03 | modified: 2019-03-17 05:25:44 | by: bookofproofs | references: [1112]

1.Proposition: Properties of Floors and Ceilings 

2.Proposition: Floor Function and Division with Quotient and Remainder 

3.Lemma: Sums of Floors 

4.Lemma: Reciprocity Law for Floor Functions