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

## 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]

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