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]

[1112] **Graham L. Ronald, Knuth E. Donald, Patashnik Oren**: “Concrete Mathematics”, Addison-Wesley, 1994, 2nd Edition

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