# Definition:Ceiling Function

## Definition

Let $x$ be a real number.

Informally, the ceiling function of $x$ is the smallest integer greater than or equal to $x$.

### Definition 1

The ceiling function of $x$ is defined as the infimum of the set of integers no smaller than $x$:

$\ceiling x := \inf \set {m \in \Z: x \le m}$

where $\le$ is the usual ordering on the real numbers.

### Definition 2

The ceiling function of $x$, denoted $\ceiling x$, is defined as the smallest element of the set of integers:

$\set {m \in \Z: x \le m}$

where $\le$ is the usual ordering on the real numbers.

### Definition 3

The ceiling function of $x$ is the unique integer $\ceiling x$ such that:

$\ceiling x - 1 < x \le \ceiling x$

## Notation

The notation $\ceiling x$ for the ceiling function is a relatively recent development.

Compare the notation $\floor x$ for the corresponding floor function.

## Also denoted as

Some sources use $\set x$, but this has too many other interpretations for it to be acceptable on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Examples

### Ceiling of $\sqrt 2$

$\ceiling {\sqrt 2} = 2$

### Ceiling of $-1.1$

$\ceiling {-1 \cdotp 1} = -1$

### Ceiling of $- \dfrac 1 2$

$\ceiling {- \dfrac 1 2} = 0$

## Also see

• Results about the ceiling function can be found here.

## Historical Note

The notation $\ceiling x$ for the ceiling function was introduced in the $1960$s by Kenneth Eugene Iverson and made popular by Donald Ervin Knuth.

## Technical Note

The $\LaTeX$ code for $\ceiling {x}$ is \ceiling {x} .

When the argument is a single character, it is usual to omit the braces:

\ceiling x