# Definition:Ceiling Function

## Contents

## 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 $x$:

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

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:

- $\set {m \in \Z: m \ge x}$

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.

## 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

- Equivalence of Definitions of Ceiling Function
- Definition:Floor Function
- Definition:Fractional Part
- Properties of Ceiling Function

- 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`

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**ceiling (least integer function)**