Definition:Ceiling Function

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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

  • 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