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 known as
The ceiling function is also known as the least integer function or lowest integer function.
Also denoted as
Some sources use $\set x$ to denote the ceiling function, 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 $-1 \cdotp 1$
- $\ceiling {-1 \cdotp 1} = -1$
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
- 1988: Dominic Welsh: Codes and Cryptography ... (next): Notation
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): ceiling or least integer function
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ceiling function
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ceiling function
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): ceiling (least integer function)