Ceiling 
Ceiling[x]
gives the smallest integer greater than or equal to x.
Ceiling[x,a]
gives the smallest multiple of a greater than or equal to x.
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- Ceiling[x] can be entered in StandardForm and InputForm as ⌈x⌉,
lc
rc
or
\[LeftCeiling]x \[RightCeiling]. - Ceiling[x] returns an integer when
is any numeric quantity, whether or not it is an explicit number.
- Ceiling[x] applies separately to real and imaginary parts of complex numbers.
- If a is not a positive real number, Ceiling[x,a] is defined by the formula Ceiling[x,a]a Ceiling[x/a]. »
- For exact numeric quantities, Ceiling internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.
- Ceiling automatically threads over lists.
Examples
open allclose allBasic Examples (3)
Scope (24)
Numerical Evaluation (6)
Specific Values (6)
Visualization (4)
Function Properties (4)
Ceiling is defined for all real and complex inputs:
Floor can produce infinitely large and small results:
TraditionalForm formatting:
Use lc
and
rc
to enter a short notation for Ceiling:
Differentiation and Integration (4)
First derivative with respect to x:
First derivative with respect to a:
Definite integrals of Ceiling:
Applications (4)
Self‐counting sequence:
Minimal number of elements in a box according to the pigeonhole principle:
Properties & Relations (10)
Negative numbers round up to the nearest integer above:
For a>0, Ceiling[x,a] gives the least multiple of a greater than or equal to x:
For other values of a, Ceiling[x,a] is defined by the following formula:
For a<0, the result is less than or equal to x:
Ceiling[x,-a] is equal to Floor[x,a]:
Denest Ceiling functions:
Reduce equations containing Ceiling:
Ceiling function in the complex plane:
Ceiling can be represented as a DifferenceRoot:
The generating function for Ceiling:
The exponential generating function for Ceiling:
Possible Issues (1)
Ceiling does not automatically resolve the value:

Neat Examples (1)
Convergence of the Fourier series of Ceiling:
Text
Wolfram Research (1988), Ceiling, Wolfram Language function, https://reference.wolfram.com/language/ref/Ceiling.html (updated 2007).
BibTeX
BibLaTeX
CMS
Wolfram Language. 1988. "Ceiling." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Ceiling.html.
APA
Wolfram Language. (1988). Ceiling. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Ceiling.html