gives the smallest integer greater than or equal to x.
gives the smallest multiple of a greater than or equal to x.
- 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.
Examplesopen allclose all
Basic Examples (4)
Round up to the nearest integer:
Round up to the nearest multiple of 10:
Plot the function over a subset of the reals:
Use lc and rc to enter a short notation for Ceiling:
Numerical Evaluation (6)
Specific Values (6)
Function Properties (9)
Ceiling is defined for all real and complex inputs:
Ceiling can produce infinitely large and small results:
Ceiling is not an analytic function:
It has both singularities and discontinuities:
Ceiling is nondecreasing:
Ceiling is not injective:
Ceiling is not surjective:
Ceiling is neither non-negative nor non-positive:
Ceiling is neither convex nor concave:
Differentiation and Integration (4)
First derivative with respect to x:
First derivative with respect to a:
Definite integrals of Ceiling:
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:
Wolfram Research (1988), Ceiling, Wolfram Language function, https://reference.wolfram.com/language/ref/Ceiling.html (updated 2007).
Wolfram Language. 1988. "Ceiling." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Ceiling.html.
Wolfram Language. (1988). Ceiling. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Ceiling.html