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Ceiling—Wolfram Language Documentation

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• See Also
  • Floor
  • IntegerPart
  • FractionalPart
  • Round
  • Chop
  • BinCounts
  • Quantile
  • FindDivisions
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  • Implementation notes: Numerical and Related Functions

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.

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:

Evaluate numerically:

Complex number inputs:

Single-argument Ceiling always returns an exact result:

The two-argument form tracks the precision of the second argument:

Evaluate efficiently at high precision:

Ceiling can deal with real‐valued intervals:

Compute the elementwise values of an array using automatic threading:

Or compute the matrix Ceiling function using MatrixFunction:

Compute average-case statistical intervals using Around:

Values of Ceiling at fixed points:

Value at zero:

Value at Infinity:

Evaluate symbolically:

Manipulate Ceiling symbolically:

Find a value of x for which the Ceiling[x]=2:

Plot the Ceiling function:

Visualize the two-argument form:

Plot Ceiling in three dimensions:

Visualize Ceiling in the complex plane:

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:

TraditionalForm formatting:

First derivative with respect to x:

First derivative with respect to a:

Definite integrals of Ceiling:

Series expansion:

Self‐counting sequence:

Minimal number of elements in a box according to the pigeonhole principle:

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]:

Convert Ceiling to Piecewise:

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:

Ceiling does not automatically resolve the value:

Convergence of the Fourier series of Ceiling:

Introduced in 1988 (1.0) | Updated in 1996 (3.0) ▪ 2007 (6.0)

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