# 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 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:

## Scope(29)

### Numerical Evaluation(6)

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 realvalued intervals:

### Specific Values(6)

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:

### Visualization(4)

Plot the Ceiling function:

Visualize the two-argument form:

Plot Ceiling in three dimensions:

Visualize Ceiling in the complex plane:

### 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:

Series expansion:

## Applications(4)

Selfcounting 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]:

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:

## Possible Issues(1)

Ceiling does not automatically resolve the value: ## Neat Examples(1)

Convergence of the Fourier series of Ceiling: