# 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.
• 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 applies separately to real and imaginary parts of complex numbers.
• Ceiling automatically threads over lists.

# Examples

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## Basic Examples(3)

Round up to the nearest integer:

Round up to the nearest multiple of 10:

Plot the function over a subset of the reals:

## Scope(24)

### 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(4)

Ceiling is defined for all real and complex inputs:

Floor can produce infinitely large and small results:

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:

Series expansion:

## Applications(4)

Selfcounting sequence:

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

## Properties & Relations(9)

Negative numbers round up to the nearest integer above:

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:

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