BUILT-IN MATHEMATICA SYMBOL

Ceiling

Ceiling[x]
gives the smallest integer greater than or equal to x.

Ceiling

gives the smallest multiple of a greater than or equal to x.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

Ceiling[x] can be entered in StandardForm and InputForm as x, Esc lc Esc Esc rc Esc or
\.

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 (2)

Round up to the nearest integer:

Round up to the nearest multiple of 10:

Round up to the nearest integer:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

In[3]:=

Out[3]=

Round up to the nearest multiple of 10:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

Scope (6)

Use exact numeric quantities:

Ceiling threads element-wise over lists:

Round up to the nearest multiple of

:

Manipulate Ceiling symbolically:

Evaluate an integral:

TraditionalForm formatting:

Generalizations & Extensions (5)

Evaluate for complex numbers:

Ceiling can deal with real-valued intervals:

Infinite arguments give symbolic results:

Series expansion:

Use Esc lc Esc and Esc rc Esc to enter a short notation for Ceiling:

Applications (4)

Self-counting sequence:

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

Properties & Relations (6)

Negative numbers round up to the nearest integer above:

Convert Ceiling to Piecewise:

De-nest Ceiling functions:

Reduce equations containing Ceiling:

Ceiling function in the complex plane:

Possible Issues (1)

Ceiling does not automatically resolve the value:

Neat Examples (1)

Convergence of the Fourier series of Ceiling: