Built-in Mathematica Symbol

Floor (...)

Floor[x]
gives the greatest integer less than or equal to x.

Floor[x, a]
gives the greatest multiple of a less than or equal to x.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

Floor[x] can be entered in StandardForm and InputForm as x, Esc lf Esc Esc rf Esc or
\[LeftFloor]x\[RightFloor].

Floor[x] returns an integer when is any numeric quantity, whether or not it is an explicit number. »

For exact numeric quantities, Floor internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.

Floor applies separately to real and imaginary parts of complex numbers.

Floor automatically threads over lists.

EXAMPLES

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Basic Examples (2)

Round down to the nearest integer:

Round down to the nearest multiple of 10:

Round down to the nearest integer:

Round down to the nearest multiple of 10:

In[1]:=

Out[1]=

Scope (5)

Use exact numeric quantities:

Floor threads element-wise over lists:

Round down to the nearest multiple of

:

Manipulate Floor symbolically:

Evaluate an integral:

Generalizations & Extensions (5)

Complex numbers:

Floor can deal with real-valued intervals:

Infinite arguments give symbolic results:

Series expansion:

Use Esc lf Esc and Esc rf Esc to enter a short notation for Floor:

Applications (3)

Find the millionth digit of 1/997 in base 10:

Properties & Relations (8)

Convert Floor to Piecewise:

De-nest Floor functions:

Get Floor from PowerExpand:

Reduce equations containing Floor:

Floor function in the complex plane:

Sum expressions involving Floor:

Possible Issues (2)

Floor does not automatically resolve the value:

Guard digits can influence the result of Floor:

Neat Examples (3)

Self-counting sequence:

Convergence of the Fourier series of Floor: