# 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.

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• Floor[x] can be entered in StandardForm and InputForm as x, lf   rf , or \[LeftFloor]x \[RightFloor].
• Floor[x] returns an integer when is any numeric quantity, whether or not it is an explicit number. »
• Floor[x] applies separately to real and imaginary parts of complex numbers.
• If a is not a positive real number, Floor[x,a] is defined by the formula Floor[x,a]a Floor[x/a]. »
• 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 automatically threads over lists.

# Examples

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## Basic Examples(3)

Round down to the nearest integer:

Round down to the nearest multiple of 10:

Plot over a subset of the reals:

## Scope(24)

### Numerical Evaluation(6)

Evaluate numerically:

Complex number inputs:

Single-argument Floor always returns an exact result:

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

Evaluate efficiently at high precision:

Floor can deal with realvalued intervals:

### Specific Values(6)

Values of Floor at fixed points:

Value at zero:

Value at Infinity:

Evaluate symbolically:

Manipulate Floor symbolically:

Find a value of x for which the :

### Visualization(4)

Plot the Floor function:

Visualize the two-argument form:

Plot Floor in three dimensions:

Visualize Floor in the complex plane:

### Function Properties(4)

Floor is defined for all real and complex inputs:

Floor can produce infinitely large and small results:

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

### Differentiation and Integration(4)

First derivative with respect to x:

First derivative with respect to a:

Definite integrals of Floor:

Series expansion:

## Applications(2)

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

## Properties & Relations(12)

Negative numbers round down to the nearest integer below:

For a>0, Floor[x,a] gives the greatest multiple of a less than or equal to x:

For other values of a, Floor[x,a] is defined by the following formula:

For a<0, the result is greater than or equal to x:

Floor[x,-a] is equal to Ceiling[x,a]:

Convert Floor to Piecewise:

Denest Floor functions:

Get Floor from PowerExpand:

Reduce equations containing Floor:

Floor function in the complex plane:

Sum expressions involving Floor:

Floor can be represented as a DifferenceRoot:

The generating function for Floor:

The exponential generating function for Floor:

## Possible Issues(2)

Floor does not automatically resolve the value: Guard digits can influence the result of Floor:

## Neat Examples(3)

Selfcounting sequence:

Convergence of the Fourier series of Floor: