gives the greatest integer less than or equal to x.
gives the greatest multiple of a less than or equal to x.
- 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.
Examplesopen allclose all
Basic Examples (4)
Round down to the nearest integer:
Round down to the nearest multiple of 10:
Plot over a subset of the reals:
Use lf and rf to enter a short notation for Floor:
Numerical Evaluation (6)
Specific Values (6)
Function Properties (9)
Floor is defined for all real and complex inputs:
Floor can produce infinitely large and small results:
Floor is not an analytic function:
It has both singularities and discontinuities:
Floor is nondecreasing:
Floor is not injective:
Floor is not surjective:
Floor is neither non-negative nor non-positive:
Floor is neither convex nor concave:
Differentiation and Integration (4)
First derivative with respect to x:
First derivative with respect to a:
Definite integrals of Floor:
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]:
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)
Neat Examples (3)
Convergence of the Fourier series of Floor:
Wolfram Research (1988), Floor, Wolfram Language function, https://reference.wolfram.com/language/ref/Floor.html (updated 2007).
Wolfram Language. 1988. "Floor." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Floor.html.
Wolfram Language. (1988). Floor. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Floor.html