Floor
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
open all close allBasic 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 :
Scope (30)
Numerical Evaluation (7)
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 real‐valued intervals:
Compute average-case statistical intervals using Around :
Compute the elementwise values of an array using automatic threading:
Or compute the matrix Floor function using MatrixFunction :
Specific Values (6)
Visualization (4)
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:
TraditionalForm formatting:
Differentiation and Integration (4)
First derivative with respect to x:
First derivative with respect to a:
Definite integrals of Floor :
Series expansion:
Applications (4)
Find the millionth digit of 1/997 in base 10:
Expand multivalued functions, giving some assumptions about variables:
Then expand the same functions without making any assumptions about variables:
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)
Self‐counting sequence:
Convergence of the Fourier series of Floor :
See Also
Ceiling Round IntegerPart Chop Piecewise BinCounts Quantile FindDivisions
Function Repository: MinMaxRounded
History
Introduced in 1988 (1.0) | Updated in 1996 (3.0) ▪ 2007 (6.0)
Text
Wolfram Research (1988), Floor, Wolfram Language function, https://reference.wolfram.com/language/ref/Floor.html (updated 2007).
CMS
Wolfram Language. 1988. "Floor." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/Floor.html.
APA
Wolfram Language. (1988). Floor. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Floor.html
BibTeX
@misc{reference.wolfram_2025_floor, author="Wolfram Research", title="{Floor}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/Floor.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_floor, organization={Wolfram Research}, title={Floor}, year={2007}, url={https://reference.wolfram.com/language/ref/Floor.html}, note=[Accessed: 16-November-2025]}