The floor function, denoted as ⌊x⌋ or floor(x), returns the greatest integer less than or equal to x. It effectively rounds a real number down to the nearest integer.
For example:
⌊3.7⌋ = 3,
⌊−2.4⌋ = −3,
⌊5⌋ = 5 (since 5 is already an integer).
The function is widely used in mathematics, computer science, and programming. In Python, it is implemented via math.floor(x). In LaTeX, it is written using \lfloor and \rfloor. The floor function is also known as the greatest integer function or integer part.
It is distinct from the ceiling function (⌈x⌉), which rounds up to the smallest integer greater than or equal to x.
functions of a real returning respectively the largest smaller and the smallest larger integer
Wikipedia
en.wikipedia.org › wiki › Floor_and_ceiling_functions
Floor and ceiling functions - Wikipedia
February 5, 2026 - In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). ...
Analytics Vidhya
analyticsvidhya.com › home › understanding floor and ceiling functions in python
Floor and Ceiling Functions in Python | Applications and Behaviour
June 20, 2023 - The floor and ceiling functions are mathematically used to round numbers to the nearest integer. This comprehensive guide will explore the floor and ceiling functions in Python, understand their formulas, and discover their various use cases and applications.
Can someone explain to me what the floor and cealing functions are actually doing numerically?
“Mathematical operations” (i.e. arithmetic) are only a small part of math. A relation is simply a mapping of elements in one set to elements in another set, and that’s what the floor and ceiling functions do. More on reddit.com
30
4
June 26, 2022Mathematical floor function - Lounge/General chat - od|forum
It sounds so simple that I'm almost ashamed of asking this. Some folks here from the programming area are doubting some results of a floor function in C#, so they asked what was the math behind the floor and ceiling functions. Does anyone know a mathematical way to convert an float to integer, fl... More on forums.odforce.net
How to represent the floor function using mathematical notation? - Mathematics Stack Exchange
I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater tha... More on math.stackexchange.com
How to integrate floor functions?
Since the floor jumps at every integer value, you need to break the interval (0,5/2) into pieces: (0,1), (1,2), and (2,5/2). Remember, the floor function is some constant on each of those intervals. More on reddit.com
5
5
December 28, 2012Videos
08:19
The Floor Function (Greatest Integer Function) - YouTube
15:16
They Named a Function after the Floor - YouTube
10:42
Floor Function or Greatest Integer Function - YouTube
12:39
Integral of floor function - YouTube
07:48
Step Function: Floor Function - YouTube
Wolfram MathWorld
mathworld.wolfram.com › FloorFunction.html
Floor Function -- from Wolfram MathWorld
September 27, 2013 - The floor function , also called the greatest integer function or integer value (Spanier and Oldham 1987), gives the largest integer less than or equal to . The name and symbol for the floor function were coined by K. E. Iverson (Graham et al.
Reddit
reddit.com › r/learnmath › can someone explain to me what the floor and cealing functions are actually doing numerically?
r/learnmath on Reddit: Can someone explain to me what the floor and cealing functions are actually doing numerically?
June 26, 2022 -
When I truncate a number what my brain actually does is ignoring the fractional part of said number. But its not doing any real math.
I understand I can express a truncate function with conditional floor and cealing functions... but thats is not what I need.
I need someone to teach me how to arrive from a number to its integer using only mathematical operations and not logical functions.
I need to know...
Plz help me someone...
Top answer 1 of 8
6
“Mathematical operations” (i.e. arithmetic) are only a small part of math. A relation is simply a mapping of elements in one set to elements in another set, and that’s what the floor and ceiling functions do.
2 of 8
5
floor and ceiling are mathematical operations. they return the highest (lowest) integer less than (greater than) their argument. I'm not sure what the problem is
MedCalc
medcalc.org › en › manual › floor-function.php
FLOOR function calculator and graph - MedCalc Manual
September 9, 2025 - FLOOR(x) rounds the number x down. The argument x can be a real number or a matrix. When it is a matrix, the function returns a matrix with the same dimensions and with the FLOOR function applied to all elements.
Davidwills
davidwills.us › math103 › trigonometry › floor.html
floor function
Math symbol chosen for obvious reason. Typically named "floor()" in text software. AKA greatest integer function.
Arduino
docs.arduino.cc › language-reference
Language Reference
Arduino programming language can be divided in three main parts: functions, values (variables and constants), and structure.
YouTube
youtube.com › watch
The Floor Function - YouTube
Short video where we talk about the floor function and give examples.My Courses: https://www.freemathvids.com/Bid on math books: https://www.ebay.com/str/the...
Published August 14, 2024
od|forum
forums.odforce.net › general › lounge/general chat
Mathematical floor function - Lounge/General chat - od|forum
November 27, 2011 - It sounds so simple that I'm almost ashamed of asking this. Some folks here from the programming area are doubting some results of a floor function in C#, so they asked what was the math behind the floor and ceiling functions. Does anyone know a mathematical way to convert an float to integer, fl...
Underground Mathematics
undergroundmathematics.org › glossary › floor-function › download › floor-function.pdf pdf
Floor function underground mathematics
July 7, 2016 - The floor of a real number 𝑥is the largest integer which is not larger than 𝑥. The notation ⌊𝑥⌋is used for this function.
Math is Fun
mathsisfun.com › sets › function-floor-ceiling.html
Floor and Ceiling Functions
The floor and ceiling functions give us the nearest integer up or down. The Floor of 2.31 is 2 The Ceiling of 2.31 is 3.
Art of Problem Solving
artofproblemsolving.com › wiki › index.php › Floor_function
Floor function - AoPS Wiki
The greatest integer function, also known as the floor function, gives the greatest integer less than or equal to its argument. The floor of is usually denoted by or .
Python
docs.python.org › 3 › library › math.html
math — Mathematical functions
2 weeks ago - Added in version 3.8. ... Return ... an Integral value. ... Return the absolute value of x. ... Return the floor of x, the largest integer less than or equal to x....
Byu
byu-dept-clsresults-prd.amazon.byu.edu › math-floor-ceil
Math Floor and Ceil Functions Explained
April 1, 2025 - We cannot provide a description for this page right now
Brilliant
brilliant.org › wiki › integer-function-floor-ceiling-sawtooth
Integer Function - Floor, Ceiling, Sawtooth | Brilliant Math & Science Wiki
the floor function returns the largest integer less than or equal to \(x\), denoted as \(\lfloor x \rfloor ;\)
Top answer 1 of 12
85
For a real number $x$, $$\lfloor x\rfloor=\max\{n\in\mathbb{Z}\mid n\leq x\}.$$ I'd like to add though, that "$\lfloor x\rfloor$" is mathematical notation, just as much as the right side of the above equation is; the right side might use more "basic" constructions, but you can then ask about $$\max,\qquad\in,\qquad \mathbb{Z},\qquad {}\mathbin{\mid}{},\qquad \leq$$ and so on. At some point you just have to start writing notation and explaining it in words and hope your readers understand. So I disagree with your phrasing of the question.
2 of 12
31
$\lfloor x \rfloor = x - \arctan(\tan(\pi x))/\pi$ ?...
GeeksforGeeks
geeksforgeeks.org › mathematics › floor-function
Floor Function - GeeksforGeeks
July 23, 2025 - The floor function is a mathematical function that returns the greatest integer less than or equal to a given number. In other words, it rounds a real number down to the largest integer less than or equal to the given number.
F-Chart Software
fchartsoftware.com › ees › eeshelp › floor.htm
FLOOR
FLOOR(Arg) will return a value equal to the largest integer value that is less than or equal to the value of Arg. Note that FLOOR differs from TRUNC when Arg is less than zero. For example TRUNC(-2.5) = -2 whereas FLOOR(-2.5) = -3. See also: CEIL · TRUNC · ROUND · Mathematical Functions
ProofWiki
proofwiki.org › wiki › Category:Floor_Function
Category:Floor Function - ProofWiki
January 8, 2018 - Conditions for Floor of Log base b of x to equal Floor of Log base b of Floor of x (3 P)
Cleveland Clinic
my.clevelandclinic.org › health › articles › 14611-kegel-exercises
Kegel Exercises: Benefits, How To & Results
December 8, 2025 - Your pelvic floor muscles support organs in your pelvis, like your bladder, bowel and vagina. Your pelvic floor muscles hold your organs in place while also assisting with bodily functions like peeing, pooping and sex.