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).

For example, for floor: ⌊2.4⌋ = 2, ⌊−2.4⌋ = −3, and for ceiling: ⌈2.4⌉ = 3, and ⌈−2.4⌉ = −2.

The floor of x is also called the integral part, integer part, greatest integer, or entier of x, and was historically denoted [x] (among other notations). However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers.

For an integer n, ⌊n⌋ = ⌈n⌉ = n.

Although floor(x + 1) and ceil(x) produce graphs that appear exactly alike, they are not the same when the value of x is an exact integer. For example, when x = 2.0001, ⌊2.0001 + 1⌋ = ⌈2.0001⌉ = 3. However, if x = 2, then ⌊2 + 1⌋ = 3, while ⌈2⌉ = 2.

The integral part or integer part of a number (partie entière in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula.

Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names "floor" and "ceiling" and the corresponding notations ⌊x⌋ and ⌈x⌉. (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.

In some sources, boldface or double brackets ⟦x⟧ are used for floor, and reversed brackets ⟧x⟦ or ]x[ for ceiling.