In mathematics and computer science, 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, ⌊2.4⌋ = 2, ⌊−2.4⌋ = −3, ⌈2.4⌉ = 3, and ⌈−2.4⌉ = −2.

Historically, the floor of x has been–and still is–called the integral part or integer part of x, often denoted [x] (as well as a variety of other notations). Some authors may define the integral part [x] as ⌊x⌋ if x is nonnegative, and ⌈x⌉ otherwise: for example, [2.4] = 2 and [−2.4] = −3. The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part.

For n an integer, ⌊n⌋ = ⌈n⌉ = [n] = n.