Floor function

In mathematics and computer science, the floor and ceiling functions map a real number to the greatest preceding or the least succeeding integer, respectively. More precisely, floor(x) = ${\displaystyle \lfloor x\rfloor }$ is the greatest integer less than or equal to x and ceiling(x) = ${\displaystyle \lceil x\rceil }$ is the least integer greater than or equal to x.

Carl Friedrich Gauss introduced the square bracket notation ${\displaystyle [x]}$ for the floor function in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth E. Iverson introduced the names "floor" and "ceiling" and the corresponding notations ${\displaystyle \lfloor x\rfloor }$ and ${\displaystyle \lceil x\rceil }$ in his 1962 book A Programming Language. Both notations are now used in mathematics; this article follows Iverson.

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