In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, the floor and ceiling function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

s map a real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 to the largest previous or the smallest following integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

, respectively. More precisely, floor(x) = is the largest integer not greater than x and ceiling(x) = is the smallest integer not less than x.
Carl Friedrich Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

 introduced the square bracket notation [x] for the floor function in his third proof of quadratic reciprocity

Quadratic reciprocity

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...

 (1808).
This remained the standard in mathematics until Kenneth E. Iverson

Kenneth E. Iverson

Kenneth Eugene Iverson was a Canadian computer scientist noted for the development of the APL programming language in 1962. He was honored with the Turing Award in 1979 for his contributions to mathematical notation and programming language theory...

 introduced the names "floor" and "ceiling" and the corresponding notations x and x in his 1962 book A Programming Language.