Notation
Carl Friedrich Gauss introduced the square bracket notation 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 and in his 1962 book A Programming Language. Both notations are now used in mathematics; this article follows Iverson.
The floor function is also called the greatest integer or entier (French for "integer") function, and its value at x is called the integral part or integer part of x; for negative values of x the latter terms are sometimes instead taken to be the value of the ceiling function, i.e., the value of x rounded to an integer towards 0. The language APL (programming language) uses ⌊x; other computer languages commonly use notations like entier(x) (Algol), INT(x) (BASIC), or floor(x)(C, C++, R, and Python). In mathematics, it can also be written with boldface or double brackets .
The ceiling function is usually denoted by ceil(x) or ceiling(x) in non-APL computer languages that have a notation for this function. The J Programming Language, a follow on to APL that is designed to use standard keyboard symbols, uses >. for ceiling and <. for floor. In mathematics, there is another notation with reversed boldface or double brackets x
The fractional part sawtooth function, denoted by for real x, is defined by the formula
For all x,
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