mathfloorx

mathfloorx refers to the floor function applied to a real number x. In mathematics, the floor of x is the greatest integer that is less than or equal to x. This value is commonly denoted floor(x) and, in programming contexts, accessed as math.floor(x) or floor(x).

Formally, floor(x) is defined for all real numbers x as floor(x) = max{ n ∈ Z : n ≤ x }. The function has a range of the integers and a domain of the real numbers. It is a piecewise constant function that exhibits jumps at each integer, remaining constant on intervals between consecutive integers.

Key properties include: floor(x) ≤ x, and x < floor(x) + 1 for all x. For integers n, floor(n) = n. There is also the identity floor(x) = -ceil(-x), where ceil is the ceiling function. The floor function is monotone: if x ≤ y, then floor(x) ≤ floor(y).

Examples illustrate its behavior: floor(3.7) = 3, floor(-2.4) = -3, floor(5) = 5, floor(-0.1) = -1. In computational contexts, some languages implement floor directly, while others use truncation toward zero and require adjustments for negative inputs to obtain the floor value.

Common applications include rounding down numbers in monetary calculations, discretizing continuous data for binning, and aiding algorithms in number theory and geometry. The concept is closely related to the ceiling function, truncation, and the greatest integer function, often denoted by ⌊x⌋ in mathematical notation.