Floorit

Floorit, in mathematical usage, is commonly identified with the floor function. It maps a real number x to the greatest integer less than or equal to x. The standard notation is ⌊x⌋, and the function is also known as the greatest integer function. Its domain is all real numbers and its range is the set of all integers.

Definition and key properties: For any real x, ⌊x⌋ ≤ x < ⌊x⌋ + 1. The fractional part {x} = x − ⌊x⌋ lies in [0, 1). The floor function is non-decreasing: if x ≤ y then ⌊x⌋ ≤ ⌊y⌋. It is idempotent: ⌊⌊x⌋⌋ = ⌊x⌋. It satisfies ⌊x + n⌋ = ⌊x⌋ + n for every integer n. A useful inequality is ⌊x⌋ + ⌊y⌋ ≤ ⌊x + y⌋ ≤ ⌊x⌋ + ⌊y⌋ + 1.

Examples: ⌊3.7⌋ = 3 and ⌊−2.4⌋ = −3. The function is a step function: it is constant on each interval [k, k+1) with value k, and it has unit jumps at each integer.

Applications and computational aspects: In programming languages, floor is a standard function, e.g., floor(3.7) = 3 and floor(−1.2) = −2. It is distinct from truncation toward zero for negative inputs. The floor function is used in indexing, discretization, rounding down in algorithms, and in conversions between continuous and discrete representations.

Generalizations and related concepts: In multiple dimensions, floor is applied coordinate-wise. Related functions include the ceiling ⌈x⌉ and various rounding modes. The preimage of an integer under the floor function is the interval [n, n+1).