floorvalue

The floor value, commonly referred to as the floor function, assigns to a real number x the greatest integer that is less than or equal to x. It is denoted by floor(x) or ⌊x⌋. Formally, ⌊x⌋ is an integer satisfying ⌊x⌋ ≤ x < ⌊x⌋ + 1 for every real x. If x is already an integer, then ⌊x⌋ = x.

Examples include ⌊3.7⌋ = 3 and ⌊−2.5⌋ = −3. The fractional part of x is defined as {x} = x − ⌊x⌋, with 0 ≤ {x} < 1.

The floor function is related to the ceiling function, which gives the least integer greater than or equal to x, as ⌈x⌉. It also underpins many rounding-down operations in algorithms and connects to floor division in programming languages, where the result of a ÷ b is ⌊a/b⌋ under appropriate numeric types.

In floating-point arithmetic, precision issues can cause values very close to an integer to yield an incorrect floor. Such edge cases are typically addressed with careful numerical handling or exact arithmetic when necessary.

Applications include rounding down quantities, index calculations in arrays, lattice point problems, and various mathematical and computational procedures that require integer-valued outputs derived from real numbers.

See also: ceiling function, truncation, rounding, floor division.