Floor and Ceiling Functions

The floor of x is defined as ⌊x⌋ = the unique integer n such that n ≤ x < n + 1. Equivalently, ⌊x⌋ is the greatest integer not exceeding x. The ceiling of x is defined as ⌈x⌉ = the unique integer n such that n − 1 < x ≤ n. Equivalently, ⌈x⌉ is the least integer not less than x. For example: ⌊3.7⌋ = 3, ⌊−1.2⌋ = −2, ⌊5⌋ = 5, ⌈3.1⌉ = 4, ⌈−1.8⌉ = −1, ⌈7⌉ = 7. Notice that for negative non-integers, the floor goes further from zero: ⌊−1.2⌋ = −2, not −1.

The fractional part of x is defined as {x} = x − ⌊x⌋. It always satisfies 0 ≤ {x} < 1. For example, {3.7} = 0.7 and {−1.2} = 0.8 (since ⌊−1.2⌋ = −2 and −1.2 − (−2) = 0.8). This means every real number can be written as x = ⌊x⌋ + {x}, a decomposition into integer and fractional parts. The fractional part is zero if and only if x is already an integer.

The floor and ceiling functions are related by a simple identity: ⌈x⌉ = −⌊−x⌋. This means you can always derive one from the other. Another key relationship is ⌈x⌉ − ⌊x⌋ = 0 if x is an integer, and ⌈x⌉ − ⌊x⌋ = 1 if x is not an integer. Combining: ⌈x⌉ − ⌊x⌋ = 1 − [x ∈ ℤ], where the bracket is 1 when x is an integer and 0 otherwise. Also note ⌊x⌋ ≤ x ≤ ⌈x⌉ always holds.

Several key properties govern how floor and ceiling interact with arithmetic. (1) Translation by an integer: ⌊x + n⌋ = ⌊x⌋ + n for any integer n. (2) Inequality characterization: ⌊x⌋ = n ⟺ n ≤ x < n + 1 ⟺ x − 1 < n ≤ x. (3) Floor of a sum: in general ⌊x + y⌋ ≥ ⌊x⌋ + ⌊y⌋, with equality when {x} + {y} < 1. (4) Hermite's identity: ⌊x⌋ + ⌊x + 1/n⌋ + ⌊x + 2/n⌋ + ⋯ + ⌊x + (n−1)/n⌋ = ⌊nx⌋ for any positive integer n. (5) The number of multiples of k in the range [1, n] is ⌊n/k⌋, a fact essential in number theory.

To solve ⌊x⌋ = k for integer k, the solution set is the interval [k, k+1). To solve ⌊2x + 1⌋ = 5, set 5 ≤ 2x + 1 < 6, giving 4 ≤ 2x < 5, so 2 ≤ x < 5/2. For inequalities: ⌊x⌋ ≥ 3 means x ≥ 3; ⌊x⌋ ≤ 3 means x < 4. To solve ceiling equations: ⌈x⌉ = k has solution set (k−1, k]. Strategy: always replace the floor or ceiling expression with its defining inequality, then solve for x in the resulting compound inequality.

• \lfloor x \rfloor — Floor of x (dimensionless (integer)): The greatest integer less than or equal to x; always satisfies ⌊x⌋ ≤ x
• \lceil x \rceil — Ceiling of x (dimensionless (integer)): The least integer greater than or equal to x; always satisfies ⌈x⌉ ≥ x
• x — Input (dimensionless (real number)): Any real number; the domain of both floor and ceiling functions is all of ℝ

Problem. Solve the equation ⌊3x − 1⌋ = 4, and then find the number of integers in the solution set of ⌈x/2⌉ ≤ 5.

Answer. Part 1: x ∈ [5/3, 2). Part 2: x ≤ 10, which contains infinitely many integers (or 10 positive integers if restricted to {1, 2, ..., 10}).

• Absolute Value Function — Like floor and ceiling, the absolute value function is a special function defined by cases; both require piecewise reasoning and appear frequently in equations and inequalities.
• Piecewise Functions — The floor and ceiling functions are formally piecewise functions — their rules change depending on which interval contains the input — making piecewise function theory directly applicable.
• Integer Division and Modular Arithmetic — The floor function is the mathematical foundation of integer division: ⌊a/b⌋ gives the quotient, and the remainder relates to the fractional part, directly connecting floor functions to modular arithmetic.
• Domain and Range of Functions — Both floor and ceiling functions have domain ℝ and range ℤ, providing a concrete example of how a function can map an uncountable set to a countable one.