Floor Function Calculator

The floor function ⌊x⌋ maps a real number x to the greatest integer less than or equal to x. It is also called the greatest integer function. For example, ⌊3.7⌋ = 3, ⌊-2.3⌋ = -3, ⌊5⌋ = 5, and ⌊π⌋ = 3.

For negative numbers, floor rounds toward negative infinity. ⌊-2.3⌋ = -3 (not -2), because -3 is the greatest integer ≤ -2.3. This is different from truncation, which would give -2. The floor function always gives the more negative result.

The floor function ⌊x⌋ rounds DOWN to the nearest integer (toward -∞). The ceiling function ⌈x⌉ rounds UP to the nearest integer (toward +∞). For example: ⌊3.7⌋ = 3, ⌈3.7⌉ = 4. ⌊-2.3⌋ = -3, ⌈-2.3⌉ = -2. For integers, both equal the number itself.

The fractional part {x} = x - ⌊x⌋, always satisfying 0 ≤ {x} < 1. For positive x, this is the part after the decimal point. For negative x, it is measured from the floor: {-2.3} = -2.3 - (-3) = 0.7.

Not exactly. In most languages, casting to int truncates toward zero (e.g., (int)-2.3 = -2). The floor function rounds toward negative infinity (⌊-2.3⌋ = -3). Use Math.floor() in JavaScript, math.floor() in Python, or std::floor() in C++ for true floor behavior.

For any integer n, ⌊n⌋ = n. The floor of an integer is the integer itself. For example, ⌊5⌋ = 5, ⌊-3⌋ = -3, ⌊0⌋ = 0. This is because an integer is already the greatest integer ≤ itself.

Key properties: (1) ⌊x⌋ ≤ x < ⌊x⌋ + 1, (2) {x} = x - ⌊x⌋ satisfies 0 ≤ {x} < 1, (3) ⌊x + n⌋ = ⌊x⌋ + n for any integer n, (4) ⌊x⌋ + ⌊y⌋ ≤ ⌊x + y⌋ ≤ ⌊x⌋ + ⌊y⌋ + 1, (5) ⌊xy⌋ ≥ ⌊x⌋⌊y⌋ for non-negative x, y.

The floor function graph is a step function. For each integer n, the value is n on the interval [n, n+1). This creates horizontal segments at each integer height, with closed dots at the left endpoint and open dots at the right endpoint. The graph looks like a staircase going up from left to right.

Yes. Applications include: (1) Converting decimal measurements to whole units (e.g., 3.7 feet = 3 feet), (2) Pagination — computing page numbers from item indices, (3) Pixel calculations in graphics, (4) Time — converting decimal hours to hours and minutes, (5) Currency — converting fractional dollars to whole dollars.

The ceiling function is the complement of floor: ⌈x⌉ = -⌊-x⌋. Also: ⌊x⌋ + ⌈-x⌉ = 0 for non-integer x, and ⌊x⌋ + ⌈x⌉ = 2x when x is an integer. The ceiling rounds up while floor rounds down.

The Floor Function Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Floor Function Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.