Floor Function Calculator — ⌊x⌋, Ceiling and Fractional Part
Enter any real number to find its floor (⌊x⌋ — greatest integer ≤ x), ceiling (⌈x⌉ — smallest integer ≥ x), fractional part, nearest-integer rounding and truncation all at once.
Greatest integer less than or equal to x
- 1
Floor ⌊x⌋ — greatest integer ≤ x
⌊3.7⌋ = 3Rounds toward −∞: ⌊3.7⌋ = 3, ⌊−2.3⌋ = −3. - 2
Fractional part {x} = x − ⌊x⌋
3.7 − (3) = 0.7
How does this calculator work?
Floor ⌊x⌋ = greatest integer ≤ x. ⌊3.7⌋ = 3; ⌊−2.3⌋ = −3. Ceiling ⌈x⌉ = smallest integer ≥ x; ⌈3.7⌉ = 4. Fractional part {x} = x − ⌊x⌋ ∈ [0, 1). For negative numbers floor differs from truncation: floor rounds toward −∞, truncation rounds toward 0.
Formula
How this is calculated
The floor function ⌊x⌋ (also called the greatest-integer function [x]) returns the largest integer that does not exceed x. For positive numbers it strips the decimal part: ⌊3.7⌋ = 3. For negative numbers it moves further from zero: ⌊−2.3⌋ = −3, not −2, because −3 ≤ −2.3 but −2 > −2.3. The fractional part {x} = x − ⌊x⌋ is always in [0, 1) regardless of the sign of x.
The ceiling function ⌈x⌉ is the smallest integer that is not less than x — the mirror image of floor. For any non-integer x, ⌈x⌉ = ⌊x⌋ + 1. For integers, ⌊x⌋ = ⌈x⌉ = x. Truncation (also called int() in many languages) rounds toward zero: trunc(3.7) = 3 and trunc(−2.3) = −2. Floor and truncation agree for non-negative numbers but differ for negatives — floor(−2.3) = −3 while trunc(−2.3) = −2.
These functions appear throughout programming (array indexing, modular arithmetic, scheduling), number theory (Gauss brackets, Beatty sequences, floor-division identities) and everyday tasks like rounding prices, calculating full boxes from a quantity, or finding the week number of a date.
Frequently asked questions
For positive x they are the same: both return the integer part. For negative x they differ: floor(−2.3) = −3 (round away from zero toward −∞) while trunc(−2.3) = −2 (round toward zero). In mathematical contexts floor is usually preferred; in C/Java/JavaScript int() and Math.trunc() both truncate.
The fractional part is defined as {x} = x − ⌊x⌋, which is always in [0, 1). For negative numbers this can seem surprising: {−2.3} = −2.3 − (−3) = 0.7, not 0.3. This ensures {x} is always non-negative and {x} + ⌊x⌋ = x always holds.
Floor appears in integer division (Python's // operator), array index computation (n = ⌊log₂ k⌋ to choose a row in a segment tree), page-count calculations (⌈items / pageSize⌉), hash-table sizing, and anywhere you need the "how many full groups fit?" answer. Math.floor() in JavaScript, floor() in C, and int() in Python all implement it.
Also known as
TG we-Calculate Editorial Team. (2026). Floor Function Calculator — ⌊x⌋, Ceiling and Fractional Part [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/floor-function-calculator
TG we-Calculate Editorial Team. "Floor Function Calculator — ⌊x⌋, Ceiling and Fractional Part." TG we-Calculate. 2026. https://we-calculate.com/calculator/floor-function-calculator.
TG we-Calculate Editorial Team, "Floor Function Calculator — ⌊x⌋, Ceiling and Fractional Part," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/floor-function-calculator
@misc{wecalculate_floor_function_calculator, title = {Floor Function Calculator — ⌊x⌋, Ceiling and Fractional Part}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/floor-function-calculator}}, year = {2026}, note = {TG we-Calculate} }
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