Ceiling Function Calculator — ⌈x⌉
Apply the ceiling function ⌈x⌉ to any real number and see the smallest integer greater than or equal to x, alongside floor ⌊x⌋, round and truncate for comparison.
Smallest integer greater than or equal to x
- 1
Floor ⌊x⌋
⌊3.7⌋ = 3 - 2
Fractional part {x}
3.7 − 3 = 0.7 - 3
Ceiling ⌈x⌉
⌊3.7⌋ + 1 = 4The fractional part is non-zero, so the ceiling is one above the floor.
How does this calculator work?
The ceiling function ⌈x⌉ returns the smallest integer greater than or equal to x — always rounding up toward positive infinity. For positive numbers it rounds up (⌈3.2⌉ = 4), for negatives it rounds toward zero (⌈−3.2⌉ = −3). The floor ⌊x⌋ is the opposite: always rounds down.
Formula
How this is calculated
The ceiling function ⌈x⌉ maps any real number to the smallest integer that is greater than or equal to it — in other words it always rounds up (away from negative infinity). For an integer input, ⌈n⌉ = n exactly, since n is already an integer that satisfies the condition. For a positive decimal such as 3.2, the ceiling is 4; for a negative decimal such as −3.2, the ceiling is −3 (which is larger than −3.2 and is the nearest integer above it).
Comparison with related functions: the floor ⌊x⌋ always rounds toward negative infinity (down for positive, up for negative); truncation rounds toward zero (same as floor for positives, same as ceiling for negatives); standard rounding goes to the nearest integer and breaks ties toward positive infinity. The fractional part {x} = x − ⌊x⌋ is always in [0, 1) and measures how far x sits above the floor.
Common uses of the ceiling function include computing the number of pages needed to fit n items (⌈n/pageSize⌉), minimum integer container sizes, cryptographic key length bounds, and any situation where you need to fit a continuous quantity into discrete units without truncation.
Frequently asked questions
The ceiling rounds toward positive infinity regardless of sign. For −3.7, the ceiling is −3 (since −3 ≥ −3.7 and is the smallest such integer). The floor of −3.7 is −4.
"Ceiling" always goes to the nearest integer at or above the value. Standard rounding only goes up when the fractional part is ≥ 0.5; otherwise it goes down. So ⌈3.2⌉ = 4 and round(3.2) = 3, but ⌈3.7⌉ = 4 and round(3.7) = 4 — they agree only when the fractional part is ≥ 0.5.
The ceiling of an integer is the integer itself, because an integer is already the smallest integer ≥ itself. ⌈5⌉ = 5, ⌈−3⌉ = −3.
TG we-Calculate Editorial Team. (2026). Ceiling Function Calculator — ⌈x⌉ [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/ceiling-function-calculator
TG we-Calculate Editorial Team. "Ceiling Function Calculator — ⌈x⌉." TG we-Calculate. 2026. https://we-calculate.com/calculator/ceiling-function-calculator.
TG we-Calculate Editorial Team, "Ceiling Function Calculator — ⌈x⌉," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/ceiling-function-calculator
@misc{wecalculate_ceiling_function_calculator, title = {Ceiling Function Calculator — ⌈x⌉}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/ceiling-function-calculator}}, year = {2026}, note = {TG we-Calculate} }
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