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Category Math & Algebra

Enter a real number Supports decimals, negatives, and scientific notation (e.g. 2.5e3)

Enter comma-separated numbers Separate values with commas. Max 100 values.

Comparison

Number Line

Input `x`	⌈`x`⌉	⌊`x`⌋	Round	Trunc

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About

The ceiling function ⌈x⌉ maps a real number x to the smallest integer greater than or equal to x. It is the counterpart of the floor function and appears in discrete mathematics, computer science (memory allocation, pagination), and combinatorics. Misapplying ceiling versus floor in pagination logic produces off-by-one errors that cascade into missing records or blank pages. This calculator handles negative inputs correctly: ⌈−2.3⌉ = −2, not −3. A common mistake.

Batch mode accepts comma-separated values for bulk computation. The comparison panel shows ceiling alongside floor, round, and truncate so you can verify which rounding strategy your application requires. The tool assumes standard real number inputs. It does not extend to complex numbers.

Formulas

The ceiling function is defined as the mapping from the set of real numbers R to the set of integers Z:

⌈x⌉ = min { n ∈ Z | n ≥ x }

Equivalently, using the floor function:

⌈x⌉ = ⌊x⌋ +

{

1 if x ∉ Z0 if x ∈ Z

The relationship between ceiling and floor for any real x:

⌈x⌉ = −⌊−x⌋

For the fractional part {x}:

{x} = x − ⌊x⌋

Where x = input real number, n = resulting integer, Z = set of all integers, ⌊x⌋ = floor of x, {x} = fractional part of x.

Reference Data

Input x	⌈x⌉ Ceiling	⌊x⌋ Floor	Round	Truncate	Fractional Part
2.1	3	2	2	2	0.1
2.5	3	2	3	2	0.5
2.9	3	2	3	2	0.9
3.0	3	3	3	3	0.0
−1.1	−1	−2	−1	−1	0.9
−1.5	−1	−2	−1	−1	0.5
−1.9	−1	−2	−2	−1	0.1
−2.0	−2	−2	−2	−2	0.0
0.0	0	0	0	0	0.0
0.001	1	0	0	0	0.001
−0.001	0	−1	0	0	0.999
99.99	100	99	100	99	0.99
−99.99	−99	−100	−100	−99	0.01
0.5	1	0	1	0	0.5
−0.5	0	−1	0	0	0.5
1000000.1	1000001	1000000	1000000	1000000	0.1
7	7	7	7	7	0.0
−7	−7	−7	−7	−7	0.0

Frequently Asked Questions

For negative non-integers, the ceiling rounds toward zero, not away from it. For example, ⌈−3.7⌉ = −3, because −3 is the smallest integer that is still ≥ −3.7. This is the opposite direction from the floor function, where ⌊−3.7⌋ = −4.

Use ceiling when you need to guarantee sufficiency. Pagination is the classic case: if you have 53 items and pages hold 10, you need ⌈53 ÷ 10⌉ = 6 pages, not 5. Resource allocation, container sizing, and billing intervals all require ceiling to avoid undercount.

If x ∈ Z, then ⌈x⌉ = x. The ceiling of any integer is itself. The fractional part is 0, so no rounding occurs.

No. Truncation removes the decimal part (rounds toward zero). For positive non-integers, ceiling rounds away from zero. trunc(2.3) = 2, but ⌈2.3⌉ = 3. They coincide only for integers and for negative non-integers (e.g., both give −2 for −2.7... actually trunc(−2.7) = −2 and ⌈−2.7⌉ = −2). So for negatives they agree, for positives they differ.

Yes. Inputs like 3.5e2 (which equals 350) and 1.2e−3 (which equals 0.0012) are parsed correctly. The ceiling of 1.2e−3 is 1.

For positive n and d, the ceiling of the division can be computed as ⌈n ÷ d⌉ = ⌊(n + d − 1) ÷ d⌋. This integer-only formula avoids floating-point issues and is standard in systems programming for buffer size calculations.