Free · Arithmetic calculators ➗

Ceiling: Smallest Integer at or Above

Round a signed decimal upward to the least integer that is no smaller than the input, including negative numbers and tiny positive fractions.

  • Formula & worked example
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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Private by default

Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

The ceiling of 4.2 is 5, while the ceiling of −4.2 is −4. An integer stays unchanged. “Up” means toward positive infinity.

y = ⌈x⌉, characterized by y − 1 < x ≤ y

Worked example

Enter these known values and leave the other values blank.

Decimal value
-4.2
Smallest integer at or above
-4

Assumptions and limitations

  • The input is a finite real decimal with magnitude below 10^50. This bound preserves exact integer arithmetic within the calculator’s precision; larger magnitudes are rejected.
  • Ceiling moves toward positive infinity. For negative fractions it can move toward zero; it does not round to the nearest integer.
  • The result is output only. Many different inputs have the same ceiling, so there is no unique reverse calculation.
  • Decimal inputs are evaluated before rounding to an integer. Unlike binary JavaScript Number arithmetic, a tiny positive decimal above an integer is not silently treated as that integer.
  • Positive and negative zero both display as 0. Missing input remains incomplete.

Common questions

What is the ceiling of a negative decimal?

For −4.2 it is −4, because −4 is the least integer that is at least −4.2. For −0.2 it is 0.

Does ceiling change an exact integer?

No. The ceiling of 7 is 7, and the ceiling of −7 is −7.

How is ceiling different from rounding to nearest?

Ceiling always selects the next integer toward positive infinity unless the value is already an integer. Thus 4.01 gives 5 rather than 4.

Can the result tell me the original decimal?

No. A ceiling result n identifies the interval n−1 < x ≤ n, not one particular decimal.

References

This calculator selects the least integer at or above an exact decimal input, with one forward-only output. Decimal arithmetic retains fractional information that binary Number rounding can lose near an integer. Inputs must have magnitude below 10^50 to preserve exact integer results.

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