Ceiling Function Calculator

Find the smallest integer greater than or equal to any real number using the ceiling function x

Calculate Ceiling Function

The ceiling function returns the smallest integer x

Example Calculations

Positive Decimals

3.2 = 4

7.9 = 8

11.1 = 12

π = 3.14... = 4

Integers

5 = 5

-3 = -3

0 = 0

100 = 100

Negative Decimals

-2.3 = -2

-5.7 = -5

-1.1 = -1

-0.5 = 0

Special Cases

0.1 = 1

-0.1 = 0

1.0 = 1

-1.0 = -1

Ceiling Function Properties

Always rounds UP to the nearest integer

For integers: n = n

For positive decimals: always goes to next integer

For negative decimals: goes toward zero

Domain: all real numbers

Range: all integers

Mathematical Notation

Symbol

x

Ceiling brackets

Programming

ceil(x)

Most programming languages

LaTeX

\lceil x \rceil

Mathematical typesetting

Related Functions

Floor Function x

Rounds DOWN to nearest integer

Round Function

Rounds to nearest integer (0.5 rounds up)

Truncation

Removes decimal part (toward zero)

Understanding the Ceiling Function

What is the Ceiling Function?

The ceiling function, denoted as x, maps any real number to the smallest integer that is greater than or equal to that number. It's essentially a "round up" function that always moves toward positive infinity.

Mathematical Definition

x = min{n ∈ ℤ : n ≥ x}

The smallest integer n such that n x

Key Properties

  • Idempotent on integers: If x is an integer, then x = x
  • Always rounds up: For non-integers, always goes to the next higher integer
  • Right-continuous: x = x⁺ for any x

Graph Visualization

Step Function Graph
The ceiling function creates a step-like graph where each step represents an integer value. The function jumps up at each integer point, creating a discontinuous but right-continuous function.

Common Applications

  • Computer Science: Memory allocation, page numbering
  • Engineering: Quantization, digital signal processing
  • Finance: Rounding up to next cent, pricing models
  • Statistics: Discrete probability distributions

Ceiling vs Other Rounding Functions

InputCeiling xFloor xRoundTruncate
3.74343
3.24333
-2.3-2-3-2-2
-2.7-2-3-3-2
5.05555