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
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
Input | Ceiling ⌈x⌉ | Floor ⌊x⌋ | Round | Truncate |
---|---|---|---|---|
3.7 | 4 | 3 | 4 | 3 |
3.2 | 4 | 3 | 3 | 3 |
-2.3 | -2 | -3 | -2 | -2 |
-2.7 | -2 | -3 | -3 | -2 |
5.0 | 5 | 5 | 5 | 5 |