1. What Is the Floor Function?
The floor function, denoted by ⌊x⌋ or floor(x), maps a real number x to the greatest integer less than or equal to it.
Formal definition:
⌊x⌋ = the greatest integer ≤ x
Examples:
⌊3.7⌋ = 3
⌊2⌋ = 2
⌊-1.2⌋ = -2
2. Properties of the Floor Function
- A step‑like, discontinuous function
- Constant on each interval [n, n+1)
- Has jump discontinuities at every integer
- Non‑decreasing (never decreases)
- For negative numbers, the value is less than the input
3. Graph of the Floor Function
The graph of f(x) = ⌊x⌋ consists of horizontal steps. Each step begins at an integer and extends to the next integer.
Graph characteristics:
- Closed circle on the left, open circle on the right
- Step increases at each integer
- Looks like a staircase
4. Domain and Range
- Domain: all real numbers (ℝ)
- Range: all integers (ℤ)
5. Behavior on Intervals
The floor function returns the integer n for all x in the interval [n, n+1).
Example:
For x in [2, 3):
⌊x⌋ = 2
6. Applications of the Floor Function
- Number theory and integer division
- Programming and algorithm design
- Discrete modeling and step functions
- Data grouping and rounding down
- Counting full units (boxes, packets, groups, etc.)
7. Important Examples
Example 1:
f(x) = ⌊x⌋
f(5.99) = 5
f(-3.1) = -4
Example 2:
f(x) = ⌊2x⌋
If x = 1.4:
2x = 2.8
⌊2.8⌋ = 2
8. Relationship with the Ceiling Function
The ceiling function behaves in the opposite direction:
⌈x⌉ = the smallest integer ≥ x
Example:
⌊2.3⌋ = 2
⌈2.3⌉ = 3
9. Conclusion
The floor function is a fundamental mathematical tool that maps any real number to the greatest integer less than or equal to it. Its step‑like graph, integer range, and wide applications make it essential in number theory, programming, and discrete mathematics.