Floor Function
The floor function, denoted as ⌊x⌋ or “floor of x”, is a mathematical function that rounds a real number down to the nearest integer that is less than or equal to x
The floor function, denoted as ⌊x⌋ or “floor of x”, is a mathematical function that rounds a real number down to the nearest integer that is less than or equal to x. In other words, it gives you the largest integer that is less than or equal to x.
For example:
⌊3.7⌋ = 3
⌊-2.5⌋ = -3
⌊0⌋ = 0
Some important properties of the floor function are:
1. ⌊x⌋ ≤ x for all real numbers x.
2. If x is an integer, then ⌊x⌋ = x.
3. ⌊x⌋ + 1 > x if x is not an integer.
4. ⌊x + n⌋ = ⌊x⌋ + n for any integer n.
5. If x and y have the same fractional part (i.e., their decimal parts are equal), then ⌊x⌋ = ⌊y⌋.
The floor function is commonly used in computer programming, number theory, and calculus, especially in problems involving inequalities, divisibility, or approximations.
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