1. What is the Floor Function?

The floor function, denoted as floor(x) or ⌊x⌋, returns the greatest integer less than or equal to x. It's a fundamental mathematical operation used in calculus, computer science, and discrete mathematics.

2. How Does the Calculator Work?

The calculator implements the floor function:

Where:

• \( x \) — Any real number input
• \( \lfloor x \rfloor \) — The greatest integer ≤ x

Examples:

• floor(3.7) = 3
• floor(-1.2) = -2
• floor(5) = 5

3. Applications of Floor Function

Details: The floor function is used in:

• Discrete mathematics and number theory
• Computer algorithms and programming
• Financial calculations (rounding down)
• Signal processing
• Cryptography

4. Using the Calculator

Tips: Enter any real number in the input field. The calculator will return the largest integer less than or equal to your input.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between floor and ceiling functions?
A: Floor returns the greatest integer ≤ x, while ceiling returns the smallest integer ≥ x.

Q2: How does floor handle negative numbers?
A: It rounds towards negative infinity (e.g., floor(-2.3) = -3).

Q3: Is floor the same as truncation?
A: Only for positive numbers. For negatives, truncation rounds toward zero while floor rounds down.

Q4: What's the relationship between floor and modulo?
A: The floor function is often used in modulo arithmetic definitions.

Q5: Are there programming equivalents?
A: Most languages have floor() functions, and some use integer casting for similar results.