1. What is the Floor Function?

The floor function, denoted \( \lfloor x \rfloor \), gives the greatest integer less than or equal to x. It "rounds down" to the nearest integer.

2. How Does the Calculator Work?

The calculator plots the floor function:

Example:

• \( \lfloor 3.7 \rfloor = 3 \)
• \( \lfloor -1.2 \rfloor = -2 \)
• \( \lfloor 5 \rfloor = 5 \)

3. Importance of Floor Function

Applications: Used in computer science (integer division), mathematics (number theory), engineering (quantization), and finance (discrete pricing).

4. Using the Calculator

Instructions: Enter minimum and maximum x-values to define the plotting range. The graph will show the step-like behavior of the floor function.

5. Frequently Asked Questions (FAQ)

Q1: How is floor different from truncation?
A: For positive numbers they're the same, but for negatives: floor(-3.2) = -4 while trunc(-3.2) = -3.

Q2: What's the relationship with ceiling function?
A: \( \lfloor x \rfloor \leq x < \lfloor x \rfloor + 1 \), while \( \lceil x \rceil - 1 < x \leq \lceil x \rceil \).

Q3: Are there programming equivalents?
A: Most languages have floor() functions (Math.floor() in JavaScript, math.floor() in Python).

Q4: What's the derivative of floor function?
A: It's 0 everywhere except at integer points where it's undefined.

Q5: How is floor used in real-world applications?
A: Common in discrete mathematics, computer algorithms, and anywhere you need whole number quantities.