1. What is Factorial?

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Factorials are used extensively in mathematics, particularly in combinatorics, algebra, and mathematical analysis.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

• \( n \) — A non-negative integer
• \( ! \) — Factorial operator

Special Cases:

• \( 0! = 1 \) (by definition)
• \( 1! = 1 \)

3. Importance of Factorial Calculation

Details: Factorials are fundamental in permutations and combinations, probability calculations, series expansions in calculus, and in many algorithms in computer science.

4. Using the Calculator

Tips: Enter any non-negative integer up to 10,000. The calculator uses arbitrary-precision arithmetic to handle very large numbers accurately.

5. Frequently Asked Questions (FAQ)

Q1: Why is 0! equal to 1?
A: This is a convention that makes many mathematical formulas work consistently, especially in combinatorics where there's exactly one way to arrange zero objects.

Q2: What is the largest factorial this calculator can compute?
A: The calculator can handle up to 10,000! (though displaying the full result may be impractical due to its enormous size).

Q3: How are factorials used in real-world applications?
A: They're used in probability calculations, statistical mechanics, quantum physics, and in algorithms for computer science problems.

Q4: What's the approximate value of 100! ?
A: 100! is approximately 9.3326 × 10¹⁵⁷, a number so large it has more digits than atoms in the observable universe.

Q5: Are there approximations for very large factorials?
A: Yes, Stirling's approximation gives a good estimate: \( n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \).