Factorial Calculator - Calculate Factorial of Any Number Online

What is a Factorial Calculator?

A factorial calculator is a mathematical tool that computes the factorial of any given number. The factorial of a number n (denoted as n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Factorials are fundamental in mathematics, particularly in combinatorics, probability theory, and calculus. They appear in permutations, combinations, and various mathematical series. Our online factorial calculator provides instant results with step-by-step calculations for better understanding.

How to Calculate Factorial

The factorial of a non-negative integer n is calculated using the formula:

n! = n × (n-1) × (n-2) × ... × 2 × 1

Special Cases:

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

Examples:

3! = 3 × 2 × 1 = 6
4! = 4 × 3 × 2 × 1 = 24
5! = 5 × 4 × 3 × 2 × 1 = 120

Applications of Factorials

1. Permutations

Factorials are used to calculate the number of ways to arrange objects. The number of permutations of n distinct objects is n!.

2. Combinations

In combinatorics, factorials are used in combination formulas: C(n,r) = n! / (r!(n-r)!)

3. Probability

Factorials appear in probability calculations, especially in counting problems and probability distributions.

4. Series and Sequences

Factorials are used in Taylor series, Maclaurin series, and other mathematical series expansions.

Properties of Factorials

Growth Rate

Factorials grow very rapidly. For example:

10! = 3,628,800
15! = 1,307,674,368,000
20! = 2,432,902,008,176,640,000

Recursive Property

Factorials can be defined recursively: n! = n × (n-1)!

Stirling's Approximation

For large n, Stirling's approximation gives: n! ≈ √(2πn) × (n/e)^n

Common Factorial Values

n	n!
0	1
1	1
2	2
3	6
4	24
5	120
6	720
7	5,040
8	40,320
9	362,880
10	3,628,800

Using Our Factorial Calculator

Our factorial calculator is designed to be user-friendly and educational:

Input Range: Supports numbers from 0 to 170
Step-by-Step Calculation: Shows the complete multiplication process
Large Number Support: Handles very large factorial results
Error Handling: Validates input and provides helpful error messages
Educational Features: Includes formula explanations and common values

Mathematical Notation

The factorial function is denoted by an exclamation mark (!) after the number. This notation was introduced by Christian Kramp in 1808.

n! = ∏(k=1 to n) k = 1 × 2 × 3 × ... × n

Computational Considerations

When computing factorials programmatically:

Integer Overflow: Factorials grow very quickly and can exceed integer limits
Floating Point Precision: For very large numbers, floating point arithmetic may lose precision
Memory Usage: Storing very large factorial results requires significant memory
Time Complexity: Computing n! has O(n) time complexity

Frequently Asked Questions

What is the factorial of 0?

The factorial of 0 is defined as 1 (0! = 1). This is a mathematical convention that makes many formulas work correctly, especially in combinatorics and probability theory.

Can factorials be calculated for negative numbers?

No, factorials are only defined for non-negative integers (0, 1, 2, 3, ...). However, there is a generalization called the gamma function that extends factorial to complex numbers, but it's not the same as the factorial function.

Why do factorials grow so rapidly?

Factorials grow exponentially because each multiplication increases the result significantly. The growth rate is faster than exponential functions, which is why factorials appear in asymptotic analysis and big-O notation in computer science.

What is the largest factorial I can calculate?

Our calculator supports factorials up to 170! (170 factorial). Beyond this point, the numbers become so large that they exceed the precision limits of standard floating-point arithmetic in JavaScript.

How are factorials used in real-world applications?

Factorials are used in:

Probability and statistics (permutations and combinations)
Computer science (algorithm analysis, sorting algorithms)
Physics (quantum mechanics, statistical mechanics)
Economics (game theory, optimization problems)
Biology (genetic analysis, population dynamics)

What is Stirling's approximation?

Stirling's approximation is a formula that approximates factorials for large numbers: n! ≈ √(2πn) × (n/e)^n. It's useful when exact calculation is impractical due to the size of the numbers involved.