Factorial Calculator

Definition and Basic Properties

The factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 to n. By definition, 0! = 1 and 1! = 1. This definition ensures that many mathematical formulas work correctly, especially in combinatorics. The factorial function is defined for non-negative integers only.

n! = \begincases 1 & \textif n = 0 \\ n × (n-1) × (n-2) × ·s × 2 × 1 & \textif n ≥ 1 \endcases

Recursive Definition and Properties

Factorials can be defined recursively: n! = n × (n-1)! for n ≥ 1, with base case 0! = 1. This leads to important properties: (n+1)! = (n+1) × n!, and n! = n × (n-1) × (n-2)! for n ≥ 2. The recursive definition is fundamental to understanding factorial properties and computational algorithms.

n! = n × (n-1)! \text for n ≥ 1, \quad 0! = 1

Mathematical Properties and Identities

Factorials satisfy several important mathematical properties: n! divides (n+k)! for any k ≥ 0, (n+1)! = (n+1) × n!, and n! = n × (n-1)! × (n-2)! × ... × 1. These properties are crucial for simplifying complex expressions involving factorials.

((n+k)!)/(n!) = (n+1)(n+2)·s(n+k), \quad \textfor k ≥ 1

Rapid Growth and Asymptotic Behavior

Factorials grow extremely rapidly, faster than any exponential function. For large n, Stirling's approximation gives n! ≈ √(2πn) × (n/e)^n. This approximation becomes more accurate as n increases and is useful for estimating large factorials. The growth rate is super-exponential.

n! ≈ √(2π n) ((n)/(e))^n \text for large n

Stirling's Approximation and Error Analysis

Stirling's approximation provides an excellent estimate for large factorials. The relative error decreases as n increases, typically within 1% for n > 10. More refined versions include correction terms for even better accuracy.

n! ≈ √(2π n) ((n)/(e))^n (1 + (1)/(12n) + (1)/(288n^2) + ·s)

Applications in Combinatorics and Counting

Factorials are fundamental in counting problems. The number of permutations of n distinct objects is n!. They appear in combinations C(n,r) = n!/(r!(n-r)!) and permutations P(n,r) = n!/(n-r)!. These formulas are essential for probability and statistics.

P(n,r) = (n!)/((n-r)!), \quad C(n,r) = (n!)/(r!(n-r)!) = \binomnr

Applications in Probability Theory

Factorials appear extensively in probability calculations, especially in discrete probability distributions. They are used in calculating probabilities of arrangements, combinations, and in the binomial distribution. The multinomial distribution also heavily relies on factorials.

P(X = k) = C(n,k) × p^k × (1-p)^n-k = (n!)/(k!(n-k)!) × p^k × (1-p)^n-k

Taylor Series and Calculus Applications

Factorials appear in Taylor series expansions: e^x = Σ(x^n/n!) for all x. They also appear in derivatives of power functions and in the definition of the exponential function. The factorial denominators ensure convergence of these series.

e^x = Σ_n=0^∞ (x^n)/(n!) = 1 + x + (x^2)/(2!) + (x^3)/(3!) + ·s

Gamma Function Extension

The gamma function Γ(z) extends factorial to complex numbers: Γ(n+1) = n! for positive integers n. This allows factorial to be defined for non-integer values and is important in advanced mathematics, including complex analysis and number theory.

\Gamma(n+1) = n! \text for n \in \mathbbN, \quad \Gamma(z) = ∫_0^∞ t^z-1 e^-t dt

Number Theory and Prime Factorization

Factorials have interesting properties in number theory. For example, n! contains all prime numbers up to n as factors. The number of trailing zeros in n! is given by the sum of floor(n/5^k) for k = 1, 2, 3, ...

\textNumber of trailing zeros in n! = Σ_k=1^∞ \lfloor (n)/(5^k) \rfloor

Computational Algorithms and Optimization

For large n, direct calculation of n! becomes impractical due to rapid growth. Special algorithms include memoization, log-space calculations, and Stirling's approximation. The largest factorial that can be stored in standard data types is typically around 170! for double precision.

\log(n!) = Σ_k=1^n \log(k) \text (for log-space calculations)

Applications in Physics and Engineering

Factorials appear in statistical mechanics (partition functions), quantum mechanics (Fermi-Dirac and Bose-Einstein statistics), and engineering (signal processing, error correction codes). They are fundamental to understanding complex systems with discrete states.

Z = Σ_n=0^∞ \fracg^n e^-β E_nn! \text (partition function)

Worked Example

Inputs

• number: 5

Result: 120

Explanation

5! = 5 × 4 × 3 × 2 × 1 = 120.

Zero Factorial

Inputs

• number: 0
• show_steps: yes
• include_approximation: no

Result: 1

Explanation

0! = 1 by definition. This convention makes many mathematical formulas work correctly.