Factorial - Mathematical Definition

Factorial Definition

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. Mathematically, it is defined as:

\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 \]

By convention, the factorial of 0 is defined as 1, i.e., \[ 0! = 1 \].

Examples

• \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
• \[ 3! = 3 \times 2 \times 1 = 6 \]
• \[ 1! = 1 \]
• \[ 0! = 1 \]

Uses of Factorial

Factorials are commonly used in permutations and combinations, series expansions, and various fields of mathematics, science, and engineering.

Recursive Definition

Factorials can also be defined recursively:

\[ n! = n \times (n-1)! \]

\[ 0! = 1 \]

In this recursive form, each factorial value is defined in terms of the previous one.

Proof that \(0! = 1\)

One way to understand why \(0! = 1\) is through the concept of the factorial function and its relationship with the number of ways to arrange objects. Here’s a simple proof:

Proof Using the Factorial Definition

The factorial of a non-negative integer \( n \) is defined as the product of all positive integers less than or equal to \( n \). Mathematically:

\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 \]

By definition, \(0!\) should fit into this pattern. Let's consider the factorial of \(1\):

\[ 1! = 1 \]

If we use the definition of factorial, we can write \(1!\) in terms of \(0!\):

\[ 1! = 1 \times 0! \]

Since \(1! = 1\), we substitute this value in the equation:

\[ 1 = 1 \times 0! \]

To satisfy this equation, \(0!\) must be equal to 1. Thus:

\[ 0! = 1 \]

Proof Using Combinatorics

In combinatorics, \( n! \) represents the number of ways to arrange \( n \) distinct objects. There is exactly one way to arrange zero objects: do nothing. Hence, we have:

\[ 0! = 1 \]

Proof Using the Gamma Function

The Gamma function \( \Gamma(n) \) is a generalization of the factorial function to complex and real number arguments. For a positive integer \( n \), the Gamma function is defined as:

\[ \Gamma(n) = (n-1)! \]

Thus, for \( n = 1 \):

\[ \Gamma(1) = 0! \]

It is also known that \( \Gamma(1) = 1 \). Therefore:

\[ 0! = 1 \]