--- title: Factorials --- ## Factorials ### Definition of Factorial The factorial is to multiply it by every intiger lower then it ending at one. If the initial number is negative the result is infinity. A factorial of

n

, a non-negative integer, is defined as:

n! = 1 * 2 * ... * (n - 1) * n

A special case arises when

n = 0

. Namely,

0! = 1

. ### Convenience of Factorials The definition above provides you with convenience in certain computations. For example, factorials inside fractions can often be simplified as follows: Example 1:

7! / 5! = (1 * 2 * 3 * 4 * 5 * 6 * 7) / (1 * 2 * 3 * 4 * 5) = 6 * 7 = 42

Example 2:

(n + 1)! / n! = (1 * 2 * ... * n * (n + 1)) / (1 * 2 * ... * n) = n + 1

### Alternative Definition Alternatively, factorials can be defined as follows:

0! = 1

n! = n * (n - 1)!

n > 0

This recursive definition means the exact same as the traditional definition. Applying this to the second example above, we get:

(n + 1)! / n! = (n + 1) * n! / n! = n + 1

### Aside: Extension to Non-Integers Note that factorial as defined above applies only to non-negative integers. Actually, there is a generalization of factorials that extends to non-integers as well, which is the Gamma function. In particular, for any natural number

n

, you have

n! = Gamma(n + 1) = n * Gamma(n)

. For more, see Extension of factorial to non-integer values of argument.

One tricky example that many may not know if 0! = 1. For futher proof see the link under More Information. #### More Information: Factorials