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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)! if 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.