A mathematical concept.
n! = n*(n-1)*(n-2)* ... * 2 * 1

The factorial function (along with the fibonacci series) is a common computer science example of recursion. For the recursive definition, consider the following:

f(n)=n*f(n-1)
and
f(0)=1.

For n negative, the factorial is usually undefined, but see also Euler Gamma Function for further info (the 'factorial of a negative integer' is always undefined, however 'factorials' of nearly all real numbers are defined using the gamma function). Another form of the above definition is the following:

         n
       -----
        | |
f(n) =  | |  i
        | |
        i=1

The capital pi symbolizes multiplication, just as a capital sigma denotes a sum. When written out, the above looks like 1*2*3*...*(n-1)*n. Here it must be explicitly stated that f(0)=1.


Reasons that f(0)=1 for f(n)=n!: "It all works out better!"

  • The nth derivative of xn is f(n)=n! (for n in the integers). Therefore, since the zeroeth (sp?) derivative of x0 is x0=1 (for all x not 0), 0!=1.
  • f(n)=Gamma(n+1). Therefore, f(0)=Gamma(1)=1.
  • The binomial theorem requires 0!=1 for consistency.
  • (as jrn points out) (x+1)!=x!(x+1), so x!=(x+1)!/(x+1). Therefore, 0!=(0+1)!/(0+1)=1!/1=1.
  • (See also 0! for more info.)

If you have any other reasons, please /msg me.


Stirling's Formula says that n! is approximately nn*e-n. This is very cool, *if* it's reasonably accurate.

The factorial can be neatly expressed in Eindhoven notation (a lot more ASCII-friendly than the pi notation): n! = (* : (iZ) ∧ (0 < in) : i).

Among other things, the factorial of n is the number of different ways n items can be ordered. For example, take 3 items, call them a, b and c. These can be arranged in 6 different ways, abc, acb, bac, bca, cab, or cba; 3! = 6. 20 items can be ordered in 20! different ways, etc..

Logically, this makes perfect sense. When choosing the first item, there are n to choose from. When choosing the second, a totally independent choice, there are (n - 1) items to choose from. Just which (n - 1) items in particular depend on the first choice, but the number of items left for the second choice is always the same. Because the two choices are independent, the numbers are multiplied. For the third choice, there are (n - 2) items to choose from, etc., and for the last choice there is only 1 item to choose from. So the total number of ways to order them are (n * (n - 1) * (n - 2) … * 1) = n!.

Fac*to"ri*al (?), a.

1.

Of or pertaining to a factory.

Buchanan.

2. Math.

Related to factorials.

 

© Webster 1913.


Fac*to"ri*al, n. Math. (a) pl.

A name given to the factors of a continued product when the former are derivable from one and the same function F(x) by successively imparting a constant increment or decrement h to the independent variable. Thus the product F(x).F(x + h).F(x + 2h) . . . F[x + (n-1)h] is called a factorial term, and its several factors take the name of factorials.

Brande & C.

(b)

The product of the consecutive numbers from unity up to any given number

.

 

© Webster 1913.

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