Factorial

5! = 5 × 4 × 3 × 2 × 1 = 120

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

Usually, n! is read as "n factorial". The current notation was introduced by the mathematician Christian Kramp in 1808.

Introduction

Factorials are often used as a simple example when teaching recursion in computer science because they satisfy the following recursive relationship (if n ≥ 1):

n! = n (n − 1)!

0! = 1

• 0! is an instance of the empty product, and therefore 1
• it makes the above recursive relation work for n = 1
• many identities in combinatorics would not work for zero sizes without this definition

Factorials are important in combinatorics because there are n! different ways of arranging n distinct objects in a sequence (see permutation). They also turn up in formulas of calculus, such as in Taylor's theorem, for instance, because the n-th derivative of the function xⁿ is n!.

When n is large, n! can be estimated quite accurately using Stirling's approximation:

A simple online factorial calculator can be obtained here.

Generalization to the Gamma function

The related Gamma function Γ(z) is defined for all complex numbers z except for z = 0, -1, -2, -3, ... It is related to the factorial in that it satisfies the same (above-mentioned) recursive relationship as the factiorial function (offset by one):

Consequently, the gamma and factorial functions equate (with an offset)

when n is any non-negative integer.

Because of this relationship, the gamma function is often thought of as a generalization of the factorial function to the domain of complex numbers. This is legitimate because:

1. Shared meaning: The canonical definition of the factioral function is the mentioned recursive relationship, shared by both.
2. Uniqueness: The gamma function is the only function which satisfies the mentioned recursive relationship for the domain of complex numbers and is holomorphic and whose restriction to the positive real axis is log-convex. That is, it is the only function that could possibly be a generalization of the factorial function.
3. Usage/context: The function is generally used in a context similiar to that of the factorial function (but, of course, where a more general domain is of interest).

Multifactorials

A common related notation is to use multiple exclamation points (!) to denote a multifactorial, the product of integers in steps of two, three, or more.

For example, n!! denotes the double factorial of n, defined recursively by n!! = n (n-2)!! for n > 1 and as 1 for n = 0,1. Thus, (2n)!! = 2ⁿn! and (2n+1)! = (2n+1)!! 2ⁿn!. The double factorial is related to the Gamma function of half-integer order by Γ(n+1/2) = √π (2n-1)!!/2ⁿ.

One should be careful not to interpret n!! as the factorial of n!, which would be written (n!)! and is a much larger number.

The double factorial is the most commonly used variant, but one can similarly define the triple factorial (!!!) and so on. In general, the k-th factorial, denoted by !^(k), is defined recursively by: n!^(k) = n (n-k)!^(k) for n > k-1, n!^(k) = n  for k > n > 0, and 0!^(k) = 1.

Hyperfactorials

Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by

H(n) = nⁿ (n-1)^(n-1) ... 3³ 2² 1¹

E.g. H(4) = 27648.

The hyperfactorial function is similar to the factorial, but produces larger numbers. The rate of growth of this function, however, is not much larger than a regular factorial.

Superfactorials

The superfactorial of n, written as n$ (a factorial sign with an S written over it) has been defined as

n$ = n!⁽⁴⁾n!

where the ⁽⁴⁾ notation denotes the hyper4 operator, or using Knuth's up-arrow notation,

External link

• The Dictionary of Large Numbers

Referenced By