From b7e6591a6c6e33dcecd8ab9b171026dec5b1e9ae Mon Sep 17 00:00:00 2001 From: Alexander Molnar <37451552+BTmathic@users.noreply.github.com> Date: Mon, 11 Mar 2019 15:34:42 -0400 Subject: [PATCH] Added content to factorial stub (#34330) * Added content to stub Added definition, examples and some uses as well as computational info and interesting formulas that occur. * Update index.md --- .../mathematics/factorial-function/index.md | 58 +++++++++++++++++-- 1 file changed, 53 insertions(+), 5 deletions(-) diff --git a/guide/english/mathematics/factorial-function/index.md b/guide/english/mathematics/factorial-function/index.md index 740413d270..a2bb815aa1 100644 --- a/guide/english/mathematics/factorial-function/index.md +++ b/guide/english/mathematics/factorial-function/index.md @@ -3,13 +3,61 @@ title: Factorial Function --- ## Factorial Function -This is a stub. Help our community expand it. +The factorial function is a useful function in [combinatorics](https://en.wikipedia.org/wiki/Combinatorics) for counting things such as [permutations](https://en.wikipedia.org/wiki/Permutation) as well as the definition of [Euler's number](https://en.wikipedia.org/wiki/E_(mathematical_constant)), the base of the natural logarithm, and appears in many other areas. -This quick style guide will help ensure your pull request gets accepted. +For any positive integer n we define the factorial of n, denoted n!, as the product - +#### n! = 1 × 2 × 3 × ... × (n-2) × (n-1) × n. -#### More Information: - +For example, +- 1! = 1, +- 2! = 2 × 1 = 2, +- 5! = 5 × 4 × 3 × 2 × 1 = 120. +Notice this function satisfies the [recurrence](https://en.wikipedia.org/wiki/Recurrence_relation) n! = n × (n-1)! which is a particularly useful viewpoint to use in [many](https://en.wikipedia.org/wiki/Gamma_function) areas of mathematics allowing the factorial to be generalized to non-integer values. (For example, this recurrence can be extended with (-1/2)! = [sqrt(π)](http://www.wolframalpha.com/input/?i=(-1%2F2)!).) +As convention, the [empty product](https://en.wikipedia.org/wiki/Empty_product), that is, the product of nothing, is usually taken to be 1, so with this definition we have 0! = 1. This convention makes sense in all the uses below. + +### Uses + +If you have n different objects and want to know how many ways they can be arranged in a row, there are n choices for the first object, then (after picking the first object) n-1 choices for the second object, n-2 choices for the third object, etc... and so we see there are n! ways to arrange the objects. + +Another common method of counting involves [combinations](https://en.wikipedia.org/wiki/Combination) which are a given by a quotient of factorials. The combinations then come up in, for example, the [binomial formula](https://en.wikipedia.org/wiki/Binomial_theorem), the coefficients in the expansion of + +#### (x + y)n + +for any integer n. + +Factorials also appear in many useful representations of functions, including approximations of derivatives in [Taylor's formula](https://en.wikipedia.org/wiki/Taylor%27s_theorem#Taylor's_theorem_in_one_real_variable), [exponential](https://en.wikipedia.org/wiki/Power_series#Examples) and [trigonometric](https://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions) functions. + +### Computation + +Computing the factorial of a positive integer is incredibly straightforward, it is simply a product of all positive integers less than or equal to itself. However, this is not an efficient approach for very large numbers, and such a product will be incredibly large as well, so it is usually better to use an approximation when looking to compute very large factorials. One simple approximation is [Stirling's approximation](https://en.wikipedia.org/wiki/Stirling%27s_approximation), namely, + +#### n! ~ sqrt(2πn)[n/e]n + +so, for example, + +#### 10100! ~ sqrt(2π10100)[10100/e]10100 ~ e-10100 × 1010102. + +### Interesting formulas + +As mentioned above, the factorial can be used to defined Euler's number, namely + +#### Σ 1/n! = e + +With a slight adjustment we have the fascinating sum + +#### Σ 1/[(n+2)n!] = 1 + +Lastly, the generalization of the factorial to non-integer values comes from the formula + +#### n! = ∫0∞ tze-t dt + +This formula is where the value for (-1/2)! above comes from, since + +#### ∫0∞ t-1/2e-t dt = sqrt(π) + +The recurrence now tells us that each half integer factorial is just a multiple of sqrt(π) as, for example + +#### (3/2)! = (3/2)*(1/2)! = (3/2)*(1/2)*(-1/2)! = 3sqrt(π)/4