# Factorial Question

**URL:** <https://boards.straightdope.com/t/factorial-question/198901>\
**Category:** Factual Questions\
**Created:** [September 1, 2003, 2:40am UTC](https://boards.straightdope.com/t/factorial-question/198901 "2003-09-01T02:40:14Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![Netbrian](https://avatars.discourse-cdn.com/v4/letter/n/71e660/32.png) [@Netbrian](https://boards.straightdope.com/u/Netbrian)\
**Post date:** [September 1, 2003, 2:40am UTC](https://boards.straightdope.com/t/factorial-question/198901/1 "2003-09-01T02:40:14Z")

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Hi! As many people are aware, “!” signifies a factorial in math, like 5! is 5 \* 4 \* 3 \* 2 \* 1. In my mathematics class, passing mention was made to factorials of things other than the natural numbers. Can one take the factorial of negative numbers? Decimals? Irrational numbers? Complex numbers? If one can, how is it done?

Thanks!

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**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\
**Post date:** [September 1, 2003, 3:13am UTC](https://boards.straightdope.com/t/factorial-question/198901/2 "2003-09-01T03:13:31Z")

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The [Gamma Function](http://mathworld.wolfram.com/GammaFunction.html) extends the factorial function for real complex numbers.

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**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\
**Post date:** [September 1, 2003, 3:14am UTC](https://boards.straightdope.com/t/factorial-question/198901/3 "2003-09-01T03:14:33Z")

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The most popular extension of factorials is based on the [Gamma function](http://www.math.toronto.edu/mathnet/plain/questionCorner/gamma.html). (Note the “1 off” value of x.)

For negative integers, factorial is undefined. The function approaches +/- infinity at those points.

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**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\
**Post date:** [September 1, 2003, 3:14am UTC](https://boards.straightdope.com/t/factorial-question/198901/4 "2003-09-01T03:14:35Z")

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real _and_ complex, obviously.

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**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\
**Post date:** [September 1, 2003, 3:31am UTC](https://boards.straightdope.com/t/factorial-question/198901/5 "2003-09-01T03:31:32Z")

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The important thing about Gamma functions is, for positive x:

Gamma(x + 1) = x · Gamma(x)

This is a lot like the factorial’s property (but again, note that it’s 1 off):

(n + 1)! = (n+1) · n!

Isn’t that handy? It means, among other things, that you only need to make a table of the values for Gamma(x) between 1 and 2. For instance, Gamma(1.6) = 0.8935. So:

Gamma(3.6) = 2.6 · 1.6 · Gamma(1.6) = 3.7170

[More than you really want to know about Gamma functions](http://mathworld.wolfram.com/GammaFunction.html)  
[Calculate real Gammas](http://www.efunda.com/math/gamma/findgamma.cfm)

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**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\
**Post date:** [September 1, 2003, 3:34am UTC](https://boards.straightdope.com/t/factorial-question/198901/6 "2003-09-01T03:34:23Z")

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My first link is redundant with **Desmostylus** ’s. Sorry!