# Inverse Factorials

1. Oct 29, 2014

### MohammedRady

Why exactly is there no such thing as an inverse factorial function? Now I am fully aware of the fact that the factorial function ($f(x) = x!$) is not one-to-one, since both 0! and 1! equal 1. But can't we circumvent this by restricting the domain of f such that it only includes values of x greater than or equal to 1?

2. Oct 29, 2014

### Staff: Mentor

There's some interesting reasoning behind the 0! = 1 definition:

http://en.wikipedia.org/wiki/Factorial

Notice the comment there is exactly 1 way to order zero objects hence the 0!=1 instead of thinking that anything times 0 is 0.

3. Oct 29, 2014

### zoki85

That comment is ridiculous . There are good reasons why 0!=1 per definition.

4. Oct 29, 2014

### PeroK

What makes you think there isn't an inverse factorial function?

5. Oct 29, 2014

### MohammedRady

This does not pertain to my question.

6. Oct 29, 2014

### MohammedRady

Is there?

7. Oct 30, 2014

### PeroK

The existence of a function is not dependent on its being given a name or on being useful.

8. Oct 30, 2014

### MohammedRady

You're right.
But what I meant was, is there a way to define such a function algebraically?

9. Oct 30, 2014

### PeroK

Here's a paper on the subject (inverse of the Gamma function, which extends factorial beyond the whole numbers).

http://www.ams.org/journals/proc/2012-140-04/S0002-9939-2011-11023-2/S0002-9939-2011-11023-2.pdf [Broken]

Last edited by a moderator: May 7, 2017
10. Oct 30, 2014

### David Carroll

I love this thread! Finding an easy inverse factorial function was an obsession of mine a couple years ago. And I have Asperger's and OCD, so that isn't good. I failed. Miserably. Logarithms of polynomials inside other logarithms...etc. It was a mess and a waste of time.

By the way, 0! = 1 for the simple reason that (n-1)! is found by dividing n! by n. And since 1! = 1, just divide 1 by 1 and you get 1.

11. Oct 30, 2014

### David Carroll

I'm not sure if I understand the intuitive principle that there is only one way to order zero objects. How many ways can you order a complex object?

12. Oct 30, 2014

### MohammedRady

I must input a username and password combination before gaining access to the pdf file.

Last edited by a moderator: May 7, 2017
13. Oct 30, 2014

### jbriggs444

Intuitively, one "orders" a set of objects by assigning them a sequence from first to last. This works for any finite number of objects which is all we need here. (The full formal definition of a "total order" on a set of objects is somewhat more general and removes the need for a first or last object).

Intuitively, the number of ways to "order" a set of objects is simply the number of different possible sequences of that set of objects.

Wrapping an intuition around the notion of an empty sequence is much like wrapping an intuition around the notion of an empty set. There is only one empty sequence. It has no elements and is identical to every other sequence with no elements.

14. Oct 30, 2014

### David Carroll

Ahhh, I see. Thank you.

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