Zero Factorial

7 July 2009

Those of you have taken math classes in high school probably learned about factorials, which are written with “! ” symbols.  The usual definition is something like the following.

Definition. For a positive integer n, we define n! (read: n factorial) as the product of all the positive integers up to and including n.

1! = 1

2! = 1\times 2 = 2

3! = 1 \times 2\times 3 = 6

4! = 1\times 2 \times 3 \times 4 = 24

5! = 1\times 2 \times 3\times 4\times 5 = 120

and so forth.

The factorials can be defined by the fact that n! is the number  of ways to put n objects in order.  They are ubiquitous in combinatorics (read: counting) and also show up in lots of other sorts of equations and formulas.  Sooner or later, it comes up that mathematicians don’t just use factorials of positive integers, and 0! shows up on the chalkboard.  Then the questions start.  Because almost all students expect 0!  to be zero.  And the exasperated teacher says something like the following.

“Okay, zero factorial is one.  It just is.  There’s doesn’t have to be a reason, there’s nothing to try to understand, it’s just a mathematical convention.  0!=1.”

But there are good reasons to decide that 0!=1, not just to take some teacher’s word for it but to know that it’s the right thing.  And I have more faith in you, fair reader, than your math teacher did.  I believe that anyone who wants to understand it can.

If you keep reading, you’ll find three ways of getting at zero factorial, including shrieks, a math koan, and the nature of nothing.

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