## 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.