Why is the factorial of 0 equal to 1?

Question

Wouldn’t this also mean that 1!=0!, why is this true?

In: 383

Answer 1

I’ll start with

>Wouldn’t this also mean that 1!=0!

Yes. Many functions are multi-valued, this is not big news. For example, (x-1)^2 has the same value, 1, for both 0 and 2.

Now for the harder bit.

One way to think of it is recursively in reverse. Meaning that n! is equal to (n+1)! divided by n. For instance, 6! = 7! / 7. Thus, 0! = 1!/1 = 1/1 = 1.

While this makes some sense, you are open to valid accusations of extrapolating outside the defined bounds of the function, and things get funky for negative integer factorials. The real reason is pretty much “because it’s defined to be”, which also makes it work with something called the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function), making Gamma(n) = (n-1)! for all positive integer n.