What Is Zero-Factorial?

Why is 0! defined to equal 1? This question arose the other day while we were discussing factorials, and I decided to find an answer by providing a very simple proof often used to support this statement. 

If n! is defined as the product of all positive intergers from 1 to n, then:

1! = 1 x 1 = 1

2! = 1 x 2 = 2 

3! = 1 x 2 x 3 = 6

4! = 1 x 2 x 3 x 4 = 24

n! = 1 x 2 x 3 x 4... x (n-2) x (n-1) x n

And so forth.

Logically, n! can also be expressed n x (n-1)!

Therefore, at n = 1, using n! = n x (n-1)! 

1! = 1 x (1-1)!

1! = 1 x 0!

This simplifies to 1 = 0!

And there you go, a mathematical explanation for 0! = 1.