# Why is zero factorial equal to one?

## Answer

###### Wiki User

###### 12/20/2010

n!/n = n!/n | reflexive property

(n)(n-1)(n-2).../n = n!/n | definition of factorial

(n-1)(n-2)... = n!/n | cancelling the common factor of n

(n-1)! = n!/n | definition of factorial

Notice that, in order for n! to be described as (n)(n-1)(n-2)... and proceed to be rewritten as (n-1)! after the n's cancel, the natural number n had to be greater than some natural number for (n-1) to be a factor in the factorial. This means that n must be at least 2, because if n were 1, (n-1) would not have been a factor of the factorial, and the proof would fail unless we assume that n is at least 2. So now you know that this rule cannot prove that 0! is 1 because 1 cannot be substituted into the rule because, as it stands, the rule is only valid for values of 2 or greater. The rule is valid for values of 1 or greater if it is assumed that 0! is 1, but that is what you are trying to prove.