Factorial of zero is 1. Why?

Question

Why is the factorial of zero, one. What is the mathematical proof behind it?

Answer 1

We generally want the factorial to satisfy: $n(n-1)!=n!$ for $n\in\mathbb{N}$ but if we put $n=1$ then we arrive at $0!=1!=1$.

Answer 2

$(n+1)! = (n+1) * n!$. Take $n=0$.

Answer 3

See this video. Two reasons are:

1. We want the functional equation $n!=(n-1)!n$ to hold when $n=1$.
2. $0!$ should be the number of different ways of arranging $0$ items in a row. There is precisely one way of doing it (the empty arrangement).

Answer 4

The Gamma function $$\Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt$$ has the property $$\Gamma(x+1)=x\Gamma(x)$$ so we find easily that $$\Gamma(n+1)=n!$$ so $$0!=\Gamma(1)=1$$

Answer 5

The proof is somewhat boring: we review the definition of the factorial

• $0! = 1$
• $(n+1)! = (n+1) n!$ for $n \geq 0$

and therefore, our proof is one line:

$0! = 1$

What you really want to ask is probably

Why is the factorial defined in such a way that $0! = 1$?

and there are a number of explanations. The simplest, if you can wrap your head around degenerate cases, is that $n!$ is equal to the number of ways to order a list of $n$ objects, and there is simply one way to order a list of zero objects. e.g. see below for that ordering: