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

Bilal Ayub
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5 Answers5

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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).
user1337
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  • Also, ${m\choose n} = m!/n!/(m-n)!$ is the number of ways of choosing $n$ objects from a set of $m$. This should work and should return $1$ for $n=m$ or $n=0$. – user3294068 Jun 24 '14 at 16:23
  • You could also say there are no ways of arranging $0$ items, because you can't arrange them. A bit like how you could argue that $x/0$ should be $x$, because you're not doing any dividing. For reasons like this, I think edge cases like this are best handled formally, rather than trying to give intuitive explanations, which tend to collapse when examined more closely. – Jack M Jun 24 '14 at 16:29
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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:

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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 $0!=1!=1$.

user71352
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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 $$0!=\Gamma(1)=1$$

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

Ned
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  • But factorial is defined in another way as n multiplied by decrementing n up-to zero. So 0 will result in zero. – Bilal Ayub Jun 24 '14 at 16:21
  • decrementing n down to 1-- which doesn't make sense for n = 0, so we seek another uniform property of factorial in order to extend the definition. Similarly, why is 4 to the -1 power equal to 1/4? "multiply it by itself -1 times" doesn't make sense, so instead we use other properties to extend the definition to negative exponents. – Ned Jun 24 '14 at 20:53