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Why is 0! = 1?

  1. Jul 29, 2010 #1
    Why is 0! = 1?
     
  2. jcsd
  3. Jul 29, 2010 #2

    arildno

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    Re: 0!

    It is defined that way. That's why.
     
  4. Jul 29, 2010 #3
    Re: 0!

    Is there no proof to that?
     
  5. Jul 29, 2010 #4

    arildno

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    Re: 0!

    All proofs have at their basis a set of axioms&definitions, and a proof is simply to show that something else follows from those very same axioms&definitions.

    Thus, neither axioms or definitions are themselves things to be proven, although it is quite possible that one may set up OTHER axioms&definitions from which the elements of the first set can be proven.

    How would you, for example prove that a+0=a for any number a?
     
  6. Jul 29, 2010 #5

    arildno

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    Re: 0!

    Not that if you have a general recurrence relation described as:

    R(n)=n*R(n-1)
    (typically part of the definition of the factorial) you could, if you ASSUME this to be valid for n>=1 insert for n=1:
    R(1)=1*R(0),
    that is R(1)=R(0).
    Now, how are you to go from this to your standard idea of the factorial?
    Clearly, by fixing the value R(1)=R(0)=1.

    This is therefore a necessary additional definition, since the relation R(n)=n*R(n-1) can have other sequences related to it, for example R(n)=0 for all n.
     
  7. Jul 29, 2010 #6
    Re: 0!

    n! is the number of possible ways to scramble up n objects & there's only one way to scramble up zero objects. It's a bit similar to showing there's only one empty set; if there were another way, what would it look like?
     
    Last edited: Jul 29, 2010
  8. Jul 29, 2010 #7

    arildno

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    Re: 0!

    n! might be INTERPRETED as that, if you like.
     
  9. Jul 29, 2010 #8
    Re: 0!

    ^ that's how I made it make sense to myself anyway
     
  10. Jul 29, 2010 #9

    alxm

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    Re: 0!

    0! = 1 is the value you get from the Gamma function, many series expansions are more compactly expressed if 0! = 1. The number of permutations of an empty set is 1.

    It's simply more convenient for most situations where factorials are used that one defines 0! to be 1.
     
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