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 January 23rd, 2008, 06:16 AM #1 Newbie   Joined: Sep 2007 Posts: 24 Thanks: 0 zero factorial (why ??) Hello Why 0! has been defined as 1 ?? Why is it so ???
 January 23rd, 2008, 07:18 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms By analogy: 10! = 10 * 9! 1! = 1 * 0! From a physical definition: n! is the number of ways to order n objects, and there is 1 way to order 0 elements. To make combinatorial identities hold -- there are a lot that depend on 0! = 1 that would have to be rewritten if it were anything else.
 January 23rd, 2008, 06:09 PM #3 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Since it's a definition, it can be whatever we feel like defining it to be. It turns out that 1 is the most useful. Don't you love math: Anything is possible, as long as your axioms are consistent!
January 29th, 2008, 01:53 PM   #4
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Re: zero factorial (why ??)

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 Originally Posted by sangfroid Hello Why 0! has been defined as 1 ?? Why is it so ???
Since 1!=1 and in general n!=n*(n-1)!, 0! has to be 1.

February 1st, 2008, 12:12 PM   #5
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Quote:
 Originally Posted by CRGreathouse By analogy: 10! = 10 * 9! 1! = 1 * 0! From a physical definition: n! is the number of ways to order n objects, and there is 1 way to order 0 elements. To make combinatorial identities hold -- there are a lot that depend on 0! = 1 that would have to be rewritten if it were anything else.
Quote:
 Originally Posted by cknapp Since it's a definition, it can be whatever we feel like defining it to be. It turns out that 1 is the most useful. Don't you love math: Anything is possible, as long as your axioms are consistent!
Games and trivia! See if you can spot the formalist and the Platonist in the quoted posts.

 February 1st, 2008, 02:27 PM #6 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Yay! Return of the fun & games section! Although I think it was previously called "games and trivia"
February 4th, 2008, 04:43 AM   #7
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 Originally Posted by cknapp Yay! Return of the fun & games section! Although I think it was previously called "games and trivia"
:hand motion: These are not the droids you are looking for.

 February 4th, 2008, 06:44 AM #8 Senior Member   Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 I find your lack of original conversation disturbing! With the joke on me for quoting a comic about over-use of quotation.
 September 15th, 2008, 04:28 AM #9 Newbie   Joined: Sep 2008 Posts: 2 Thanks: 0 Re: zero factorial (why ??) The reason for choosing 0!=1 is not base on some personal choice or just because it works; it a based on a fact that most mathematicians overlook. The fact is that, unlike any other number, zero exist in a multidimensional state. It is dimensionless. This means that the factorial of every other number is defined to be dimension dependent; it exist in only one plane. But the number of ways zero can be arranged in any plane is just one. This theory I believe has not been brought up by any mathematician yet. I'm still working of the formal prove that axiom seem to be inexplicable just because they fall into multidimensional arenas. I hope others will take it up too.

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