Re: What is zero divided by zero anyway?

The fact that 0! == 1 is a logical extension of the definition of factorial: n! == n * (n - 1)!.

    1 == 1! == 1 * (1 - 1)! == 1 * 0! == 0!
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It's not an arbitrary value, it's a logical one.

But we can't do so for 0/0. Consider:

        0 
    lim - == 0
   x->0 x 
 
but,

        x
    lim - == 1
   x->0 x

It would also mean the function x / 0 is undefined, except for the case x == 0.

Abigail

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Re: Re: What is zero divided by zero anyway? by John M. Dlugosz (Monsignor) on Oct 11, 2002 at 19:00 UTC
The fact that 0! == 1 is a logical extension of the definition of factorial: n! == n (n - 1)!.* So 0! == 0 * (-1)!, 1 == 0*(-1)! What is the value of (-1)! that when multiplied by 0 gives 1?	[reply]
Re: What is zero divided by zero anyway? by Abigail-II (Bishop) on Oct 14, 2002 at 11:35 UTC
The Gamma function is an extension of the factorial function to real and complex arguments. Gamma (n) == (n - 1)! Hence, (-1)! == Gamma (0). But for non-positive integers, the Gamma function isn't analytic (it is elsewhere). http://mathworld.wolfram.com/GammaFunction.html. But there are other extensions, like the Roman Factorial, which would give 1 as value for -1 http://mathworld.wolfram.com/RomanFactorial.html. Abigail	[reply]