Why does 0! equal 1?

Because it is defined that way. It happens to be convenient in most cases where factorials are used.

it''s a convention - it makes some formulas nicer
e.g. you can write the taylor series as a sum starting with index 0 without making a special case for the first element.
and it''s consistent to the analytic extension of the factorial.

Also you can define factorial as the number of permutations of a given set i.e. there is 1 way you can have the NULL set, one order of the set { 1 }, two ways for the set { 1, 2 ] etc..

You''d also get a lot of divide by zero issues if 0! = 0. I could be wrong, but i remember it being somehow related to the fact that x⁰ = 1

Because i! = (i - 1)! * i, so if 0! is defined, it must be 1, or else, this property wouldn''t work (put i = 1 to see it). (-1)! can''t be defined consistenly with this logic, so it''s not.

And, of course, it''s very convenient to define 0! in numerous problems. I wrote this just to show that 0! couldn''t be anything else than 1, logically.

Cédric

Another explanation is that factorials can be defined via the gamma function

(x-1)! = Γ(x) = ₀∫^∞ t^x-1 e^-t dt

If you substitute a natural number n in that, you get what you'd expect n! = n*(n-1)* ... * 2*1

Furthermore this defines a factorial for any integer, real number or complex number. If you stick x = 1 in it you get 0! = 1

[edited by - sQuid on March 20, 2003 10:17:45 PM]

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Furthermore this defines a factorial for any integer, real number or complex number. If you stick x = 1 in it you get 0! = 1

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Not exactly every real number. The Gamma function, gf(x), is only defined for x>0.

regards

/Mankind gave birth to God.

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Original post by silvren
Not exactly every real number. The Gamma function, gf(x), is only defined for x>0.

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good point. thanks

Nope, gamma is defined over the reals and complex plane, except for negative integral values. Check out wolfram: http://mathworld.wolfram.com/GammaFunction.html

Brendan

Also, there is one way to take no objects by taking nothing! That''s why 0! is 1