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dstebbins

factorial of a decimal

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When you perform the expression n! (read "n factorial"), you multiply all the positive integers between one and n. They have to be positive, so n>0, but they also have to be integers, so you would think that factorial of a decimal is impossible.

 

However, when I put it into my computer's calculator, I do get a real answer. For example, .01! = 0.99432585119150603713532988870511..., and 2.1! = 2.1976202783924770541835645379483..., so there must be a way to perform a factorial of a decimal.

 

What is it? How do you do that?

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My guess would be the gamma function.

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Gamma Function

 

[math]\Gamma(z)=\lim_{n\rightarrow\infty}\frac{n!n^z}{z(z+1)...(z+n)}[/math]

 

 

 

edit: 4,000th post....oh yea

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This is based entirely from my memory and I'm about to go to bed so CBA to verify:

[math]n!=\int_0^{\infty} x^n e^{-x} dx [/math]

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The gamma function is the extension of the factorial to complex and real numbers.

 

[math] \Gamma(n) = (n-1)![/math]

 

and the Gamma function is defined as:

 

[math]\Gamma(n) = \int_0^{\infty} t^{n-1}e^{-t} dt [/math]

 

You can see a lot more at Mathworld: http://mathworld.wolfram.com/GammaFunction.html

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The gamma function is the extension of the factorial to complex and real numbers.

It is an extension (admittedly the commonly-mentioned one, hence my guess).

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Also, you should note that the Gamma function is not defined for any negative integer (I believe there are poles at each one of those points).

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Can't say I understand what a factorial of a non-natural number would mean.

I always understood that the definition was the amount of ways to order n objects and any formula was merely a way of calculating that. And ways to order 2.5 distinct objects doesn't make much sense to me.

 

Although I guess I could have it very backwards.

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Can't say I understand what a factorial of a non-natural number would mean.

I always understood that the definition was the amount of ways to order n objects and any formula was merely a way of calculating that. And ways to order 2.5 distinct objects doesn't make much sense to me.

 

Although I guess I could have it very backwards.

 

No, you are correct- the "factorial" is only defined for positive itegers. What others are saying here is that the "gamma function" has the property that it is identical to the factorial for positive integer values of the argument but is defined for other numbers as well. (It is NOT defined for negative integers because the integral does not exist in that case.)

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