non integer factorial
I'm pretty sure that the definition of the factorial operation in mathematics requires an integer. Yet google seems to disagree. I messed up the formatting of a calculation I was using google to compute and it ended up calculating the factorial of an irrational number. I'm just wondering how exactly would one do that?
The gamma function is a generalization of (n-1)! onto the real set. That is, gamma() is one function such that f(n+1)=n! ∀ n∈N and dom f⊃N.
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This is a level of mathematics that my body isn't quite ready for. lol, well at least I know that they weren't pulling crap out of a hat.
well 0! is defined to be equal to one, and if they can do that, they can do anything they want.
0! := 1 for simplicity of notation. It's perfectly fine to give it that value as it isn't defined otherwise.
Yes, you could. The definition of 0! is arbitrary, but not whimsical. 0!=1 was chosen because it has useful properties.
Ahem, useful!=arbitrary. I would stick to the "useful" adjective.
The (Newton) binomial coefficient formula says that
where
is the binomial coefficient, defined to be equal ton!/(i!(n-i)!)<- Bullshit! The *ONE AND TRUE* definition of the binomial coefficient
is x(x-1)...(x-1+i)/i!. The difference is that the one and true definition works for x not being an integer as well.
Now, if you want the (Newton) binomial formula to be correct, the only way to do it is to have 0!=1.
The (Newton) binomial coefficient formula says that
|
|
where
|
|
is the binomial coefficient, defined to be equal to
|
|
is x(x-1)...(x-1+i)/i!. The difference is that the one and true definition works for x not being an integer as well.
Now, if you want the (Newton) binomial formula to be correct, the only way to do it is to have 0!=1.
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| useful!=arbitrary |
| 0!=1 |
I'm sure it wouldn't be impossible to find a different formula that would only make sense if 0!=0. Then what? Sure, they wouldn't both be equally useful, but there would be no logical way to choose one definition over the other.
Still, useful is an English language word, rather than a mathematical term.
"Useful" is what people [edit] kind of[/edit] agree on, not what could be theoretically possible if one were inclined to the dark side.
[Edit:] I finally got the pun in your post! Hahahahaaa cheers mate...
"Useful" is what people [edit] kind of[/edit] agree on, not what could be theoretically possible if one were inclined to the dark side.
[Edit:] I finally got the pun in your post! Hahahahaaa cheers mate...
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