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Offline Munchor

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Factorials
« on: January 24, 2011, 09:42:18 am »
Hello everybody,

I know the basics of factorials:

5! = 5*4*3*2*1 = 120

However, I do not know about negative numbers and how to use them in calculus and what they are useful for.

I know floats won't work ;D

Can someone tell me some more about them?
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Offline Xeda112358

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Re: Factorials
« Reply #1 on: January 24, 2011, 09:45:33 am »
Negative factorials do not work, apparently. But also keep in mind that 0!=1

*As a side note, I am working on making a way to find things like 3.1! :D That will take a few years of playing with numbers, though.
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Offline AngelFish

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Re: Factorials
« Reply #2 on: January 24, 2011, 11:23:31 am »
Xeda, I'm not sure the factorial operation is defined for decimals. Isn't it just the product of all positive integers less than or equal to a number N?

Even if you do extend the definition to decimals, as I did in below screenshot, it still gives you odd results.

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∂²Ψ    -(2m(V(x)-E)Ψ
---  = -------------
∂x²        ℏ²Ψ

Offline Xeda112358

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Re: Factorials
« Reply #3 on: January 24, 2011, 11:31:48 am »
Ah, but I came up with a way to find the sum of, say, X^2 from 3.1 to 4.6.
(2X^3+3X^2+X)/6
Just plug in 4.6 and 3.1

I've been working with Pascals Triangle and I have methods of defining nCr in terms of sums as opposed to factorials. It is going to be a VERY complicated process, but I am working on a method to find, say 3.1 C 4.2

Believe me, it is a really messed up, convoluted process that doesn't use 3.1*2.1*1.1. Many mathematicians have done similar things and they started with an easy, finite equation and resulted in an infinite equation when they applied decimals.
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Offline Builderboy

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Re: Factorials
« Reply #4 on: January 24, 2011, 12:06:40 pm »
Might I interest you in the Gamma function?
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Offline Xeda112358

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Re: Factorials
« Reply #5 on: January 24, 2011, 12:24:16 pm »
You may...
Also note that I do a lot of things just for fun and so that I can figure it out... I've also done work that leads to the Zeta functions.

So about this Gamma function...
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Offline Builderboy

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Re: Factorials
« Reply #6 on: January 24, 2011, 12:30:54 pm »
The gamma function gives out the same answers as the Factorial function, but works for all Real and Complex numbers :)
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Offline Xeda112358

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Re: Factorials
« Reply #7 on: January 24, 2011, 12:34:16 pm »
ooooh! That sounds very nice! When I get back from my classes I'll have to check out Wolfram... For now, I have to go :'(
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Offline Builderboy

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Re: Factorials
« Reply #8 on: January 24, 2011, 12:35:30 pm »
Just know that Omnicalc has a build in Gamma function :D
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Offline AngelFish

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Re: Factorials
« Reply #9 on: January 24, 2011, 01:40:13 pm »
Oh yeah, I'd forgotten about the Gamma function  :P
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∂²Ψ    -(2m(V(x)-E)Ψ
---  = -------------
∂x²        ℏ²Ψ

Offline Xeda112358

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Re: Factorials
« Reply #10 on: January 30, 2011, 12:18:10 am »
Hmm, Omnicalc has proven quite useful for that kind of thing. I always seem to think of it as a sprite program and whatnot, but then again, I started using it before I appreciated all the math functions.
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Offline phenomist

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Re: Factorials
« Reply #11 on: January 30, 2011, 02:53:44 am »
A small caveat to the gamma function: it still does not take negative integers.
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Offline Xeda112358

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Re: Factorials
« Reply #12 on: January 30, 2011, 02:55:26 am »
Ah, so there is still something for me to figure out :D
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Offline merthsoft

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Re: Factorials
« Reply #13 on: January 30, 2011, 02:58:51 am »
The wikiedpia article on factorials maybe be an enjoyable read if you're in to this sort of stuff. The link I posted is specifically talking about some negative stuff.
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Offline phenomist

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Re: Factorials
« Reply #14 on: January 30, 2011, 03:01:45 am »
Maybe I should have rephrased a bit differently: The gamma function for positive integers follows the following property: gamma(x)=(x-1)!. Hence gamma(1)=0!=1, for instance.

However, what is gamma(0)? It would equal to (-1)!. But using the factorial recurrence formula would give us 0(-1)!=0!=1. In other words, (-1)! = 1/0. This is bad; hence, (-1)! is not defined, and so is gamma(0).

Most other numbers still have a gamma function attached to it, for example gamma(-1/2), gamma(3+4i), and gamma(1337.42i), but zero and negative numbers when plugged into gamma generate an undefined result, because their respective factorials are undefined as well.

EDIT: Aw darn, sniped
« Last Edit: January 30, 2011, 03:02:26 am by phenomist »
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