0!

[Math Is Hard]

Why is it that zero factorial is equal to 1?

This came up in class tonight (just sort of as a side-note) and nobody knew.

Thanks,

M.I.H.

[cookiemonster]

I shouldn't stick my nose in here... I'm just gonna screw it up again. =\

So I'm just going to refer you to the Gamma Function!

Gamma Function:
http://mathworld.wolfram.com/GammaFunction.html

cookiemonster

[outandbeyond2004]

It's just a definition, to simplify certain mathematical procedures.

[phoenixthoth]

3!=6
2!=2
1!=1
let x=0!

note that 3!/2!=3
note that 2!/1!=2
then 1!/0!=1 to keep the same pattern.

therefore, 1!/x=1. if you solve for x, you get x=1.

therefore, 0!=1.

[matt grime]

THe gamma function isn't analytic at 0, and by pheonixthoth's reasoning the factorial of all negative numbers is 1, and that doesn't hold. (It would in particular create issues with Pascal's triangle etc).

Think of it this way, if you have 0 objects, there is exactly 1 way to arrange them - the empty ordering.

[cookiemonster]

Aha! I never said anything about it! Doesn't count in my mistakes today!

And wouldn't you have to evaluate the Gamma Function at 1 in order to find 0! ?

cookiemonster

[HallsofIvy]

"by pheonixthoth's reasoning the factorial of all negative numbers is 1, and that doesn't hold. "

I don't see that. pheonixthoth's "reasoning" is that 4!/3!= 4,
3!/2!= 3, 2!/1!= 2 (and in general (n+1)!/n!= n+1 for n any positive integer) so we should have 1!/0!= 1 and therefore 0!= 1!= 1. Extending that one more time, we would have 0!/(-1)!= 0 or 1= 0(-1)! which tells us that (-1)! does not exist.

[matt grime]

Yes, sorry, another mistake. never do maths before breakfast.

[Damned charming :)]

If 0! =1 we can say
nCr = n!/((n-r)!*r!)

if 0! was something else we would have to say
nCr = n!/((n-r)!*r!)
except for nCn = nC0=1.

I know what I would rather say.


Also can't you use the rule that
n! = \Gamma(n+1)

to get
0! = \Gamma(1)=1

As far as I can tell \Gamma(x) goes straight to hell
at x= 0 but \Gamma(1) =1 since
\int_0^{\infty} e^{-t} dt = 1

[matt grime]

yep it has a pole there with residue 1/0!, of all things. it has poles at all non-postive integers -k with residue (-1)^k/k! or something.

[Math Is Hard]

Thank you for the insight.
Cookie, I appreciate that link.

[Organic]

http://mathforum.org/dr.math/faq/faq.0factorial.html

[Math Is Hard]

Awesome - thanks! I am sending this link to my math prof.

[matt grime]

That link just reiterates what Phoenixthoth (and others) said (and what I misunderstood) and contains some frankly dubious claims: to say that because the gamma function has poles at negative integers, factorials do not exist for negative integers is slightly misleading. The square root is defined (naively) only for positive numbers, that doesn't stop us saying i = sqrt(-1). You'd need to show there was no analytic function that agreed with the gamma function on the integers, and always had poles at the negative integers. Remember this is just a generalization - another function wouldn't satisfy all of the functional equations the gamma function does. If you can 'define' "i!" why can't you 'define' "-1!"?

[DrMatrix]

If we define n! as the product of all numbers from 1 to n, then n! is only defined for integral values of n such that n > 0. We observe that for all n > 0 we have:
(n+1)! = n! (n+1)

If we are to extend the defintion of factorial to include 0!, we would like the above to apply. Substituting we have:
1! = 0! (0+1) = 0!
This allows us to (re)define factorial thus:

0! = 1
n! = (n-1)! n for n>0

[Hessam]

ooh! this was a topic in one of my math classes.... i thought simply it was because of the formulas for log's.... w/ 0, it would result in a divide by zero, thus they just threw an exception for log(0) to = 1..., anyone?

[Chen]

Actually log(0) is undefined, Hessam...