Math Is Fun Forum
You are not logged in.
- Topics: Active | Unanswered
#1 2008-06-21 08:36:32
- ZHero
- Real Member
- Registered: 2008-06-08
- Posts: 1,889
0!
All of us know that 0! is 1 but i just am eager enough to know "how"? Also, does 0! really "EXIST"? Means, if it does then why not the factorials of -ve numbers?
How can one "DEFINE" it??
Last edited by ZHero (2008-06-21 08:39:56)
If two or more thoughts intersect, there has to be a point!
Offline
#2 2008-06-21 09:01:56
- mathsyperson
- Moderator
- Registered: 2005-06-22
- Posts: 4,900
Re: 0!
When thinking about combinations and permutations, it makes sense that 0! = 1.
The number of way of choosing r objects from a choice of n is nCr, or
If we use 0!=1, then the number of ways of picking no objects turns out to be
This makes sense, because there's only one way of picking no objects and that's to not pick anything.
Another way of rationalising it is by considering this pattern:
3! = 4!/4
2! = 3!/3
1! = 2!/2
∴ 0! = 1!/1 = 1/1 = 1.
If you try to continue that to the negatives, then you'd get that (-1)! = 0!/0, which isn't allowed.
All factorials of other negative integers involve (-1)! in their definition (possibly very indirectly), and so no negative integer has a factorial.
You might be interested in the Gamma function, which is an "expansion" of factorials.
This relates to factorials because:
The Gamma function of 1 is equal to 1, giving another argument to why 0! = 1.
Why did the vector cross the road?
It wanted to be normal.
Offline
#3 2008-06-21 10:41:19
- ZHero
- Real Member
- Registered: 2008-06-08
- Posts: 1,889
Re: 0!
Oh! That was such a beautiful explanation 'mathsyperson'! I bet i could not have known this from anywhere else!
This definitely makes Great Sense!
Many a Thanks!! 
If two or more thoughts intersect, there has to be a point!
Offline
#4 2008-06-21 11:38:06
- MathsIsFun
- Administrator
- Registered: 2005-01-21
- Posts: 7,711
Re: 0!
mathsy: you should write pages for mathsisfun.com, that was pretty good.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
Offline
#5 2008-06-21 19:21:49
- luca-deltodesco
- Member
- Registered: 2006-05-05
- Posts: 1,470
Re: 0!
also note that by using the gamma function you can define the factorial for any non-negative-real-integer over the whole complex domain.
http://mathworld.wolfram.com/images/eps-gif/Factorial_1000.gif
http://mathworld.wolfram.com/images/interactive/FactorialReImAbs.gif
Last edited by luca-deltodesco (2008-06-21 19:23:32)
The Beginning Of All Things To End.
The End Of All Things To Come.
Offline