# Formal definition of what the symbol 0!

1. Sep 20, 2004

### abc

hellooooooooooo everybody !!!!!!!!!!!!!!!!!
can anyone plz prove the following :
0! = 1 :surprised :surprised :surprised :surprised

cheers abc

2. Sep 20, 2004

### matt grime

3. Sep 20, 2004

### abc

i really didn't understand that ....... could u plz reexplain it in an easier way ... and i will be so thankful to u
cheers
abc

4. Sep 20, 2004

### matt grime

Well, I guess that means as a FAQ definitive answer it is defective. However I don't think that you've actually spent sufficiently long considering what the definition of factorial is. So, why don't you write what you understand factorial to mean; it may at least improve the faq type answer.

That 0!=1 is pretty much a formal definition of what the symbol 0! means and it is consistent with n! for all positive n.

5. Sep 20, 2004

### abc

dear matt
as u said i have just today studied the factorial at class ..... and when the teacher explained the definition of ......then said that 0!=1 ...... it was weird to me and i didn't have the time to ask him about it ....... so if u would explain ... plz start from the zero ..... so i could understand well
thanx
abc

6. Sep 20, 2004

### matt grime

that doesn't tell me what you think factorial means, in fact it appears that you 've not remembered the definition of n! so if you've not remembered the definition of n! for positive n how can you expect to understand why 0!=1? get your notes from class and look at the definition, and then post it so we can see what your working from.

7. Sep 20, 2004

### abc

do u mean this
n! = n(n-1 ) (n-2 ) .................... 3*2*1

8. Sep 20, 2004

### matt grime

Let's go with that. That doesn't tell us what 0! is, so we can simply declare 0! to be 1, and there is no problem there at all. this is common, and widely accepted. factorials of negative numbers aren't defined. this then allows us to say that n!=n*(n-1)! for all n greater than or equal to 1 (and that 0!=1).

9. Sep 20, 2004

### Tide

You could use either of these arguments:

(1) The factorial tells how many ways there are of arranging N items. You can arrange 5 items in 5! ways. There is exactly one way of arranging NO items or 0! ways.

(2) The factorial is a special case of the gamma function with $x! = \Gamma (x+1)$.

10. Sep 20, 2004

### HallsofIvy

Staff Emeritus
You don't "prove" that 0!= 1, any more than you "prove" that 3!= 3*2*1. That's the definition of 0!. You could ask WHY that is the definition and the best answer is that it's because so many formulas involving n! also work for n= 0 as long as 0! is defined to be 1.