Formal definition of what the symbol 0

In summary, the concept of 0! (zero factorial) can be confusing, as it seems counterintuitive that 0! = 1. However, this is a formal definition of the symbol and is consistent with the definition of n! for all positive n. The concept of factorial is often used to represent the number of ways to arrange a certain number of items, and thus 0! makes sense as there is exactly one way to arrange no items. This definition also allows for simpler equations and consistency in formulas involving factorial.
  • #1
abc
22
0
hellooooooooooo everybody !
can anyone please prove the following :
0! = 1

cheers abc
 
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  • #3
i really didn't understand that ... could u please reexplain it in an easier way ... and i will be so thankful to u
cheers
abc
 
  • #4
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
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 ... please start from the zero ... so i could understand well
thanx
abc
 
  • #6
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
do u mean this
n! = n(n-1 ) (n-2 ) ...... 3*2*1
 
  • #8
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
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 [itex]x! = \Gamma (x+1)[/itex].
 
  • #10
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.
 

1. What is the formal definition of the symbol 0?

The formal definition of the symbol 0 is a numerical digit that represents the absence of a quantity or value. It is considered a placeholder or a null value in the base 10 number system.

2. How is 0 used in mathematics?

In mathematics, 0 is used as a number to indicate that there is no value or quantity present. It is also used as a starting point for counting and as a reference point for other numbers.

3. Is 0 a positive or negative number?

0 is considered neither positive nor negative. It is a neutral number that falls between the positive and negative numbers on the number line.

4. What is the significance of 0 in algebra?

In algebra, 0 is important because it represents the additive identity, meaning that when added to a number, it does not change the value of the number. It is also used as a placeholder in algebraic equations.

5. Can 0 be divided by any number?

No, 0 cannot be divided by any number. Division by 0 is undefined and not allowed in mathematics. It is considered an indeterminate form and leads to an error in calculations.

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