Formal definition of what the symbol 0

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Discussion Overview

The discussion centers around the formal definition of the factorial of zero, specifically the claim that 0! equals 1. Participants explore the reasoning behind this definition, its implications, and seek clarification on the concept of factorial.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant requests a proof that 0! = 1, indicating a lack of understanding of the concept.
  • Another participant references a previous post that suggests the definition of factorial provides the answer.
  • A participant expresses confusion and asks for a simpler explanation of the definition of factorial and why 0! = 1.
  • It is suggested that 0! = 1 is a formal definition that is consistent with the factorial of positive integers.
  • One participant proposes that the definition of factorial should be revisited to clarify the understanding of 0! = 1.
  • Another participant explains that 0! can be understood as the number of ways to arrange zero items, which is one way to justify the definition.
  • A different perspective is presented, stating that 0! is defined as 1 because it allows for consistency in formulas involving factorials when n = 0.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and acceptance of the definition of 0!. Some agree on the definition being a formal one, while others seek further clarification and express confusion, indicating that the discussion remains unresolved.

Contextual Notes

Participants reference the definition of factorial and its implications but do not fully agree on the clarity of the explanation provided. There is a noted dependence on the understanding of factorial for positive integers to grasp the concept of 0!.

abc
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hellooooooooooo everybody !
can anyone please prove the following :
0! = 1

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

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