Is the statement 0! = 1 actually wrong or just ill-defined?

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In summary: I could be wrong on this, but that's my intuition.In summary, the proof for the statement 0! = 1 is based on the definition of factorial, which states that 0! = 1 by recursive definition. This is consistent with the Peano Axioms and the properties of parity. While there may be alternative approaches or motivations for this definition, it is ultimately a matter of convention and cannot be considered wrong or incorrect.
  • #1
johncena
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what is the proof for the statement 0! = 1??
 
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  • #2
a number neither odd nor even cannot be equal to an odd number.
 
  • #3
monty37 said:
a number neither odd nor even cannot be equal to an odd number.
But zero is an even number since it has a parity of 0.
 
  • #4
Assuming you mean the factorial of 0, then factorial is usually defined recursively by,
0! = 1
n! = (n-1)! * n for n > 0
So it's true by the definition of the factorial. If you mean why 0 doesn't equal 1 then you have to state explicitly some formal properties of the integers. For instance a popular way to describe the non-negative integers is the Peano Axioms which among other things state that 0 is a non-negative integer, there is no natural number whose successor is 0 and 1 is defined as the successor to 0. Hence if 0 = 1 then 0 would be the successor to 0 which contradicts the axiom that 0 isn't the successor of any non-negative number.

Alternatively if you are allowed to use properties like parity, and the fact that 0 and 1 have different parity, then they can't be equal because parity is uniquely determined. Note: 0 has even parity while 1 has odd parity; 0 is NOT neither odd nor even.
 
  • #5
gunch said:
n! = (n-1)! * n for n > 0

you said n! = (n-1)! * n for n > 0
so taking n = 1,
1! = (1-1)! * 1 = 1
0! * 1 = 1
thus, 0! = 1/1 = 1
is this proof correct?
 
  • #6
There is no proof, it's by definition. It gives a basis for a recursive definition of n! as n! = n*(n-1)! and 0! = 1.
 
  • #7
Like others said, this is by definition. You might be interested in the http://en.wikipedia.org/wiki/Gamma_function" .
 
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  • #8
I'll also note a definition can never be wrong. It may be useless, but it's never wrong.
 
  • #9
A motivation(maybe) behind defining 0!=1, might be if we look at the combinations of class k taken from a set of n elements.
[tex]C_{k}^{n}[/tex]


THen, since this is nothing else but the set of all subsets of k elements taken from a set of n elements, if we have:

[tex]C_0^n=\frac{n!}{(n-0)!0!}[/tex] then since there is only one set that contains 0 elements taken from any set of n elements (the empty set), it follows that

C_0^n should equal 1, for this to happen 0! should be 1.
 
  • #10
Tac-Tics said:
I'll also note a definition can never be wrong. It may be useless, but it's never wrong.

... unless a useless definition is defined to be something that is incorrect or wrong. :tongue:
 
  • #11
derek e said:
... unless a useless definition is defined to be something that is incorrect or wrong. :tongue:
But that would be a useless definition!
 
  • #12
derek e said:
... unless a useless definition is defined to be something that is incorrect or wrong. :tongue:
A definition cannot be incorrect or wrong. What a group of definitions can be is inconsistent, which is subtly different :) Determining if a set of axioms is consistent is a difficult problem (and consistency is the cornerstone for godel's theorem as with an inconsistent set of axioms you can prove stupid things like 0=1, 1=2, etc)
 
  • #13
I was kinda playing. But what I think is more subtle is the use of the words "wrong" and "incorrect." If one were trying to make a definition of something containing the essence of an idea, such as curvature, then I could see how some definitions can be considered wrong or incorrect. Something being ill-defined often carries connotations of incorrectness or inconsistency, as its name implies. However, stating that the truth/validity in the defining of definition X is false is something I find somewhat meaningless.
 
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What is factorial and what is 0 factorial?

Factorial is a mathematical operation denoted by an exclamation mark (!) and is used to find the product of a number and all the positive integers that are smaller than it. 0 factorial (0!) is defined as 1.

How can 0! equal to 1 when there are no numbers being multiplied?

While it may seem counterintuitive, the definition of factorial (n!) includes the number 1 as a factor. Therefore, when n = 0, 0! is equal to 1.

What is the mathematical proof that shows 0! = 1?

The proof for 0! = 1 is based on the mathematical principle of induction. By using the definition of factorial, it can be shown that 1! = 1, and then using the induction step, it can be shown that (n+1)! = (n+1) * n! for all positive integers n. Substituting n = 0 in this equation gives 1 = (0+1) * 0!, which simplifies to 1 = 1 * 0!. Therefore, 0! = 1.

Is there a real-world application for the concept of 0 factorial?

Yes, the concept of 0 factorial is used in various areas of mathematics and science, such as in the binomial theorem, combinatorics, and probability calculations. It also has applications in computer science and physics.

Are there any exceptions to the rule that 0! = 1?

No, the definition of factorial and the mathematical proof both show that 0! = 1. Therefore, there are no exceptions to this rule.

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