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

• johncena
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.
johncena
what is the proof for the statement 0! = 1??

a number neither odd nor even cannot be equal to an odd number.

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.

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.

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?

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.

Like others said, this is by definition. You might be interested in the http://en.wikipedia.org/wiki/Gamma_function" .

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I'll also note a definition can never be wrong. It may be useless, but it's never wrong.

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.
$$C_{k}^{n}$$

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:

$$C_0^n=\frac{n!}{(n-0)!0!}$$ 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.

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:

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!

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)

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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