Why is 0!=1?

[AvgStudent]

Hi, I'm looking for an explanation of why 0!=1.
Through my preliminary research, I found that the main reason mathematicians define 0!=1 is to be consistent with the rest of mathematics. Though, it is a somewhat unsatisfying answer. If mathematicians can define 0!=1, can I define 0! as something else?

Follow up question: why can't you have negative factorials?

I hope you guys can enlighten me. Thanks.

 

[Deleted member 4993]

Hi, I'm looking for an explanation of why 0!=1.
Through my preliminary research, I found that the main reason mathematicians define 0!=1 is to be consistent with the rest of mathematics. Though, it is a somewhat unsatisfying answer. If mathematicians can define 0!=1, can I define 0! as something else?

Follow up question: why can't you have negative factorials?

I hope you guys can enlighten me. Thanks.

It comes from permutation/combination. Have you studied those operations permutation/combination.

Have you studied those two topics? Have you asked Googol "why is 0! = 1" ?

 

[AvgStudent]

It comes from permutation/combination. Have you studied those operations permutation/combination.

Have you studied those two topics? Have you asked Googol "why is 0! = 1" ?

Yes, I have learned combination and permutation. More specifically, nCr if that's what you're referring to. I see that setting 0!=1 makes the equation works, but is there a more profounding reason?

 

[Dr.Peterson]

Hi, I'm looking for an explanation of why 0!=1.
Through my preliminary research, I found that the main reason mathematicians define 0!=1 is to be consistent with the rest of mathematics. Though, it is a somewhat unsatisfying answer. If mathematicians can define 0!=1, can I define 0! as something else?

Follow up question: why can't you have negative factorials?

I hope you guys can enlighten me. Thanks.

The key is in the details. Surely you found some mention of what those potential inconsistencies are! What are they? Which part of the rest of mathematics would be messed up?

Then, if you defined 0! as, say, 0, or 2, what specifically would go wrong? This is what you need to look into if you want to feel satisfied.

On the other hand, didn't you find any other approaches to the question, such as definitions of the factorial from which this answer can be derived, or explanations of what the factorial means in terms of permutations or combinations? Show us what you actually found.

After that, we can look into "negative factorials" (probably you mean "factorials of negative numbers"). That may come directly from one of the answers to your main question.

 

[blamocur]

Hi, I'm looking for an explanation of why 0!=1.
Through my preliminary research, I found that the main reason mathematicians define 0!=1 is to be consistent with the rest of mathematics. Though, it is a somewhat unsatisfying answer. If mathematicians can define 0!=1, can I define 0! as something else?

Follow up question: why can't you have negative factorials?

I hope you guys can enlighten me. Thanks.

I don't know how to define factorials of negative numbers. Disappointingly, the gamma function (https://en.wikipedia.org/wiki/Gamma_function) extends the definition of factorials to pretty much everything except negative integers

 

[Deleted member 4993]

Follow up question: why can't you have negative factorials?

Where would the factorial of negative number be needed?

I have not seen any for defining factorial of negative number.

Please tell us - show us - the need for calculating factorial of negative numbers

 

[AvgStudent]

The key is in the details. Surely you found some mention of what those potential inconsistencies are! What are they? Which part of the rest of mathematics would be messed up?

Then, if you defined 0! as, say, 0, or 2, what specifically would go wrong? This is what you need to look into if you want to feel satisfied.

On the other hand, didn't you find any other approaches to the question, such as definitions of the factorial from which this answer can be derived, or explanations of what the factorial means in terms of permutations or combinations? Show us what you actually found.

After that, we can look into "negative factorials" (probably you mean "factorials of negative numbers"). That may come directly from one of the answers to your main question.

Hi @Dr.Peterson again, I found the following arguments:

1. "There's only one way to arrange zero objects". I don't entirely agree with this argument because, for example, you have the set S={a,b}. There are two ways to arrange the elements, namely ab and ba. However, if S is an empty set (no elements). Why does the number of ways to arrange an empty has to be 1, not 0 or infinity?
2. The second argument justifies setting 0!=1 to make binomial coefficient (nCr) works. Sure, it makes the equation works, but does it make ALL equations work?
3. The third argument I found on chilimath.com is: If n!=n(n-1)! then 1!=1(0!) => 0!=1. My issue with this argument is that with the definition of factorial. It is defined as: "the product of all positive integers from 1 to n". I don't think it tells us anything about n<1, i.e. 0 or negatives.

With these three arguments so far, I see that it's convenient to have 0!=1, but not necessary hence why question can I define my own 0! ?

Where would the factorial of negative number be needed?

I have not seen any for defining factorial of negative number.

Please tell us - show us - the need for calculating factorial of negative numbers

I haven't seen where factorial of a negative would be needed, but it's just an entertaining thought. Much like mathematicians entertained the idea of the square root of negative numbers, maybe it'll lead to something useful.

 

[Deleted member 4993]

I haven't seen where factorial of a negative would be needed, but it's just an entertaining thought. Much like mathematicians entertained the idea of the square root of negative numbers, maybe it'll lead to something useful.

That is a noble thought and the door is wide open. Please feel free to develop the theory/practice of negative factorials for future use.

 

[Dr.Peterson]

Yes, I have learned combination and permutation. More specifically, nCr if that's what you're referring to. I see that setting 0!=1 makes the equation works, but is there a more profound reason?

The fact that C(n,0) and C(n,n) make sense if 0! = 0 is probably the consistency argument you referred to. It's also true that P(n,n) = n! makes sense this way, though not quite as obviously; there is only one way to arrange an empty set.

Here's another argument:

We can define the factorial recursively:

1! = 1​

n! = n*(n-1)! for n > 1​

If we solve that second line for (n-1)!, we get

(n-1)! = n!/n​

and letting n = 1,

0! = (1-1)! = 1!/1 = 1/1 = 1​

So although the basic definition doesn't apply to 0!, it is easy to extend it.

But if we try to extend it to a negative number, it doesn't work:

(-1)! = (0-1)! = 0!/0 = 1/0 is undefined​

(And if you look up the gamma function, you'll see that that, which is the natural way to extend to non-integers, gives the same result for negative integers.)

Anyway, this leads to a better definition, that does include 0:

0! = 1​

n! = n*(n-1)! for n > 0​

Does that work better for you? This is the alternative argument I suggested you would find in your search.

 

[Dr.Peterson]

Hi @Dr.Peterson again, I found the following arguments:

1. "There's only one way to arrange zero objects". I don't entirely agree with this argument because, for example, you have the set S={a,b}. There are two ways to arrange the elements, namely ab and ba. However, if S is an empty set (no elements). Why does the number of ways to arrange an empty has to be 1, not 0 or infinity?
2. The second argument justifies setting 0!=1 to make binomial coefficient (nCr) works. Sure, it makes the equation works, but does it make ALL equations work?
3. The third argument I found on chilimath.com is: If n!=n(n-1)! then 1!=1(0!) => 0!=1. My issue with this argument is that with the definition of factorial. It is defined as: "the product of all positive integers from 1 to n". I don't think it tells us anything about n<1, i.e. 0 or negatives.

With these three arguments so far, I see that it's convenient to have 0!=1, but not necessary hence why question can I define my own 0! ?

What I just posted was what I wrote before you sent this; the site had frozen up on me so I couldn't post until now.

Ultimately, this is a definition, which in principle is something we do on the basis of convenience (because a word for some particular group of ideas is useful). The basic definition you refer to can only apply to positive integers, so it has to be extended to a new definition if you want to apply it elsewhere. When we extend a definition, there should be a reason for doing so -- and the definition 0! = 1, as you've seen, is useful for several reasons.

What would be the reason for your imagined definition? Would you have any better reason to call it 0, or 2, or whatever? You haven't answered that question.

As to specifics:

1. This is not a particularly strong argument, I agree; but I'd say 0 and 1 are the only reasonable possibilities.
2. What other applications do you have in mind that might not work? And why would you be satisfied with combination formulas that don't work? That one thing seems like a pretty strong argument to me.
3. You're right. That's not an argument against extending the definition; it's the reason we need to! You aren't really answering the argument at all.

 

[AvgStudent]

What I just posted was what I wrote before you sent this; the site had frozen up on me so I couldn't post until now.

Ultimately, this is a definition, which in principle is something we do on the basis of convenience (because a word for some particular group of ideas is useful). The basic definition you refer to can only apply to positive integers, so it has to be extended to a new definition if you want to apply it elsewhere. When we extend a definition, there should be a reason for doing so -- and the definition 0! = 1, as you've seen, is useful for several reasons.

What would be the reason for your imagined definition? Would you have any better reason to call it 0, or 2, or whatever? You haven't answered that question.

As to specifics:

1. This is not a particularly strong argument, I agree; but I'd say 0 and 1 are the only reasonable possibilities.
2. What other applications do you have in mind that might not work? And why would you be satisfied with combination formulas that don't work? That one thing seems like a pretty strong argument to me.
3. You're right. That's not an argument against extending the definition; it's the reason we need to! You aren't really answering the argument at all.

Through your "extension of the definition" explanation, I think it's something I can live with.
To answer your question, I have no specifics in mind or any situation where I would need to redefine 0!. It's just a thought if we defined 0!=1 for convenience, then can someone refine it? I know now the answer is yes, but I don't have a better reason or the need to redefine it yet.
Until then, 0!=1

 

[Dr.Peterson]

To answer your question, I have no specifics in mind or any situation where I would need to redefine 0!. It's just a thought if we defined 0!=1 for convenience, then can someone refine it? I know now the answer is yes, but I don't have a better reason or the need to redefine it yet.
Until then, 0!=1

Well, if you had the authority to redefine 0!, and did so, you'd have a lot of people angry with you (and telling you that "convenience" doesn't begin to express what the existing definition provides). You'd better have a really good reason!

 

[blamocur]

I haven't seen where factorial of a negative would be needed, but it's just an entertaining thought. Much like mathematicians entertained the idea of the square root of negative numbers, maybe it'll lead to something useful.

Interesting comparison of factorials and square roots for negative numbers. But I see an important difference there: square -- and other -- roots are already defined for anything for which integer powers are defined. Thus there is nothing arbitrary in the definition of the imaginary unit [imath]i[/imath].

I agree with Subhotosh about the nobility of building extensions, even if I am not sure that extending factorials to negative numbers would be a fruitful exercise. But extending square roots is one of the most fundamental events in math. Complex numbers alone are hugely important, but the idea was generalized even further to extending other fields using irreducible polynomials (just like the field of real numbers is extended using [imath]x^2+1[/imath]).

 

[JasonTurner]

What an interesting question.
A number's factorial is the function that multiplies it by each natural number below it. Factorial can be symbolized by the letter "!". The product of the first n natural numbers is n factorial, which is written as n!
In mathematics, the product of all positive numbers smaller than or equal to a number is called the factorial.
However, because there are no positive values smaller than zero, the data set cannot be sorted, which qualifies as one of the available ways to arrange data (it cannot).
As a result, 0! = 1.
An exclamation mark indicates the factorial of a number. Because the factorial of a number only works with natural numbers, 0 is not included. Any factorial is multiplied all the way down to one, not zero. The first reason zero factorial is equal to one is that this is what the definition specifies, which is a mathematically true explanation (if a somewhat unsatisfying one). Still, keep in mind that a factorial is defined as the product of all integers with values equal to or less than the original number—in other words, a factorial is the number of feasible combinations with numbers less than or equal to that number. Because zero contains no numbers smaller than it yet is still a number in and of itself, there is only one feasible arrangement of that data set: it cannot. Factorials are most commonly employed to solve permutations and combinations. A zero factorial is a mathematical term that equals one for the number of ways to organize a data set with no values. In general, a number's factorial is a shorthand way of writing a multiplication expression in which the number is multiplied by each number smaller than it but higher than zero.