Why Does 0! Equal 1? A Puzzling Question Explained

  • Thread starter Euphoriet
  • Start date
In summary: The latter group is obviously smaller.The full permutation group is known as the unrestricted permutations and the smaller group are known as permutations with compact support. In this case the compact support permutations are known as the even permutations.In summary, the reason why zero factorial equals one is because there is one way to scramble up zero things, which is the empty ordering. This is consistent with the definition of factorial and allows for the coefficient of x^n in the binomial expansion to be defined as n choose zero. There are also logical and mathematical justifications for this definition, such as the properties of the Gamma function and the need for consistency in defining the factorial operation. There may be other definitions and perspectives on this topic, such as the idea of constructing sets and
  • #1
Euphoriet
105
0
Why does zero factorial equal one?... I don't get it..

1! = 1 now that makes sense but not 0! = 1

=-/

can someone explain the reason for this to me?
 
Mathematics news on Phys.org
  • #2
by definition... try looking at the integral definition of the factorial, and it should work out...
 
  • #4
Dr. Transport - I'm pretty sure that 0! was defined before anyone kbew about the Gamma function tho'.
 
  • #5
0! = 1 because there's 1 way to scramble up 0 things. There's 1 way to scramble up 1 thing, there are 2 ways to scramble up 2 things (AB and BA), there are 6 ways to scramble up 3 things, etc etc...
 
  • #6
if 0! weren't 1 (which it logicall is since there is one way to order no objecsts - the empty ordering) then you'd have no way of defining n choose zero, which is clearly the coeff of x^n in the expansion of (1+x)^n
 
  • #7
could be, I have been wrong in the past...the ordering arguments are the most logical ones I've seen for the definition.

dt
 
  • #8
I've heard the ordering argument.. but how do you order soemthing that is not there?.. that's my big question...
 
  • #9
Euphoriet said:
I've heard the ordering argument.. but how do you order soemthing that is not there?.. that's my big question...
Well:
{}
{1}
{1,2} {2,1}

and so on certainly makes sense. (This is using n! as the number of bijections A -> A where A is a set with cardinality n).

There's also that we like:
n!=(n-1)! * n
to be true
and that it works out for the binomial formula.

It turns out to be consistenly correct (unlike [tex]n^0=1[/tex]) so that's the way it is.
 
  • #10
jcsd said:
Dr. Transport - I'm pretty sure that 0! was defined before anyone kbew about the Gamma function tho'.

That may or may not be true, Euler discovered (or invented, depending on who you are) both factorial notation and the Gamma function, and I think he would have been the 1st to notice many of the properties of the Gamma function. I'll look it up see what came first.
 
  • #11
Euphoriet said:
I've heard the ordering argument.. but how do you order soemthing that is not there?.. that's my big question...

That's easy- you just leave it alone! :wink:

Another way of looking at it:

5!/4!= 5, 4!/3!= 4, 3!/2= 3, 2!/1!= 2, so we want 1!/0!= 1 which requires that 0!= 1.

Of course we would also "want" 0!/(-1)!= 0 but that is impossible- which is why (-1)! is not defined to be anything.
 
  • #12
It all depends on how we define the factorial operation, doesn't it?

If we define it as the product of all positive integer's up to and including n (ie: n! = 1x2x3x...n), then obviously 0! is not one, by definition. As it happens, we don't define factorial exactly that way...
 
  • #13
DeadWolfe said:
It all depends on how we define the factorial operation, doesn't it?

If we define it as the product of all positive integer's up to and including n (ie: n! = 1x2x3x...n), then obviously 0! is not one, by definition. As it happens, we don't define factorial exactly that way...

Actually, an emtpy product is usually considered to be 1, although undefined is also acceptable.
 
  • #14
I think my answer is best. :)
 
  • #15
I always figured that 0! = 1 because if 0! were to equal 0, then I'm saying that "there is no way to do nothing!" which clearly there's a way to do nothing: do nothing (which is one way). I know this is weird, but this is how I remember it...lol...
 
  • #16
ok, then what about this:

infinity ! = ?

? infinity ! = infinity ?


or


? [ infinity ! = 1(infinity) ] = ( 0! = 1 ) ?
? [ infinity ! = 0(infinity) ] = ( 0! = 1 ) ?
 
Last edited:
  • #17
??
 
  • #18
mikesvenson said:
ok, then what about this:
infinity ! = ?

I'm not sure what you mean by that. By the context of your post, I'm guessing that you're not familiar with cardinal arithmetic.

If we define:
[tex]X ! = \{f: X \rightarrow X s.t. f bij.\}[/tex]
then we get
[tex]|X !|=|X|![/tex]
for finite sets.

In that sense, it's not difficult to show that:
[tex]| \mathbb{N} ! | = | \mathbb{R}|[/tex]

For more esoteric cardinalities, you'll have to work things out for yourself.
 
  • #19
HA, yeah, i don't really know anything about what I'm trying to explain, its been years since I've even taken algebra (which is the highest math I've ever taken)! All i was trying to ask is "how many ways are there to arrange an infinite set of numbers"? infinite?, or just 1? Like how there's only 1 way to arrange an empty set, is there only one way to arrange an infinite set as well? is it impossible to arrange an inifinite set?, since its end is undefinable?
Thats all I am asking, I am out of my league when i try to mathematicaly right it out.
 
  • #20
Mikesvenson,

This may not help much, but there is (or was) a type of mathematician known as a 'constructivist,' and to that type it was anathema to even talk about infinite sets, if I have understood properly. A constructivist would probably feel nauseated at just the thought of rearranging elements in an infinite set.

Anyway, about two-thirds of the way down this page belonging to John Baez, you will find a mention of constructivists:

http://math.ucr.edu/home/baez/topos.html

The relevant quote is, "Suppose you're a constructivist and you only want to work with 'effectively constructible' sets and 'effectively computable' functions. Then you want to work in the 'effective topos' developed by Martin Hyland.

ADDED: Maybe even more to your point is another item in the same Baez page: "Suppose you're a finitist and you only want to work with finite sets and functions between them. Then you want to work in the topos FinSet."
 
Last edited:
  • #21
Another way of permuting an infinite set that is useful is to use things with compact support.

In the main example of N the natural numbers there are two permutation groups, the full perms and the ones that only swap a finite set of numbers.
 

What is 0 factorial and why does it equal 1?

0 factorial, denoted as 0!, is a mathematical concept used in permutations and combinations. It represents the number of ways to arrange 0 objects, which is only 1 way. Therefore, 0! is defined as equal to 1.

How can 0! equal 1 when factorial values are supposed to increase?

Factorial values do typically increase, but at 0!, the pattern breaks. This is because 0! represents the number of ways to arrange 0 objects, which is only 1 way. It is a special case and does not follow the same pattern as other factorial values.

Why is 0! necessary in mathematical calculations?

0! may seem unnecessary since it always equals 1, but it is actually a crucial part of mathematical calculations, particularly in combinatorics and probability. It simplifies equations and makes them more efficient to solve.

Can 0! ever equal a number other than 1?

No, 0! will always equal 1. This is because the concept of factorial is defined in a way that makes 0! equal to 1. It represents the number of ways to arrange 0 objects, which is only 1 way.

Is there a real-world application for 0! equaling 1?

Yes, there are several real-world applications of 0! equaling 1. It is used in various fields such as physics, computer science, and economics. For example, it is used to calculate the number of ways to arrange 0 elements in a set, the probability of an event with 0 outcomes occurring, and the number of ways to choose 0 objects from a larger set.

Similar threads

  • General Math
Replies
2
Views
763
Replies
55
Views
3K
  • General Math
2
Replies
47
Views
2K
  • General Math
2
Replies
66
Views
4K
Replies
2
Views
1K
  • General Math
Replies
31
Views
1K
  • General Math
Replies
7
Views
428
  • General Math
2
Replies
45
Views
3K
  • General Math
Replies
1
Views
1K
Back
Top