# Homework Help: Proof that 0!=1?

1. Nov 4, 2007

### rock.freak667

Proof that 0!=1??

1. The problem statement, all variables and given/known data
How does one prove that 0!=1, which I've only been told to accept as true.

2. Relevant equations

n!=n(n-1)(n-2)(n-3)*...*3*2*1

3. The attempt at a solution

well using the def'n for n!

0!=0(-1)(-2)(-3)(-4)(-5)*...*3*2*1

which should be zero as....any number by 0 is 0

2. Nov 4, 2007

### arildno

It is a matter of definition. Period.

For n>=1, we have the recursive relationship $$n!=n*(n-1)!$$

3. Nov 4, 2007

### Avodyne

Yes, it's a matter of definition, but this is a useful defintion. Note that arildno's recursion relation can be written as (n-1)!=n!/n. Plugging in n=1 yields 0!=1.

Also, formulas like $e^x = \sum_{n=0}^\infty x^n/n!$ only work if 0!=1.

4. Nov 4, 2007

### rock.freak667

hm...but then the same goes for 1!...as that is the same as 0! by the definition of n!

5. Nov 4, 2007

### ZioX

How many ways can you order 0 things? 1 way!

I had originally thought that you were asking why 0 was not equal to 1. A very interesting question to pose compared to other more tired equalities.

6. Nov 4, 2007

### ZioX

Well, no. Formally these things are defined recursively starting at 1.

7. Nov 4, 2007

### Avodyne

Yes, 1!=1 as well.

Do you know about the gamma function?
$$\Gamma(z) = \int_0^\infty dt\,t^{z-1}\,e^{-t}.$$
This has the interesting property that for an integer n, $\Gamma(n)=(n-1)!$
For more, see http://en.wikipedia.org/wiki/Gamma_function

8. Nov 5, 2007

### rock.freak667

never heard of this gamma function...shall read now

9. Nov 5, 2007

### dynamicsolo

We also define 0! = 1 to provide consistency in the equations for nPr and nCr : when r = 0 or r = n, the formula should give values of 1. This will only be possible if (n-n)! equals 1.

There are a number of such situations in mathematics where an operation originally defined only for positive integers ("counting numbers"), as evolved from ordinary human uses, is extended to larger sets of numbers. A widely-used example is x^n .

10. Nov 5, 2007

### rock.freak667

So then I guess there is no way to prove it but it is just taken as 1 just to make things simpler

11. Nov 5, 2007

### Avodyne

Yes, that's basically it.

12. Nov 5, 2007

### symbolipoint

Actually, there is an algebraic proof. Try a search of physicforums. I know I saw the proof somewhere, but I cannot remember exactly where, and I do not remember this proof myself.

13. Nov 5, 2007

### stewartcs

http://www.jimloy.com/algebra/zero-f.htm [Broken]

Last edited by a moderator: May 3, 2017
14. Nov 6, 2007

### arildno

What do you think a "proof" is?

It is a valid deduction from ACCEPTED AXIOMS OR DEFINITIONS.

Thus, that some statement is an axiom or a definition is NOT some flaw with it as if ideally we would have liked to prove it.

Of course, we have the freedom to choose another set of axioms&definitions than the "standard" set; in that case, axioms in another set might be needed to be proved, and vice versa.

Specifically, if we CHOOSE to define the factorial in terms of the gamma function, then we may prove the statement 0!=1.

15. Nov 6, 2007

### Avodyne

I can't agree with that. We'd like to use the minimal number of axioms. If one can be shown to be derivable from the others, that's an improvement. People spent 2000 years trying to show that Euclid's fifth postulate was derivable before concluding that it couldn't be done.

16. Nov 6, 2007

### arildno

I can agree to that; I should have inserted the word "necesssarily" in "...is NOT necessarily a flaw..", which should cover the instances you mentioned.

17. Jan 11, 2010

### rjoshi1906

Re: Proof that 0!=1??

To Prove:- 0! =1.

Known:
y! = y x (y-1)!

So,
1! = 1 x (1-1)!
1! = 0!
or
0! = 1! -(a)
Proof:
As We know,

n! = n x n-1 x n-2 x n-3 x n-4 x .......... 2 x 1
So,
4! = (3+1)! = 4 x 3! = 4 x 3 x 2 x 1
3! = (2+1)! = 3 x 2! = 3 x 2 x 1
2! = (1+1)! = 2 x 1! = 2 x 1
1! = (0+1)! = 1 x 0! = 1 -(b)

Hence, from (a) and (b)
We get, 0! = 1

18. Jan 11, 2010

### Fightfish

Re: Proof that 0!=1??

Personally I tend to prefer the combinatorial approach. There is just one way to choose 0 objects from a set of n objects, hence:
$$^{n}C_{0} = \frac{n!}{(n-0)!(0!)} = \frac{1}{0!} = 1$$
which thus implies that 0! = 1

19. Jan 11, 2010

### HallsofIvy

Re: Proof that 0!=1??

This is "known" for what values of y? And how is it known? Exactly what definition of factorial are you using?

By exactly the same "proof" then, 0!= 0(0-1)!. But 0 times any number is 0 so whatever (-1)! is, 0!= 0.

20. Jan 12, 2010

### Mentallic

Re: Proof that 0!=1??

Well then couldn't (-1)! simply be undefined to keep the consistency that 0!=1.

In the same way that $$lim_{x \rightarrow 0}\left(x.\frac{1}{x}\right)=1$$you are similarly saying that the first factor x=0 but whatever the next factor 1/x is equal to, the RHS should equal 0 since 0 times any other number is 0. This is not the case however.

21. May 1, 2011

### joaquince

Re: Proof that 0!=1??

f(x)= x^5 = [5!/5!] x5
f(x)(1)= 5x^4 = [5!/4!] x^4
f(x)(2)= 20x^3 = [5!/3!] x^3
f(x)(3)= 60x^2 = [5!/2!] x^2
f(x)(4)= 120x^1 = [5!/1!] x^1
f(x)(5)= 120x^0 = [5!/0!] x^0 [Following the observed pattern… ]

f(x)(k)= n!/((n-k)!) [x^(n-k)] […in general, for coefficient of x^n = 1, and n a positive real integer ]

Since f(x)(5) = 120, we must conclude that 0! = 1

22. Jul 12, 2011

### dragonslayer

Re: Proof that 0!=1??

alright, consider the gamma function:
Γ(z)=∫(t^(z-1)e^(-t)dt, 0, infinity)

which has the property Γ(z+1)=zΓ(z) (too long to prove, look it up),
in turn implying that Γ(z+1)=z!

so 0!=Γ(0+1)=Γ(1)

Γ(1)=∫(t^(1-1)e^(-t)dt, 0, infinity)
=∫(e^(-t)dt, 0, infinity)
= -e^(-t), 0, infinity
=0-(-1)
=1

sorry for the poor formatting, but I hope you get the idea here. The gamma function can also show that (1/2)!=sqrt(pi)

23. Jul 12, 2011

### stefounet

Re: Proof that 0!=1??

Agreed. There are some sets containing X such that X*a=a (X==1) and X+a=a (X==0) for all a. Depending on the definition of '*' and '+', I suppose there could be more than one such set.
So, I don't know what the OP means, really. I mean, if we're talking Peano's integers, then 1!=0 because 1 is the successor of 0. If we're talking real numbers, then those are an extension of the former...

24. Jul 12, 2011

### stefounet

Re: Proof that 0!=1??

lol, I get it now. Not "prove that 0 is different from 1", but "prove that factorial of 0 is 1".
Just refer to the definition of factorial, then, 0! = 1 does not contain ANY information whatsoever.

25. Jul 12, 2011

### stefounet

Re: Proof that 0!=1??

You've been told to accept it as true because it is a convenient convention. Just like any other word, n! means nothing unless we agree on a meaning for it. Well, n! means n*((n-1)!) if n >0 and 0! means 1. People could have agreed that n! means n*((n-1)!) if n >1, that 1! means 1, and that 0! means nothing, but that would be inconvenient in most applications even though both conventions coincide for any n>0.