maths proof?

Question

does anyone have any good proofs why 0!=1
tried everything i know to to expalin it to my AS class, but a couple of them still think its just a con.

Answer 1

We sometimes see that 0!=1. Why is that? It would seem that we cannot do the multiplications from one "up to" zero. And, if we go backwards (1x0), we get zero.

Experimenting with factorials, we come up with n!=n(n-1)!. For example 17!=17x(16!):

16!=1x2x...x16
17!=(1x2x...x16)x17

That equation (n!=n(n-1)!) just dictated to us where to put the parentheses. By making n=1, we can find 0!:

1!=1(0!)
0!=1

And, it turns out that 0!=1 works very well in many situations (in probability, for example).


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Addendum:

The above proof that 0!=1 is based upon n!=n(n-1)!, which is in turn based upon the definition of factorial. So, it would seem to be a valid proof. But, 0! cannot be defined directly from the definition of factorial. So, mathematicians like to define 0! as 1, without proving it. So, the proof just amounts to a demonstration that defining 0! as 1 is consistent with the definition of factorial.

I received email saying that my proof that 0!=1 is invalid because 0! is a constant, and you cannot solve for a constant. Wrong. I can find 4! (in a number of ways), and 4! is a constant. I can solve for the square root of 7, probably using my calculator, and it is a constant, too. There are numerous ways of finding pi, and it too is a constant.

n! is also the number of permutations (ways of arranging) exactly n things. It makes sense to say that there is one way to arrange zero things. So again, 0!=1.

Answer 2

It's the only value that lets you use the sum of x^k/k! =1+x+x^2/2+... as a power series for e^x. The "0th" term is x^0/0!=1.

Answer 3

The factorial function only works for integer values.
A function which works for all values of the variable, and coincides with factorials at the integers is the gamma function.

GAMMA (n) = (n – 1)!

You'll see from the link below that GAMMA (1) = 1

http://mathworld.wolfram.com/GammaFunction.html

Answer 4

This is a convention, a definition. The reason for definition is it makes things easier in combinatorial analysis and other branches of Math.

Answer 5

n! is the number of ways in which n objects can
be permuted
the permutation of {1,2} is (1,2)(2,1)
=2!
{1,2,3} has six permutations =3!
=6
there is only a single permutation of
zero elements{0},namely 1
therefore,we have the axiom
0!=1

i hope that this helps

Answer 6

So in holding with all of your assumptions, you assert you assume God is actual. that's no longer precisely an evidence. God is purely one in all many imaginary issues. Your info ought to also tutor how magical faeries are actual.

Answer 7

This is due to definition of Gamma function.

gamma(n) = int(0 to inf) [exp(-x) * x^(n-1)]dx
and n! = gamma(n+1)

So,

0! = gamma(n+1) = int(0 to inf) [exp(-x) * x^(1-1)]dx
=>0! = int(0 to inf) [exp(-x) * x^(1-1)]dx
=>0! = int(0 to inf) [exp(-x) * x^(0)]dx
=>0! = int(0 to inf) [exp(-x) ]dx
=>0! = [-exp(-x) ] limit(0 to inf)
=>0! = [-exp(-inf) ] - [-exp(-0) ]
=>0! = [- 0 ] - [-1 ] Since exp(-inf) = 0 and exp(-0) = 1
=>0! = - [-1 ]
=>0! = 1

Thats the argument for 0! = 1.