0! = 1 ???????????????????????????????????????

Question

why?

Answer 1

What kind of math is that?

Answer 2

You cannot reason that x^0 = 1 by thinking of the meaning of powers as
"repeated multiplications" because you cannot multiply x zero times.
Similarly, you cannot reason out 0! just in terms of the meaning of
factorial because you cannot multiply all the numbers from zero down
to 1 to get 1.

Mathematicians *define* x^0 = 1 in order to make the laws of exponents
work even when the exponents can no longer be thought of as repeated
multiplication. For example, (x^3)(x^5) = x^8 because you can add
exponents. In the same way (x^0)(x^2) should be equal to x^2 by
adding exponents. But that means that x^0 must be 1 because when you
multiply x^2 by it, the result is still x^2. Only x^0 = 1 makes sense
here.

In the same way, when thinking about combinations we can derive a
formula for "the number of ways of choosing k things from a collection
of n things." The formula to count out such problems is n!/k!(n-k)!.
For example, the number of handshakes that occur when everybody in a
group of 5 people shakes hands can be computed using n = 5 (five
people) and k = 2 (2 people per handshake) in this formula. (So the
answer is 5!/(2! 3!) = 10).

Now suppose that there are 2 people and "everybody shakes hands with
everybody else." Obviously there is only one handshake. But what
happens if we put n = 2 (2 people) and k = 2 (2 people per handshake)
in the formula? We get 2! / (2! 0!). This is 2/(2 x), where x is the
value of 0!. The fraction reduces to 1/x, which must equal 1 since
there is only 1 handshake. The only value of 0! that makes sense here
is 0! = 1.

And so we define 0! = 1.

Answer 3

In higher level math there is this thing called the gamma function.
It satisfies the following propriety

Gamma(x+1)=x*(gamma(x))

so for example
gamma(2)=1*(gamma(1))

and gamma(3)=2*gamma(2)

if you look at this function you might realize that
gamma(x+1) is x! but this will only work if gamma(1)=0!=1
because 1!=gamma(2)=1*gamma(1)=1
so we say that 0!=1.

The gamma function is really use full in doing stuff like solving partial differential equations. If you don't know what those are don't worry, but you will just have to take my word for it.

Answer 4

4! = 3! * 4 = 1 * 2 * 3 * 4
3! = 2! * 3 = 1 * 2 * 3
2! = 1! * 2 = 1 * 2
1! = 0! * 1 = 1

divide both sides of this last equality to get 0! = 1

Answer 5

! means how many different ways you can arrange something. 0 can be arranged in 1 way, hence 0!=1.

Answer 6

! is a sign used for how many ways you can move the mnumbers around, so you can only move zero around once. Or something like that...

Answer 7

Because you're mucking about with infinities here. Apparently an infinite amount of zeroes will get you to 1. But only if it's an infinite amount. ;-)

Answer 8

Sorry, I'm not good in math