Can you prove to me now that 0! is equal to 1?

Question

Why is 0 factorial equal to 1? Please prove it.

I know that.. I mean... WHY IS zero factorial 1? That's the question.

Answer 1

You are correct that 0! = 1 for reasons that are similar to why
x^0 = 1. Both are defined that way. But there are reasons for these
definitions; they are not arbitrary.

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.

I hope that helps.

Answer 2

x^0=1 bcuz x^0= x^(n-n)=x^n/x^n=1!!!!!!!!!
Why does 0! = 1 ?
Usually n factorial is defined in the following way:

n! = 1*2*3*...*n
But this definition does not give a value for 0 factorial, so a natural question is: what is the value here of 0! ?

A first way to see that 0! = 1 is by working backward. We know that:

1! = 1
2! = 1!*2
2! = 2
3! = 2!*3
3! = 6
4! = 3!*4
4! = 24


We can turn this around:
4! = 24
3! = 4!/4
3! = 6
2! = 3!/3
2! = 2
1! = 2!/2
1! = 1
0! = 1!/1
0! = 1


In this way a reasonable value for 0! can be found.
How can we fit 0! = 1 into a definition for n! ? Let's rewrite the usual definition with recurrence:

1! = 1
n! = n*(n-1)! for n > 1


Now it is simple to change the definition to include 0! :
0! = 1
n! = n*(n-1)! for n > 0


Why is it important to compute 0! ?
An important application of factorials is the computation of number combinations:

n!
C(n,k) = --------
k!(n-k)!


C(n,k) is the number of combinations you can make of k objects out of a given set of n objects. We see that C(n,0) and C(n,n) should be equal to 1, but they require that 0! be used.
n!
C(n,0) = C(n,n) = ----
n!0!


So 0! = 1 neatly fits what we expect C(n,0) and C(n,n) to be

Answer 3

Absolutely wrong! It was decided long time ago that 0 is a landmark among numbers to establish a basis for somethign that does not exist. It is the only number that if you add, subtract, multiply, divide will not give more than itself or the initial number. Example: 1 x 0 = 0 0/1 = 0 1+0 = 1 1-0 = 1 How is such a number to be equal to 1? No way pal.

Answer 4

0!=1 is taken as an instance of the fact that the product of no numbers at all is 1. This fact for factorials is useful, because

1)the recursive relation (n+1)!=n!x (n+1) works for n = 0;
2)this definition makes many identities in combinatorics valid for zero sizes.
3)In particular, the number of combinations or permutations of an empty set is, clearly, 1.

Answer 5

0 is factorial to 1,.. example x^0=1 because any number multiplied by the power of zero is 1...^^,

Answer 6

0! = 1 for reasons that are similar to why
x^0 = 1. Both are defined that way. But there are reasons for these
definitions; they are not arbitrary.

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.

Answer 7

basically it is to make combinatorics work.

We know that the number of ways to choose k things from a set of n things is n!/k!(n-k)!

So, lets say n=2 and k=2
clearly there is only one way to pick two things from two objects. so 2!/2!0! = 1 or 1/0! = 1
In order for this to make sense, 0! must =1

Answer 8

there is no proof:
by definition, 0! = 1

u can intuitively get an idea from combinatorics:
Q) in how many ways can 0 number of (distinct) people be made to sit in a row?
A) 1 way (there is no one to seat actually! :))
but the combinatorial answer would be 0!, thus
0! = 1

Answer 9

http://www.zero-factorial.com/whatis.html

Answer 10

I think it is defined that way.