# Why is the factorial of 0 equal to 1?

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Jamil Kasan 2016년 10월 4일
답변: Abhishek Jain 2016년 10월 18일
Can someone explain here the above question. If possible can you prove it via Matlab code? ;)

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### 답변(6개)

Thorsten 2016년 10월 4일
"In mathematics, an empty product, or nullary product, is the result of multiplying no
factors. It is by convention equal to the multiplicative identity 1."
0! is such an empty product. You cannot prove it using Matlab, it's a mathematical convention.
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José-Luis 2016년 10월 4일
편집: José-Luis 2016년 10월 4일
And since it sorts of boils down to it being defined as 1, then the code TMW uses for computing it follows that definition.
edit factorial
at the command line to see for yourself.
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Jamil Kasan 2016년 10월 7일
No, I know the answer but I just liked to see for many perceptions for different persons

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gubertoli 2016년 10월 4일
You might go to definition of factorial:
n! = n*(n-1)! for n>0
Calculating for n = 1, to prove 0! = 1 1! = 1*(1-1)! //by def. 1! = 1 1 = 0! 0! = 1
Considering definitions, you can implement:
function fac=fact(n)
fac = 1
if (n==1)
fac = 1
elseif n >= 1
fac = n*fact(n-1)
else
fac = 1
end
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Jamil Kasan 2016년 10월 7일
I thought no one would answer this (y)

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ayyappa rudrasimha yedida 2016년 10월 7일
편집: ayyappa rudrasimha yedida 2016년 10월 7일
Simple program:
n = 0;
f = 1;
for i = 1:n
f = f*i;
end
f
you can get 0!=1 for n=0; Theoratical proof: he rigorous answer to this question is that the factorial operation is extended to non-integer arguments by what is called the Gamma function, defined as
Γ(x)=∫z^(x−1)*e^−z dz.limits 0 to infinity.
A bit of integration by parts can get you to the relation that for n∈N (natural numbers, where the factorial operation is naturally defined)
n!=Γ(n+1).
Therefore
0!=Γ(1)=∫e^−z dz=1.
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Walter Roberson 2016년 10월 8일
Factorial is the number of distinguishable ways you can arrange a particular number of objects. The empty set (zero objects) is distinguishable from sets that are not empty, so the number of ways to arrange the empty set has to be at least 1. But there are not multiple distinguishable ways to arrange the emptiness, so the count is not more than 1. Therefore the factorial must be 1.
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Abhishek Jain 2016년 10월 18일
A Factorial of a number n can defined as the total number of ways of selecting n different objects. For Example, total number of ways of selecting 5 different objects is 5!=120.
Since, there is only one way of selecting none of the objects.
Hence, 0!=1.
Hope that helps.

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