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I see proof below for 0! = 1. Is this true in mathematical scientists' idea?

(n+1)! = (n+1)n! → (n+1)! ÷ (n+1) = n!

Now if we put 0 to n, we will have:

1! ÷ 1 = 0! → 1 = 0!

Is this true in math scientists' opinion?

Reza Hosseinzadeh
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  • Yes it is correct that $0!=1$, and this is the reason – Henry Sep 27 '21 at 13:17
  • No, the first line $(n+1)!=(n+1)n!$ is valid only for $n\ge 1$, because $n!$ has no definition for $n=0$, so we cannot write it down for $n=0$. – Dietrich Burde Sep 27 '21 at 13:17
  • @DietrichBurde $(n+1)!=(n+1)n!$ is correct when $n=0$ – Henry Sep 27 '21 at 13:18
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    No, it isn't. $n!=1\cdot 2\cdot \cdots n$ is not defined for $n=0$. So $0!$ is a convention. I cannot be proved from first principles, see this duplicate. – Dietrich Burde Sep 27 '21 at 13:19
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    See Factorial: "In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. The value of 0! is 1, according to the convention for an empty product." – Mauro ALLEGRANZA Sep 27 '21 at 13:19
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    Unrelated, but I don't agree with your denomination of ‘mathematical scientist’. I don't think mathematics are really a science. – Bernard Sep 27 '21 at 13:22
  • "$0!=1$" is true per definition, it need not be proven. To understand why this was defined, the best way is to think of the empty product which has value $1$ allowing many induction proofs starting with $n=0$ instead of $n=1$ – Peter Sep 27 '21 at 13:29
  • @Bernard In my opinion, who has knowledge in a science in scientist. Not just in experimental sciences. In my opinion math and philosophy are science too. – Reza Hosseinzadeh Sep 27 '21 at 21:00
  • @RezaHosseinzadeh: For me, mathematics has to do with poetry. – Bernard Sep 27 '21 at 21:30

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