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In mathematics there are some data , we have took them by convention and mathematics is

not able to show us them proves , now want just to know if the "convention" term enough

mathematics condition to say a student if he asked about the prove of $0! =1$ , "should

be for u to take it as convention".

I would like someone to show me where is the science that proved $0!=1$ while mathematics

not able to prove $0!=1$ ? is it applied in chemistry field ?

just i want how i can deal with a student in high school if he asked ?

i would interest for any replies or any comments

salimmath15
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  • It may be explained as saying $x^0 = 1$ for any $x \ne 0$. So, the "empty product" which is $0!$ should be $1$ also. Another explanation, it makes the subsequent formulas (like the binomial coefficients) consistent. – PA6OTA Jun 11 '14 at 14:25
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    How about \begin{align} 3!=3\cdot2!\qquad&\Longrightarrow\qquad\frac{3!}{2!}=3\ 2!=2\cdot1!\qquad&\Longrightarrow\qquad\frac{2!}{1!}=2\ 1!=1\cdot0!\qquad&\Longrightarrow\qquad\frac{1!}{0!}=1\ \end{align} From the last step we get $0!=\cfrac{1!}{1}=1$. – Tunk-Fey Jun 11 '14 at 14:26
  • @Tunk-Fey The only problem is that it is inconsistent with $\Gamma(0)=(-1)!$ – Hakim Jun 11 '14 at 14:28
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    @حكيمالفيلسوفالضائع: You need to remember that analytic extensions don't have to agree with certain rules that can be deduced on discrete functions. (For example $1+\frac12+\frac13+\ldots$ is NOT $-\frac1{12}$, but it can be proved to be, if we take these rules and apply them to a particular analytic continuation.) – Asaf Karagila Jun 11 '14 at 14:30
  • Thanks @AsafKaragila , I forgot that. – Hakim Jun 11 '14 at 14:31
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    @حكيمالفيلسوفالضائع The OP will immediately notice it that it only holds for integer $n\ge0$. – Tunk-Fey Jun 11 '14 at 14:31
  • @PA6OTA , in the fact x^0 is not by convention , there is a prove for it ( just to use exponent law ) , it's seems that you are used just the definition of $n!=n.(n-1)......(n-h+1)$ in $ then u use 0!=1 as it's verified , thanks – salimmath15 Jun 11 '14 at 15:02

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