I am a high school student and I learnt Factorials today. The Professor told that you can only factorialize natural numbers. Why is it so??
Is there a way to get the factorial of all Real numbers???
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3$\begingroup$ The gamma function generalizes factorials. $\endgroup$– Sean RobersonCommented Nov 23, 2020 at 16:24
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$\begingroup$ The gamma function is not defined on integers that are not positive $\endgroup$– J. W. TannerCommented Nov 23, 2020 at 16:27
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2$\begingroup$ This article describes how Euler answered to your question in 1728. $\endgroup$– BumblebeeCommented Nov 23, 2020 at 16:55
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$\begingroup$ The Davis reference there is a good read $\endgroup$– J. W. TannerCommented Nov 23, 2020 at 17:00
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1$\begingroup$ en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem $\endgroup$– Will JagyCommented Nov 23, 2020 at 17:17
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1 Answer
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To see why, consider this: Take $\pi$ things. In how many different ways can you order them? Does that make sense?
If you discard the combinatorial aspect, and focus on the algebraic aspect (namely $0!=1$ and $n!\cdot(n+1)=(n+1)!$), then there are many ways to generalize this to be valid for (almost) all real numbers, but the most common is the so-called $\Gamma$ function: $$ \Gamma(x)=\int_0^\infty t^{x-1}e^{-t}dt $$ with $n!=\Gamma(n+1)$.
The $\Gamma$ function is undefined on $\{0,-1,-2,\ldots\}$. This is basically unavoidable if you want the defining algebraic property above to hold. On the other hand, we have $\Gamma(\pi+1)\approx 7.188$, giving something like an answer to the question in the first paragraph.
But note that the $\Gamma$ function is not the factorial. The factorial works on the natural numbers and only the natural numbers.
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$\begingroup$ all real numbers except integers that are not positive $\endgroup$ Commented Nov 23, 2020 at 16:31
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