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In: Mathematics
Something that I haven’t seen mentioned yet is the following. The factorial only works for positive integers. But lets say that you want a version of the factorial that works for decimals? Or negative numbers? or even complex numbers? If you want to keep all the important features of the factorial (the shape you expect and the recurrence relation [(5 + 1)! = (5 + 1)(5!) = 6(5!) = 6!] being two), then it turns out that there is only one function that works, the gamma function shifted by one, [written: Γ(n + 1)]. It turns out that Γ(n + 1) is equal to all the factorials for positive integers, and produces a value for every positive and negative real and complex number except for the negative integers, which is really cool (if you try to use the negative real integers, then you get a divide by zero). But it turns out that 0! = Γ (0 + 1) = 1. So, if you want to use the gamma function (which does some pretty cool things and is pretty simple and elegant) then you need to think that 0! = 1.
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