I looked it up on google but my extent of math goes to algebra 2, but it immediately went into wacky stuff like “the gamma function”, simply put, what is the gamma function and why is it stopping stuff like -9! From being -362880
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The Gamma function (shifted by 1) is considered the “best” way to extend the concept of factorial from natural number into complex number. But it’s not the only way.
If you use Gamma function shifted by 1 as the extended definition of factorial, then it has a particular restrictive property: “factorial” of z+1 is “factorial” of z times z+1. Unfortunately, this means “factorial” of 0 is “factorial” of -1 times 0, but factorial of 0 is 1, so “factorial” of -1 has to be infinity. By repeating this argument, we can do the same for -2, -3, …
If you use other functions as extension of factorial, however, then you can have “factorial” of negative number. In fact, many early mathematicians were uncomfortable with the idea that “factorial” of negative integers is infinite. For example, the Hadamard’s Gamma function was a historical contender to the Euler’s Gamma function (which we now know as just Gamma function), and Hadamard’s Gamma function is not infinite at negative integer. And that’s not the only one function, there are many others. However, for rather deep mathematical reasons, the Hadamard’s Gamma function fell out of favor. But it took a long time before mathematicians were able to settle into the Gamma function as we know today.
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