![]() Also, this is actually technically the Gamma function displaced by 1, but the difference won’t become important here.) (The integral should be taken from 0 to ∞, but I can’t figure out how to get WordPress to allow me to do this. It’s weird and mysterious, and tremendously useful in a whole bunch of very different areas. The Gamma function is one of my favorite functions. Now, the lover of abstraction looks at the factorial and notices that it is only defined for positive integers. We can say that 5! = 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 120, but what about something like ½!, or (-√2)!? Enter the Gamma function! The basic reason for this comes down to the fact that there are N! different ways of putting together N distinct objects into a sequence. The factorial also turns out to be extremely useful in probability theory and in calculus. ![]() This function turns out to be mightily useful in combinatorics. The definition of the factorial is the following: I want to take you down one of these little rabbit holes of abstraction in this post, starting with factorials. This gives rise to quotes like Feynman’s “Physics is to math what sex is to masturbation” and jokes like the one above. He spend the rest of his life generalizing the results for the table with N legs (where N is not necessarily a natural number).Ī key feature of mathematics that makes it so variously fun or irritating (depending on who you are) is the tendency to abstract away from an initially practically useful question and end up millions of miles from where you started, talking about things that bear virtually no resemblance whatsoever to the topic you started with. Then he designs a table with infinitely many legs. ![]() A mathematician is asked to design a table.
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