What's the intuition that helps to understand factorial of real number?

Question

I just learn that gamma function can be used for real number factorial. How to intuitively interpret (1/2)! ?

Answer 1

When we talk about it as a "real factorial", we are not really saying that it should have an easy way of finding it for every real number, but instead talking about a thing called extensions.

It means "okay, we have this function in the natural numbers, and it would be awesome to find a real function that is both nice and follows things we would expect"

If you ask the real functions for one such that $f(x+1)=x*f(x)$ (which works as a nice way of generalising the factorial, as $(n+1)!=n*(n)!$), that $f(1)=1$, that it is logarithmic-convex (roughly meaning it keeps growing faster, something you expect seeing the factorial) and that it is analytic (a technical but very important condition for Calculus), a theorem called the Bohr-Mollerup Theorem tells you there is only one function following all those things

So we called it the Gamma function and say it is the real factorial, and now the "factorial" of 1/2 is just the value of this function. It's not like it has a reason to take the value it takes, it just happens to be the value that the only function nice enough for our standards takes, and it works amazingly well when using it this way