How do you compute the factorial of something like 3/2 or -2? Wolfram Alpha gives an answer, but how does it arrive at that point?
This is the Gamma function. The gamma function is defined by an integral, and we define n! = Γ(n+1).
For factorials of half-integers, you can start with (1/2)! = Γ(3/2) = ∫0∞ t1/2 e-t dt. Substituting u = t1/2 turns this into the "Gaussian integral" (integral of e^(-x^2), possibly with some constants), and you get (1/2)! = sqrt(π)/2.
The recurrence n! = n * (n-1)! holds for the Γ function, so you get (3/2)! = (3/2) * (1/2)! = 3 sqrt(π)/4.
(-2)! actually doesn't exist. If it did, then we'd have
1! = 1 * 0! = 1 * 0 * (-1)! = 1 * 0 * (-1) * (-2)!
so 1! = 0 * (-2)!, but 1! = 1.
Just to add a footnote, one might ask why the gamma function and not some other function that agrees with factorial (when shifted by one). There are other ways one might interpolate between the values of factorial. One justification is the Bohr-Mollerup theorem.