Does (9/2)! have a real answer or not?

Question

The TI-84 says 52.342777 but other calculators says domain error.

Answer 1

factorial is defined only for positive integers. But it is generalized to real numbers using the gamma function. For each positive integer n, you have $\Gamma(n+1)=n!$. You have that $\Gamma(5.5)=52.342777..$

Answer 2

Using the Gamma function we have $$(9/2)!=\Gamma(11/2)=\frac{945}{32}\sqrt{\pi}=52.3427777 $$

Answer 3

You're getting the answer for $\Gamma(11/2)$, where

$$\Gamma(n)=(n-1)!$$

The gamma function $\Gamma(x)$ is defined for all $x$ except $0, -1, -2, -3, ...$. The factorial function, however, is defined only for positive integers.