# Does (9/2)! have a real answer or not? [duplicate]

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

## marked as duplicate by MJD, Dietrich Burde, Leucippus, Ben Sheller, user147263 Apr 18 '16 at 23:38

• I think technically it doesn't since factorial is defined for integers. But we extend the definition to real numbers using the Gamma function. – abiessu Apr 18 '16 at 17:39
• It is quite common for calculators to use the gamma function to do approximations to factorials. It seems true in this case: wolframalpha.com/input/?i=gamma%289%2F2%2B1%29 – mathreadler Apr 18 '16 at 17:39
• For non negative real numbers the factorial is extended by the gamma function which makes (9/2)! well defined, en.m.wikipedia.org/wiki/Gamma_function (it's also defined for non integer negative reals). Likely some calculators don't use gamma functions and so return errors. It's the same reason as a simple calculator might return an error for square rooting a negative number. – Triatticus Apr 18 '16 at 17:44

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

• I'd say this is an abuse of notation. Gamma function and factorials are not identical things. – Omar Nagib Apr 18 '16 at 18:01
• Yes, sure. But this is the way to interpret factorials of rational numbers. I do not see any abuse in it. – Dietrich Burde Apr 18 '16 at 18:42

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 $\Gamma(5.5)=52.342777..$

• Thank you for creating Akinator, I have played it a lot with friends and family. – Jorge Fernández Hidalgo Apr 18 '16 at 17:42

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.