Factorial Calculation for Non-Integers? I was playing with numbers on calculator and to my amaze i could see that calculator calculated $(4.5)!$ or any real numbers but factorial is defined for integers how is this done any advanced function. (Note I am grade $11$ student)
In general, $~n!~=~\displaystyle\int_0^\infty\exp\Big(-\sqrt[n]x\Big)~dx,~$ which for $~n=\dfrac12~$ yields $~\Big(\tfrac12\Big)!~=~\displaystyle\int_0^\infty e^{-x^2}~dx.~$ 

But the value of the Gaussian integral is known to be $\sqrt\pi~,~$ implying that $~\Big(\tfrac12\Big)!~=~\dfrac{\sqrt\pi}2,~$ 

since the integrand is even. Now all that's left to do is to repeatedly employ the well-known 

factorial property $(n+1)!=(n+1)~n!~$ for $~n+1=4+\dfrac12,~$ and the result follows.
Along with the gamma function, it is much easier to approximate to good accuracy using Stirling's approximation.

It is defined as:

$$n!\approx \sqrt{2\pi n}\left(\frac ne\right)^n$$