So, basically, factorial gives us the arrangements. Now, the question is why do we need to know the factorial of a negative number?, let's say -5. How can we imagine that there are -5 seats, and we need to arrange it? Something, which doesn't exist shouldn't have an arrangement right? Can someone please throw some light on it?.
The theorem that $\binom {n} {k} = \frac {n!} {k! (n-k)!}$ already assumes $0!$ is defined to be $1$. Otherwise this would be restricted to $0 <k < n$. A reason that we do define $0!$ to be $1$ is so that we can cover those edge cases with the same formula, instead of having to treat them separately. We treat binomial coefficients like $\binom {5} {6}$ separately already; the theorem assumes ...
Why is this? I know what a factorial is, so what does it actually mean to take the factorial of a complex number? Also, are those parts of the complex answer rational or irrational? Do complex factorials give rise to any interesting geometric shapes/curves on the complex plane?
Some theorems that suggest that the Gamma Function is the "right" extension of the factorial to the complex plane are the Bohr–Mollerup theorem and the Wielandt theorem.
I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$. Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\\times1$, but how do we e...
12 I've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck whatsoever.
And 0! = 1. However, this page seems to be saying that you can take the factorial of a fraction, like, for instance, 1 2!, which they claim is equal to 1 2√π due to something called the gamma function. Moreover, they start getting the factorial of negative numbers, like − 1 2! = √π How is this possible? What is the definition of the factorial of a fraction? What about negative numbers ...
He describes it precisely for the purpose of contrasting with the factorial function, and the name seems to be a play on words (term-inal rather than factor-ial).
