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16 Μαΐ 2014 · The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Eg:- 4!=1*2*3*4 . 0!=1 states that factorial of 0 is 1 and not that 0 is not equal to 1. The value of 0! is 1, according to the convention for an empty product.
3 Αυγ 2022 · Definition of a Factorial. The factorial of a number is the multiplication of all the numbers between 1 and the number itself. It is written like this: n!. So the factorial of 2 is 2! (= 1 × 2). To calculate a factorial you need to know two things: 0! = 1. n! = (n - 1)! × n.
(1/2)! factorial is not (√π)/2 - it's undefined. If you extend factorial to non-counting numbers using gamma (yes i get it it's a unique "natural" extension) then fine. But for the factorial n!, the thing that counts the number permutations of a set of n elements, (1/2)! simply does not make sense.
The interesting thing about this integral is that it also works for $n$ that is not a natural number (and the result is a nice smooth function of $n$). The factorial of one half ($0.5$) is thus defined as $$ (1/2)! = ∫_0^∞ x^{1/2}e^{-x}\,dx $$ We will show that:
The factorial function is defined for all positive integers, along with 0. What value should 0! have? It's the product of all integers greater than or equal to 1 and less than or equal to 0. But there are no such integers. Therefore, we define 0! to equal the identity for multiplication, which is 1.
29 Νοε 2021 · $$(-1)!=\lim_{n\to\infty}\frac{n!(n+1)^{-1}}{\color{red}{(-1+1)}\dots(-1+n)}$$ Ah... so negative integers result in division by zero. Go figure if you read Akiva Weinberger's answer.
The factorial of a number is the product of all the integers from 1 to that number. For example, the factorial of 6 is 1*2*3*4*5*6 = 720. Factorial is not defined for negative numbers, and the factorial of zero is one, 0! = 1.
