# What is Factorial: Definition and 162 Discussions

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n:

For example,

The value of 0! is 1, according to the convention for an empty product.The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there are n!.
The factorial function can also be extended to non-integer arguments while retaining its most important properties by defining x! = Γ(x + 1), where Γ is the gamma function; this is undefined when x is a negative integer.

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1. ### A "spiral" in the Complex plane

I understand that the "spiral" converges to 1+i-1/2-i/3!+1/4!+i/5!-1/6!-i/7!... . It splits into two: one for Re, 1-1/2+1/4!-1/6!..., and the other for Im, 1-1/3!+1/5!-1/7!... . Any hints on how to compute them?
2. ### Finding the Last Non-Zero Digit of a Repeated Factorial Expression

Without using computer programs, can we find the last non-zero digit of $$(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$$? What I know is that the last non-zero digit of ##2018!## is ##4##, but I do not know what to do with that ##4##. Is it useful that...
3. ### B Cool fact about number of digits in n!

This may have already been found by many people but I discovered the pattern on my own out of curiosity with some coding. There are only 4 natural numbers whose factorial contains the same number of digits as the number itself. That is to say n = digits_in(n!). The trivial case is obviously...
4. ### How do we get from one step to another in these factorial equations?

this is the answer, but i don't get why k factorial multiplies the bracket, what i did was k factorial divided by the bracket
5. ### MHB Find A,B,C for Factorial Equation

find A,B,C such that: ABC= A!+B!+C!
6. ### Stirling's Approximation for a factorial raised to a power

Using log identities: ##log((\alpha - 1)!^2) = 2(log(\alpha - 1)!)## Then apply Stirling's Approximation ##(2[(\alpha - 1)log(\alpha - 1) - (\alpha - 1)## ## = 2(\alpha -1)log(\alpha -1) - 2\alpha+2## Is this correct? I can't find a way to check this computationally.
7. ### I Prove series identity (Alternating reciprocal factorial sum)

This alternating series indentity with ascending and descending reciprocal factorials has me stumped. \frac{1}{k! \, n!} + \frac{-1}{(k+1)! \, (n-1)!} + \frac{1}{(k+2)! \, (n-2)!} \cdots \frac{(-1)^n}{(k+n)! \, (0)!} = \frac{1}{(k-1)! \, n! \, (k+n)} Or more compactly, \sum_{r=0}^{n} (...
8. ### MHB What are the Positive Integer Solutions to the Factorial Equation?

Determine all positive integers $a,\,b$ and $c$ that satisfy equation $(a+b)!=4(b+c)!+18(a+c)!$.
9. ### I Define the double factorial as being a continous, non-hybrid function

The double factorial, ##n!## (not to be confused with ##(n!)!##), can be defined for positive integer values like so: $$n!=n(n−2)(n−4)(n−6)...(n-a)$$ Where ##(n−a)=1## if ##n## is odd or ##(n−a)=2## if ##n## is even. Additionally, the definition of the double factorial extends such that...
10. ### MHB Inequality involves radical, square and factorial expression 3√{x}+2y+1z^2⩽ 13

If $x^2+y^2+z^2+xyz=4$ and that $x,\,y,\,x\ge 0$, prove $3!\sqrt{x}+2!y+1!z^2\le 13$.
11. ### A Analytical function from numerable point set

Sometimes there are functions that are initially defined for only integer values of the argument, but can be extended to functions of real variable by some obvious way. An example of this is the factorial ##n!## which is extended to a gamma function by a convenient integral definition. So, if I...