# Eli5 why we cant have negative factorials?

40 views
0

I looked it up on google but my extent of math goes to algebra 2, but it immediately went into wacky stuff like “the gamma function”, simply put, what is the gamma function and why is it stopping stuff like -9! From being -362880

In: 2 By (the basic) definition, for any number **n**, we define the factorial (**n!**) as **n***(**n**-1)*(**n**-2)*…*2*1

If you started with an **n** less than 1, you would never get a 1, so would, in a way, go off to +/- infinity.

If you wanted to have -9! = -362880, just rewrite it as -(9!). The gamma function is what you get when you try to extend factorials to fractions, real numbers (positive and negative) and complex numbers….

After all, the basic rule of factorials is that n! = n times (n-1)!.

We can reverse this: (n-1)! = n! / n.

Given 1! = 1, that tells us that 0! = 1! / 1 = 1 also. But then things go downhill.

(-1)! should be 0! / 0 = 1 / 0, but we can’t divide by 0. (-1)! is not defined (even using gamma functions), and nor is (-2)!, (-3)! etc for any negative integer. In mathematics, the factorial symbol is used to represent the product of all positive integers less than or equal to a given number. So, 5! = 12345 = 120.

The gamma function is an extension of the factorial function to negative numbers. It is defined as:

Γ(x) = (x-1)!

where x is a real number.

The gamma function is not defined for negative integers, because the product of all positive integers less than or equal to a negative integer is 0. For example, (-9)! = 012345678*9 = 0.

The gamma function can be used to calculate the factorial of any non-negative integer. For example, Γ(5) = 120.

Here is a table of some values of the gamma function:

x | Γ(x)

——- | ——–

1 | 1

2 | 1

3 | 2

4 | 6

5 | 24

6 | 120

As you can see, the gamma function is always positive for non-negative integers. This is why we cannot have negative factorials. You can.

The Gamma function (shifted by 1) is considered the “best” way to extend the concept of factorial from natural number into complex number. But it’s not the only way.

If you use Gamma function shifted by 1 as the extended definition of factorial, then it has a particular restrictive property: “factorial” of z+1 is “factorial” of z times z+1. Unfortunately, this means “factorial” of 0 is “factorial” of -1 times 0, but factorial of 0 is 1, so “factorial” of -1 has to be infinity. By repeating this argument, we can do the same for -2, -3, …

If you use other functions as extension of factorial, however, then you can have “factorial” of negative number. In fact, many early mathematicians were uncomfortable with the idea that “factorial” of negative integers is infinite. For example, the Hadamard’s Gamma function was a historical contender to the Euler’s Gamma function (which we now know as just Gamma function), and Hadamard’s Gamma function is not infinite at negative integer. And that’s not the only one function, there are many others. However, for rather deep mathematical reasons, the Hadamard’s Gamma function fell out of favor. But it took a long time before mathematicians were able to settle into the Gamma function as we know today. The gamma function is really just an extension of the factorial to non whole numbers. It’s used in probability quite a bit.

The way it is defined, the gamma function just doesn’t allow for whole negative numbers but you can have negative non whole numbers. The way you defined it would violate one of the properties of factorials: (n-1)! = n!/n. Even if you defined -1 factorial (where the problems begin) to be something else (say 1) it wouldn’t equal a whole number. It would be a fraction.