Why is 0! = 1 Veeky Forums?

you couldn't even do something as simple as that huh
czeched

They felt sorry for it.

Is it pathetically sad that I never knew this? This whole time I had zero idea why factorials even existed, just knew how to do them.

nah i didnt know til 3rd year of my math degree

You can prove it with only basic algebra. Assume you have a ring where the multiplicative identity is equal to the additive identity, e, then x*e= x*(e+e) =x*e+x*e thus x = x*e = e, the only ring that works in is the trivial ring thus for any non trivial ring we have 0!=1.

one way to arrange 0 things

Really? Show it. Like in

x

meant
for

But that is 1 thing, not 0. You've shown that 1! = 1.

empty set=[]

Take the set of zero things

That set exist, even though nothings contained in it.

If that set didn't exist, there'd be no way for there to be zero of something.

Because Zero Factorial Is A Stupid Question To Answer.

0! = 0i+c = 01/00

I literally don't know how much more information I can give you guys without wanting to beat HUMANITY to fucking death.

ENSLAVE US SIR!

`.Prof. Far"WHATIT'SWORTH" These Glasses Are AMAZEING! That Smell-O-Scope! Good thinking!

The set exists. But there is nothing in it which can be arranged in any way.

Awful lot of hot opinions with nothing backing them up, user. Feel free to write a paper about how "but you can't physically arrange it!!!!!" and see if literally anyone cares about your nonsensical autistic fit.

How can you arrange nothing? You can't. Just like you can't divide by 0. It is just the result of analytical continuation. But it makes as much sense as saying that [math]1 + 2 + 3 + 4 + ... = -\frac{1}{12}[/math].

>continuing to just be autistic in a post after being told to put your money where your autistic mouth is
You can just keep replying to me while I continue to dismiss you because you're fucking insane if you're a little bitch, or you can actually show some conviction and go out there and tell every mathematician in the world that they're wrong because it hurts your autism.

>"Can you explain why 0! = 1?"
>"Because XYZ"
>"That doesn't make much sense to me"
>"HURR HOT OPINIONS WHY NOT WRITE A PAPER ABOUT IT NOBODY CARES DURR"
You could just admit that you don't actually know it either. Would safe you the embarrassment.

(n-1)! * n = n!
0! * 1 = 1!
0! = 1

This is news to me as well. A lot of formulas in probability(counting methods, etc) mean a lot more to me now.

Saying >hurr when you're the one who's wrong just makes you look sillier.

because modern mathematics is a joke that is a competition between schizophrenics that focuses on whoever can make the most retarded set of rules, instead of solving problems

Wrong about what? Not knowing why 0! = 1?
There are many mathematical methods to prove it (some were already posted). But that particular one just isn't as intuitive as you seem to believe. Which gets obvious when starting from . You can go down to 1 thing and show that there is 1 way to arrange it. But if you go with less, there is just nothing and you can't show that 0! = 1 that way.

>there is just nothing and you can't show that 0! = 1 that way.
Yes you can. We showed it here

just because a set of nothing exists doesnt mean that nothing can be arranged in 1 way
nothing is nothing, it cannot be arranged
factorial is the number in which you can arrange ELEMENTS of the set, but you can't arrange nothing
imagine a function that takes a molecule and returns the number of it's atoms
and then there comes some autist that puts nothing into the function and gets one, even though it makes no sense

EMPTY PRODUCTS

>imagine a function that takes a molecule and returns the number of it's atoms

This analogy is wrong.

What your function is measuring is a certain amount of a physical thing. A factorial doesn't tell you how many things there are, it tells how many ways those things can be arranged. These are two completely distinct concepts. Arrangement is not a measurement of elements, it's a measurement of ordering.

What university did you go to? I was told this in our high school probability classes

because it's convenient

so that nCr is always defined

what shit uni did you attend ? LMFAO

There is one bijection from {} to {}

0 things can't be arranged in any way.

yeah they can, observe:

0 things can be arranged 0 ways, which is one way
>think about it

...

1st year: calc I, II, III & Ode
2nd year: linear algebra, intro proof class, pde, modelling, number theory
wasnt til 3rd year i took combinatorics. thats not strange

I went to Cornell and never learned this until today, at age 27.

By definition

Next

You seem to be awfully focused on the "have to be able to change the arrangement" bit when the whole point is that you can't, there's only one way to put it.

Circular logic. The gamma function was created with the purpose of being a differentiable (and therefore continous) version of the factorial function

you frigged it up

Wow, this board truly is full of brainlets. Here's the real reason.

There's only one way to arrange zero things, if you had nothing there's only one state for it be arranged in which is nothing. One and zero factorial are both one since there's only one way to arrange each.

This, factorial is more of a logical operation that a mathematical operation.

factorial can be defined recursively with the stopping case 0!=1
so by definition

is there a case where the multiplicative and additive identity are the same?

my algebra game was always weak af

Yes, Z1

damn son you are right. thanks

actual best answer tho