Why 0! = 1?

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Anonymous Sun Dec 2 16:29:43 2018 No.10183425 [Reply] [Original]

Why 0! = 1?

>>	Anonymous Sun Dec 2 16:31:04 2018 No.10183429 >>10183425 because (((maths))) is a scam

>>	Anonymous Sun Dec 2 16:31:50 2018 No.10183430 >>10183425 because the number of ways to arrange the null set is 1 you could have googled this though so you're a bitch

>>	Anonymous Sun Dec 2 16:31:54 2018 No.10183431 You can one arrange 0 into 1 configuration.

>>	Anonymous Sun Dec 2 16:35:17 2018 No.10183442 >>10183425 >>10183430 >>10183431 if n! = n(n-1) ! then 0! = 0(0 - 1)! ????

>>	Anonymous Sun Dec 2 16:37:00 2018 No.10183449 >>10183442 n! =/= n(n-1)! n! = (n-1)! * n This is just a case of simple algebra for the "proof" (it's insanely trivial)

>>	Anonymous Sun Dec 2 16:38:02 2018 No.10183451 >>10183449 n(n-1)! and (n-1)!*n is the same exact thing senpai

>>	Anonymous Sun Dec 2 16:38:29 2018 No.10183452 >>10183442 Go to bed joe

>>	Anonymous Sun Dec 2 16:40:01 2018 No.10183461 >>10183452 I actually want to know why since I'm spending my time into proofs, would you mind to explain why there is a difference?

>>	Anonymous Sun Dec 2 16:40:54 2018 No.10183464 >>10183451 n(n-1)! = (n^2 - n) ! =/= (n-1)! * n example: 5(5-1)! = 5(4)! = 20! =/= 5! = (5-1)1 * 5

>>	Anonymous Sun Dec 2 16:41:44 2018 No.10183467 >>10183425 I don't need to know why, it's enough for me that when I type 0!=1 into the Javascript console, it says the result is true. That's proof for you right there.

>>	Anonymous Sun Dec 2 16:41:58 2018 No.10183470 >>10183464 whoops the (5-1)1 should be (5-1)!

>>	Anonymous Sun Dec 2 16:42:35 2018 No.10183472 >>10183464 Holy shit you're retarded

>>	Anonymous Sun Dec 2 16:44:18 2018 No.10183478 >>10183425 using n! = n(n-1)! 1! = 10!

>>	Anonymous Sun Dec 2 16:44:46 2018 No.10183481 File: 115 KB, 682x900, 1542600281317.jpg [View same] [iqdb] [saucenao] [google] >>10183472 you just made proof of why exactly the product is commutative

>>	Anonymous Sun Dec 2 16:45:52 2018 No.10183484 >>10183481 shit i meant him >>10183464

>>	Anonymous Sun Dec 2 16:46:32 2018 No.10183485 File: 17 KB, 400x366, three-worlds-roger-penrose.jpg [View same] [iqdb] [saucenao] [google] >>10183430 Decreasing energy increases the empty set of a set. Most undergrads learn this in their 2nd year...

Anonymous Sun Dec 2 16:46:40 2018 No.10183486

>>10183425
factorial is taken to be the nonnegative integer restriction of a shift of the gamma function, which is an integral. the reason we take it to be this specific function instead of some other is because it is the analytic continuation of factorial and it's meromorphic on the complex plane. and gamma(n) = (n-1)!, while gamma(1) = 1, so it makes sense to define 0! = 1.
the reason this is useful is because almost any series representation of a function/constant which is written using factorial has the n=0 (first) term to have a 1 where all the other terms have an n!, so letting 0! = 1 is consistent across pretty much any math you'll do.
in fact, gamma is defined for all non-positive integers, where it has poles. so defining 0! = 0*(0-1)! doesn't even make sense, since (0-1)! = gamma(0) which is a pole (and can be interpreted as infinity. So 0! = 0*infinity, which of course is an indeterminate and can assume 1 in the limit.

>>	Anonymous Sun Dec 2 16:46:44 2018 No.10183487 >>10183467 >0!=1 It says true because assertions returns true if they go right. Also it should return some error code.

>>	Anonymous Sun Dec 2 16:52:12 2018 No.10183502 >>10183487 It is not an assertion, it's a simple expression. It returns true because the result of the inequality operator (!=) with the operands 0 and 1 is true.

>>	Anonymous Sun Dec 2 16:53:25 2018 No.10183505 >>10183467 >the absolute state of computer "scientists"

>>	Anonymous Sun Dec 2 16:53:29 2018 No.10183506 >>10183464 >>10183472 >>10183481 welp i messed up now im gonna hang myself

>>	Anonymous Sun Dec 2 16:53:29 2018 No.10183507 0 is null set? I didn't knew it was empty set, I tough it was an identity.

>>	Anonymous Sun Dec 2 16:59:35 2018 No.10183527 >>10183505 Why do you think it's called science? You do empirical experiments with the computer instead of trying to solve problems with some kind of abstract theory.

>>	Anonymous Sun Dec 2 17:17:48 2018 No.10183574 File: 2.56 MB, 300x424, 1542866393899.gif [View same] [iqdb] [saucenao] [google] >>10183505 we computer scientist know shit like D'morgan law anon, we are fucking superior

>>	Anonymous Sun Dec 2 17:24:51 2018 No.10183587 >>10183485 actually most undergrads learn this in the first week of their first year. You must have gone to a special community college or something lol

>>	Anonymous Sun Dec 2 17:36:35 2018 No.10183617 >>10183425 Gamma function

>>	Anonymous Sun Dec 2 17:51:22 2018 No.10183655 >>10183425 because it's useful and x!=y is just dependent on some definition.

>>	Anonymous Sun Dec 2 17:53:04 2018 No.10183665 >>10183425 why not

>>	Anonymous Sun Dec 2 17:56:56 2018 No.10183681 The binary opposite of 0 is 1. I regret nothing by saying this.

>>	Anonymous Sun Dec 2 18:01:57 2018 No.10183700 >>10183505 he's le epic trolle you. != is not equals, 0 is not equals 1 is true.

>>	Anonymous Sun Dec 2 18:16:27 2018 No.10183743 >>10183617 >implying the gamma function and factorial are the same thing

>>	Anonymous Sun Dec 2 18:32:14 2018 No.10183793 it's a convention...the sum of 0 items is 0, the product of 0 items is 1, no reason except that it makes making formulas easier

>>	Anonymous Sun Dec 2 18:36:15 2018 No.10183806 >>10183442 -1! is undefined.

>>	Anonymous Sun Dec 2 18:37:27 2018 No.10183811 >>10183442 n! = n*(n-1)! is valid only for n>0

Anonymous Sun Dec 2 19:11:30 2018 No.10183879

It's not a convention if you start from a set theoretic definition. The factorial function can be defined as the cardinality of the set of bijective functions from a set of cardinaility n to itself. Now, the only set that has cardinality 0 is the empty set, so how many bijective functions can you make from the empty set to itself? The idea here is to understand that a function it's defined formally as a relation of two sets that satisfies certain conditions, so like anything in a set theoretical construction, a function is a set itself. I.e if you have sets [math]A,B[/math] a function is a subset of [math]A\times B[/math] . But because [math]\emptyset\times\emptyset = \emptyset[/math] The only possible subset is the empty set itself which trivialy satisfies the properties of a function. Now that doesn't mean the set of bijections is itself the empty set, the empty set is a function itself so the set is actually [math]\{\emptyset\}[/math], i.e. the set that contains as it's only element the empty set. So the cardinality is 1.

>>	Anonymous Mon Dec 3 00:08:04 2018 No.10184516 >>10183743 The gamma function is the analytic continuation of factorials.

>>	Anonymous Mon Dec 3 00:14:20 2018 No.10184535 >>10184516 They're not the same thing, though. Saying that they're the same thing is like saying [math]\sum_{n=1}^\infty n=-\frac{1}{12}[/math].

>>	Anonymous Mon Dec 3 00:14:49 2018 No.10184536 >>10183743 >but factorial is literally defined by the gamma function imagine being this uneducated

>>	Anonymous Mon Dec 3 00:15:52 2018 No.10184539 >>10184535 you dont have a reason to say that summation is wrong, you just dont like it.

>>	Anonymous Mon Dec 3 00:19:02 2018 No.10184550 >>10184539 I do have a reason, though. 1 = 1. 1 + 2 = 3. 1 + 2 + 3 = 6. Now take the limit of that sequence. What is it, -1/12? What kind of fuckways topology of the reals are we using here?

>>	Anonymous Mon Dec 3 00:21:17 2018 No.10184554 >>10184550 thats a bad reason tho 1 is an integer 2 is an integer 3 is an integer now take the limit infinity is an integer

Anonymous Mon Dec 3 00:24:24 2018 No.10184562

>>10184550
induction only proves things for the set of natural numbers, it doesn't prove anything for infinity.

step 1: add 10 balls labeled 1 to 10 into a container, remove ball 1
step 2: add 10 balls labeled 11 to 20 into a container, remove ball 2
...
for all finite steps the number of balls increases without bound

if you do it infinitely often, however, there are 0 balls in the container

all sorts of things are discontinuous at infinity

>>	Anonymous Mon Dec 3 00:27:17 2018 No.10184576 >>10184554 >>10184562 I'm not using infinite induction, I'm using the literal definition of infinite series. In case you didn't know, it's the epsilon-delta limit of partial sums.

Anonymous Mon Dec 3 00:27:33 2018 No.10184578

>>10184550
It's easy to confuse yourself with this shit but it's quite simple.

All that the ramanujan summation stuff, cutoff and zeta regularization does, is look at the smoothed curve at x = 0.
What sums usually do is look at the value as x->inf.

It's just a unique value you can assign to a sum, really they have many such values.

https://en.wikipedia.org/wiki/File:Sum1234Summary.svg

[math] \displaystyle
\zeta \neq \Sigma
[/math]

https://youtu.be/sD0NjbwqlYw?t=10m

>>	Anonymous Mon Dec 3 00:28:58 2018 No.10184583 >>10184535 Did I disagree?

Anonymous Mon Dec 3 00:29:45 2018 No.10184587

>>10184576
>I'm using the literal definition of infinite series.

no, you are using _a_ definition, you're using the only definition you like.

partial sum is intuitive, but thats it, its not better than the other definitions, you can just understand what its saying easier.

>>	Anonymous Mon Dec 3 00:39:29 2018 No.10184612 >>10184587 Dipshit, you're supposed to be wrong. That's how internet arguments are supposed to work. Why aren't you fucking wrong?

Anonymous Mon Dec 3 00:47:00 2018 No.10184635

lets say you have a series [math] A = a_0 +a_1 + a_2 + a_3 +... [/math]
if you were trying to increase the rate of convergence, then you might look at the partial sums
[math]\\
a_0\\
a_0 + a_1\\
a_0 + a_1 + a_2\\
...\\
[/math]
and you'd say, "why not look at these partial sums instead"

[math]
a_0 + \frac{1}{2}*a_1\\
a_0 + a_1 + \frac{1}{2}*a_2\\
a_0 + a_1 + a_2 + \frac{1}{2}*a_3\\...\\
[/math]

if the first one converges, then so does this new one. But this new sequence works really really well for alternating series, and if you try grandis series, you get

[math]

1 - \frac{1}{2}*1 = \frac{1}{2}\\
1 - 1 + \frac{1}{2}*1 = \frac{1}{2}\\
1 - 1 + 1 - \frac{1}{2}*1 = \frac{1}{2}\\
...\\

[/math]

So now grandis series is equal to [math]\frac{1}{2}[/math]

And that's why you don't only use one definition, because improving the ones you have is really useful.

>>10184612
oh fuck sorry dude, uhhh what about "niggers are good for society"

>>	Anonymous Mon Dec 3 01:56:28 2018 No.10184747 >>10183574 Nigga I'm a biochemist and I know De Morgan's law

>>	Anonymous Mon Dec 3 02:27:38 2018 No.10184781 >>10183431 very elegant i like it, next time smartass freshmen ask me this question i'll pull this one out >>10183507 the set with 0 elements in it is { } the empty set

>>	Anonymous Mon Dec 3 02:47:08 2018 No.10184804 >>10183464 you knew what he meant, why be an asshat

>>	Anonymous Mon Dec 3 04:02:29 2018 No.10184912 >>10183425 Because their is only one way to arrange zero things.

>>	Anonymous Mon Dec 3 04:47:34 2018 No.10184977 >>10184539 That rests on the assumption that 0+1-1+1-1... = 1/2, does it not? That means that whatever operation you are using it is no longer strictly summation.

>>	Anonymous Mon Dec 3 04:48:39 2018 No.10184978 >>10184912 that's actually a pretty meaningless statement. Define "arrange".

Anonymous Mon Dec 3 09:18:43 2018 No.10185371

>>10184977
>That means that whatever operation you are using it is no longer strictly summation.
Yeah, but using ZFC, addition is defined in terms of cardinalities of sets, so adding anything besides non negative integers is also no longer strict summation.

But strict summation is just another name for partial sums, which is just _a_ definition anyway, not the best definition

>>	Anonymous Mon Dec 3 10:08:43 2018 No.10185439 >>10184978 low IQ

>>	Anonymous Mon Dec 3 15:55:11 2018 No.10186314 It's defined as such.

>>	Anonymous Mon Dec 3 16:50:05 2018 No.10186455 >>10186314 yeah, its kinda like asking why 1 - 1 = 0

>>	Anonymous Mon Dec 3 16:52:35 2018 No.10186461 >>10183485 lol what a retard

>>	Anonymous Mon Dec 3 16:55:09 2018 No.10186469 >>10183467 >"5" - 3 == 2 >"5" + "3" == "53" The absolute state.

>>	Anonymous Mon Dec 3 16:56:11 2018 No.10186474 File: 26 KB, 390x310, 32.jpg [View same] [iqdb] [saucenao] [google] >>10183467

>>	Anonymous Mon Dec 3 17:08:09 2018 No.10186509 >>10184535 [math] \displaystyle \sum_{n=1}^ \infty n = - \dfrac{1}{12} [/math] Optimized.

>>	Anonymous Mon Dec 3 18:34:18 2018 No.10186723 >>10183442 if n!= n(n-1)! then 1!=1(1-1)!=10! we agree for 0!=1 so that there is a condition that stops the recursion whole formula is: n!= n(n-1)! for n>0; 1 for n=0. note the definition for n<0 would make it go to -infinity.

>>	Anonymous Mon Dec 3 18:36:30 2018 No.10186728 >>10186723 >inb4 what do you mean by we, peasant?

>>	Anonymous Mon Dec 3 20:09:39 2018 No.10186941 >>10184535 But those two are the same thing, by any reasonable definition of "same". I mean, do you use a separate symbol for finite and infinite summation? For conditionally and unconditionally convergent series? See where I'm going here?

>>	Anonymous Mon Dec 3 20:15:44 2018 No.10186956 >>10183527 Its not a science

>>	Anonymous Mon Dec 3 20:16:12 2018 No.10186959 >>10184747 lol you're so dumb

>>	Anonymous Mon Dec 3 20:19:35 2018 No.10186968 >>10183879 For anyone who's wondering, the correct answer is this.

Anonymous Mon Dec 3 20:22:09 2018 No.10186976

>>10184576
You're right that 1+2+3+... doesn't *converge* to -1/12, but so what? There's a very real sense in which 1+2+3+... IS -1/12. We don't artificially distinguish between conditional and unconditional convergence, for example, even though the former is not invariant to rearrangements!

>>	Anonymous Mon Dec 3 20:23:44 2018 No.10186980 >>10183574 >>10184747 kek

>>	Anonymous Tue Dec 4 04:45:25 2018 No.10188001 >>10186723 >note the definition for n<0 would make it go to -infinity. Which is exactly what the gamma function does.

>>	Anonymous Tue Dec 4 13:52:09 2018 No.10188973 3!=1×2×3 2!=1×2=3!/3 1!=1=2!/2 0!=1=1!/1

Anonymous Tue Dec 4 13:59:01 2018 No.10188989

Imagin you have 5 coins, you can arrange them in 5! Different ways, therefore 120 ways. 4 coins can be arranged in 24 different ways or 4!ways. 3 coins can be arranged in 6 different ways or 3! Ways like coin a and coin b and coin c can be abc acb bac bca cab and cba and two coins can be arranged in 2 ways one on the lef and the other on the right or the other way, one coin can be only arranged in one way, just put it on the table and that's it. And no coins can only arranged in one way. Just imagin your broke ass don't have change, and can't put it the vending machine, so boom you fuck off. You're welcome

>>	Anonymous Tue Dec 4 16:25:50 2018 No.10189437 >>10183879 /thread Ffs brainlets.

>>	Anonymous Tue Dec 4 16:33:41 2018 No.10189460 File: 297 KB, 836x1136, 1537003859573.png [View same] [iqdb] [saucenao] [google] >>10183574

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