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has anybody ever summed all real numbers?

Anonymous Sun 10 Nov 21:58:48 2019 No.11134674 View Reply Original Report

Quoted By: >>11134683 >>11134704 >>11134728 >>11134763 >>11134828 >>11137564

or does a set have to be countable to extract a meaningful sum? Where can I read about this?

Anonymous

Anonymous Sun 10 Nov 2019 22:00:05 No.11134679 Report

Quoted By: >>11134682 >>11134708

Sum of all real numbers = -1/12.
familiar post haha...
https://www.youtube.com/watch?v=w-I6XTVZXww

Anonymous

Anonymous Sun 10 Nov 2019 22:01:08 No.11134682 Report

Quoted By: >>11134689

>>11134679
those are the natural numbers you dumb apple

Anonymous

Anonymous Sun 10 Nov 2019 22:01:18 No.11134683 Report

Quoted By: >>11134697 >>11134716 >>11136518

>>11134674
the sum of all real numbers is zero

Anonymous

Anonymous Sun 10 Nov 2019 22:02:10 No.11134689 Report

Quoted By: >>11134697

>>11134682
What's the difference?

Anonymous

Anonymous Sun 10 Nov 2019 22:03:47 No.11134695 Report

Quoted By: >>11134716

>sum all real numbers.

I don't think it's possible anon. There are subsets of the reals that contain infinite elements who themselves contain subsets of infinite elements etc.

Anonymous

Anonymous Sun 10 Nov 2019 22:04:24 No.11134697 Report

Quoted By: >>11134807

>>11134689
there are way more real numbers and you cant but them in an algorithm like they do in that video
>>11134683
What about [0,1] or the positive reals. I assume you just canceled out the positive and negatives, but do you have a paper to back that up?

Anonymous

Anonymous Sun 10 Nov 2019 22:06:45 No.11134704 Report

Quoted By: >>11134716

>>11134674
$\sum_{x \in \mathbb{R}}x = 0 + \sum_{x>0}x + \sum_{x<0}x = \sum_{x>0}x + \sum_{x>0}-x= \sum_{x>0}(x-x) = \sum_{x>0}0 = 0$

Anonymous

Anonymous Sun 10 Nov 2019 22:08:59 No.11134708 Report

Quoted By: >>11134710

>>11134679
Math is broken, why isn't anyone worried about this?

Anonymous

Anonymous Sun 10 Nov 2019 22:10:29 No.11134710 Report

Quoted By: >>11134721

>>11134708
It's not broken. The sum of all real numbers (or natural numbers) is NOT -1/12.

Anonymous

Anonymous Sun 10 Nov 2019 22:12:58 No.11134716 Report

Quoted By: >>11134739

>>11134704
>>11134683
now what about the uncountable set of all positive reals?

>>11134695
>I don't think it's possible anon
why is it not possible? The only reason i can think of is its uncountable, but i dont see exatly how that contributes to it being unsummable

Anonymous

Anonymous Sun 10 Nov 2019 22:14:47 No.11134721 Report

Quoted By: >>11134735

>>11134710
Using generalized Summing, and analytic continuation, the sum of all naturals is indeed -1/12. But thats not my question, Im wondering if anybody found a way to do this with the reals.

Anonymous

Anonymous Sun 10 Nov 2019 22:19:25 No.11134728 Report

Quoted By:

>>11134674
The sum 1-1+1-1+1-1+.....=1/2. Therefore when you sum 1 - 1 + 1.0...01 - 1.0...01 + 1.0...02 - 1.0...02 +..... you also get 1/2. So the sum of the real numbers is 1/2.

Anonymous

Anonymous Sun 10 Nov 2019 22:21:46 No.11134733 Report

Quoted By: >>11134753

I think I have a proper amswer for you.
let's say you sum an uncountable set of numbers I wilm call a_i, for i in I some huge index set.
Then you look at how many of the a_i are greater than 1/n. If you find out that for some n there are infinitely many a_i > 1/n, then the sum of all a_i must be bigger than the infinite sum of these a_i>1/n, but 1/n *infinity = infinity so the sum of all a_i is infinite.
So if you want the sum of all a_i to be finite, you need that for all n, at most finitely many a_i can be bigger than 1/n.
But then the set of all a_i is the union over n of the sets of a_i > 1/n, which are each finite, so all the a_i are countable.

Anonymous

Anonymous Sun 10 Nov 2019 22:22:52 No.11134735 Report

Quoted By: >>11134747

>>11134721
"Generalized summing" is so far removed from actual sums that it's essentially dishonest to call it summing.
>But thats not my question, Im wondering if anybody found a way to do this with the reals.
There is no need to, so no. -1/12 was not discovered after someone decided to sum all natural numbers, it was rather by investigating a special function which only happen to coincide in meaning somewhat with regular summation of natural numbers.
Even if there was a number that could be interpreted in a way which is close to summing all reals, it would come from asking a completely different, unrelated question. As it stands, your question is ill-defined and pretty much useless.

Anonymous

Anonymous Sun 10 Nov 2019 22:25:29 No.11134739 Report

Quoted By:

>>11134716
>now what about the uncountable set of all positive reals?
$\sum_{x\geq0}x = \sum_{x\in[0,1)}\sum_{n=1}^{\infty} nx = \sum_{x\in[0,1)} x \sum_{n=1}^{\infty}n = \sum_{x\in[0,1)}x\cdot \tfrac{-1}{12} = -\tfrac{1}{12}\sum_{x\in[0,1)}x$

Anonymous

Anonymous Sun 10 Nov 2019 22:30:52 No.11134747 Report

Quoted By:

>>11134735
>"Generalized summing" is so far removed from actual sums
its really not
>There is no need to, so no
theres no need to do anything in pure math, thats a terrible excuse. It is only natural to wonder if you extend that notion of infinite sum to uncountable sets.
>-1/12 was not discovered after someone decided to sum all natural numbers, it was rather by investigating a special function which only happen to coincide in meaning somewhat with regular summation of natural numbers
yes, thats my point. Im wondering If there is some further generalization
>Even if there was a number that could be interpreted in a way which is close to summing all reals, it would come from asking a completely different, unrelated question. As it stands, your question is ill-defined and pretty much useless.
You arent understanding what im saying. Im asking a common thing mathematicians do. There is obviously a reason why there are no meme videos about uncountable infinite sums. This leads me to believe there is a reason we cannot sum infinite reals.

Anonymous

Anonymous Sun 10 Nov 2019 22:35:05 No.11134753 Report

Quoted By:

>>11134733
>So if you want the sum of all a_i to be finite, you need that for all n, at most finitely many a_i can be bigger than 1/n.
this is what im confused with, why exactly is that the case

Anonymous

Anonymous Sun 10 Nov 2019 22:40:09 No.11134763 Report

Quoted By:

>>11134674
Just integrate them like a normal person.

Anonymous

Anonymous Sun 10 Nov 2019 23:00:03 No.11134807 Report

Quoted By: >>11134825

>>11134697
>I assume you just canceled out the positive and negatives, but do you have a paper to back that up?
no i dont, do you think it might not be zero?

Anonymous

Anonymous Sun 10 Nov 2019 23:04:02 No.11134825 Report

Quoted By: >>11134888

>>11134807
If someone found some function that takes in the reals and spits out zero, aswell as returning similar consistent results with other subsets

Anonymous

Anonymous Sun 10 Nov 2019 23:04:33 No.11134828 Report

Quoted By:

>>11134674
Uncountable sums don't make sense. Countably infinite sums only do because they're defined as a limit of finite sums.

Anonymous

Anonymous Sun 10 Nov 2019 23:31:57 No.11134888 Report

Quoted By:

>>11134825
there should be some theorem that the sum of all the reals on an interval [-a, a] is zero
now, if the interval is [-inf, inf] maybe it doesnt work

Anonymous

Anonymous Mon 11 Nov 2019 13:41:29 No.11136518 Report

Quoted By: >>11137446

>>11134683
Depends on the order in which you sum them

Anonymous

Anonymous Mon 11 Nov 2019 20:03:02 No.11137446 Report

Quoted By: >>11137994

>>11136518
no it doesn't.

if you have the same number of negatives as positives they sum to zero. if you're not getting zero then what you have isn't the sum but some bullshit number you got by employing some fancy little tricks.

for all A and B A + B = C

NOT A + B = C OR D "depending on how you sum"

Anonymous

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Anonymous Mon 11 Nov 2019 20:53:09 No.11137564 Report

Quoted By:

>>11134674
You've just created a Riemann sum.

Anonymous

Anonymous Mon 11 Nov 2019 23:09:43 No.11137994 Report

Quoted By: >>11138036

>>11137446
lol, how's high school going?
https://en.wikipedia.org/wiki/Riemann_series_theorem

Anonymous

Anonymous Mon 11 Nov 2019 23:20:33 No.11138036 Report

Quoted By:

>>11137994
don't need to read that bullshit.

like i said. you can do some fancy little tricks to get a number that isn't zero but what get is not the sum of the set.

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