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# Sum of All Positive Numbers is -1/12?

11/04/2015 7:13 AM

Let S = 1+2+3+4+.....∞ = -1/12, does this sound right?

Here's the proof for it.

Man, arithmetic might be just the positive part of solution of the reality Insane.

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#1

### Re: Sum of all positive numbers is -1/12?

11/04/2015 7:43 AM

Seems like Roger Pink had a post on that awhile back...

http://cr4.globalspec.com/blogentry/24184

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#2

### Re: Sum of all positive numbers is -1/12?

11/04/2015 7:48 AM

Damn it, Roger, you got there first! But well, I found the weirdness of reality on my own.

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#13

### Re: Sum of all positive numbers is -1/12?

11/04/2015 1:33 PM

Lol, first on CR4 maybe. Numberphile were all over this well before I posted...

It is a really cool result. It does take a while to accept though. At least for me it did.

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#14

### Re: Sum of all positive numbers is -1/12?

11/04/2015 2:28 PM

Is it not more likely that our math is not yet mature (or is even wrong)?

The fact that we create a "cesaro sum" makes me think we just don't know how to deal with divergent sums.

When I took advanced calculus in college, I felt many times like the answer was never accurate, just very, very, very close. Something always bothered me about that, and I questioned (and still do) whether we are off base.

But then again, I have the exact same concerns regarding Pi.

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#3

### Re: Sum of all positive numbers is -1/12?

11/04/2015 8:08 AM

42

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#4

### Re: Sum of all positive numbers is -1/12?

11/04/2015 8:11 AM

Now thats even weirder dont you think?

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#23

### Re: Sum of all positive numbers is -1/12?

11/05/2015 7:01 AM

It's still 42!

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#10

### Re: Sum of all positive numbers is -1/12?

11/04/2015 9:44 AM

I agree....42

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#5

### Re: Sum of all positive numbers is -1/12?

11/04/2015 8:19 AM

The only thing they are proving is that mathematicians make irrational assumptions, mathematical sleight of hand tricks and huge rounding errors to act like they know more than they do.

Basic and obvious mathematical logic fail by my standards.

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#9

### Re: Sum of all positive numbers is -1/12?

11/04/2015 9:24 AM

This is a legitimate solution, you know. I wont believe it first but, got convinced along the way.

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#6

### Re: Sum of all positive numbers is -1/12?

11/04/2015 8:24 AM

It seems to me like the first step is a bit sketchy. He has a series that has two possible sums, so he picks a third value as the sum which is not equal to either (0 or 1), but to the average of the two (1/2). The rest of the argument is based on this value.

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#8

### Re: Sum of all positive numbers is -1/12?

11/04/2015 9:05 AM

It's with this grandi's series, that they arrived on the total value of series =1/2

Seem like there is a hope for the previous post of me and kulas before about (Infinity and Zero)

0 = N/∞, so N = ∞*0 where N - represents any numbers (represented in physical quantity)

Because this just means if you sum up infinity, it would equal a definite real number.

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#12

### Re: Sum of all positive numbers is -1/12?

11/04/2015 1:17 PM

The first step is where the "slight of hand" misdirection happened. As the Wikipedia article on Grandi's series states in the introduction:

It is a divergent series, meaning that it lacks a sum in the usual sense. On the other hand, its Cesàro sum is 1/2.

The Cesàro sum creates an identical numeric value to an actual infinite series sum if that infinite series is convergent. If it is a divergent series it may still create a numeric value that is useful in some cases but it is not the sum of a divergent infinite series.

Clearly the sum of all positive integers is a divergent infinite series.

Another perspective of this was Evelyn Lamb's write up in Scientific American.

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#17

### Re: Sum of all positive numbers is -1/12?

11/04/2015 4:29 PM

Here is one of the flies in this ointment. From the Wikipedia page on Cesaro summation I cited earlier.

"In other words, the Cesàro sum of an infinite series is the limit of the arithmetic mean (average) of the first n partial sums of the series, as n goes to infinity."

So for summation of 1,0,1,0,1,.... to the 1000 iteration (n=1000) I get a result of:

1000*1/2= 500

That makes sense to me.

For another fly, the Wolfram Mathworld page on divergent series reveals this convenient comment:

"The Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge."

... made to converge to any desired value. I wonder why they desired -1/12?

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#15

### Re: Sum of all positive numbers is -1/12?

11/04/2015 3:17 PM

If you look at the video and the link you see they give 3 different answers for the sum of the series 1+1-1+1-1+1....

That's the problems with infinities. Dealing with them can sometimes be only quasi-mathematical. If you're not rigorous, then you can wind up dividing by zero.

Since there are an infinite number of elements in the series, one could just as easily pull forward (let's say) 5 of the 1s and write it as 1+1+1+1+1-1+1-1+1-1.... (which is equivalent to the original series) and then go through the process again, and get yet a new result.

The 'reason' this analysis sort-of works in physics is because physical processes can average and round off when handling large series.

Take for example a square wave signal driving an LED.

The LED is powered only when the current is at +1 (black sq wave), yet after a few moments the brightness of the LED is equivalent to a steady-state current at +1/2 (the blue line). Which is what you'd expect and is one of the results they achieve. (Unless you examine the LED with a photocell that reacts very fast, in which case the LED would be detected as blinking, and its net output would depend on when you switched off the detector.)

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#18

### Re: Sum of all positive numbers is -1/12?

11/04/2015 5:32 PM

Here's another way to write the sum:

1-1+1-1+1-1+1-1+1-1... is equivalent to 1 - (1-1) - (1-1) - (1-1) - (1-1) - (1-1) -...

In other words the series can be written as 1 - 0 - 0 - 0 - 0 ... etc.

BUT, continuously subtracting a number is equivalent to dividing by that number. So this is equivalent to 1 ÷ 0, which is undefined; dividing by zero is forbidden in mathematics.

So you see one must be very rigorous in dealing with infinite series, or the results can be meaningless.

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#19

### Re: Sum of all positive numbers is -1/12?

11/05/2015 4:53 AM

"BUT, continuously subtracting a number is equivalent to dividing by that number".

What?!! You mean that 1-2-2-2-2-2...=1/2 and 1-5-5-5-5-5...=1/5??

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#33

### Re: Sum of all positive numbers is -1/12?

11/10/2015 4:51 AM

May be adding 1+∞ is not anymore infinity.

This could mean that adding 1 to the sum of the whole real number set changes everything, conversely also by subtracting 1, because infinity you know is the set of all real numbers with out end, but it should be that its subset can not be repeated or unique.

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#20

### Re: Sum of all positive numbers is -1/12?

11/05/2015 6:41 AM

Assume x squared = 1. Either x=1 or x=-1, I don't know which, but I will assume x=0, the average of the two.

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#7

### Re: Sum of all positive numbers is -1/12?

11/04/2015 8:37 AM

It does if you're part of the Ontario government Finance department(\$300 Billion in the hole and trying to spend their way out!).

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#11

### Re: Sum of All Positive Numbers is -1/12?

11/04/2015 12:13 PM

To me this is perfect example of where taking a large number of mathematical steps and trying to reduce them down to a simple equivalent or such that does not actually represent the real answer causes a gross degree of variation and or inaccuracy in another answer later on that is part of another set of equations that are highly dependant on the accuracy or the first set of equations answer.

In the first step the number set and resulting answer creates a sort of perpetual two state alternating high low resultant that can not be simplified to the 1/2 average that this guy pulls out of his butt. The correct answer is either 1 or 0 going on for infinite steps. 1/2 is not a valid approximation.

In the second step I have no explanation for the purpose of the copying and then offsetting the copy by one step to the left or right in order to make the two equations factorable in an up and down reference when they are of such original sets that they need to be factored left to right. To me that's about as logical as taking a written sentence and copying it just below itself but offsetting it one way or the other so that new words and sentences can be made up with completely different meanings and contexts than what the original sentence states as when read the correct way in order to justify a totally different meaning than what the original sentence wording states.

Step three. Um well. Just run it it in reverse and see if -1/12 is equal to the first equation without making a few huge unrelated and illogical mathematical jumps to get there.

All I can say is that if these guys are mathematical geniuses I definitely do not want to meet the mathematical idiots from the opposite end of the spectrum and see what they do with numbers.

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#24

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 8:31 AM

GA. My thoughts (almost) exactly for step 2. Look at the beginning of the video and the reference to the "text" book. I think everyone is aware that just because it is written down does not make it a fact or better,,, true.

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#25

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 9:04 AM

Since these guys don't cite their textbook I suspect that this simplification is true only when an analysis meets certain unstated conditions.

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#41

### Re: Sum of All Positive Numbers is -1/12?

11/11/2015 4:43 AM

Agree with others re: 42

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#16

### Re: Sum of All Positive Numbers is -1/12?

11/04/2015 3:19 PM

Only in academia can something like this be thought of as correct.

The solution is flawed as it goes against sound math fundamentals.

1. Adding positive integers can not result in a negative integer (unless you are using a computer with limited number of bits where you will get wrap around to negative numbers; but then that's not really math...that's math being performed in limited work space).
2. Adding only integers can not result in a fraction or decimal number.

Even my 10 year old son knows this.

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#21

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 6:55 AM

What a bunch of nonsense....I could only watch about five minutes of it and the multiple assumptions and presumptions are too ridiculous to address, the first and most blatant absurdity is when they say "OK, what we did was add up all the positive numbers from one to infinity, every one of them, and the sum we came up with is -1/12".

Me thinks if they added up all the numbers from to one to infinity we would not have heard their result yet, as they would still be adding numbers from now to infinity.

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#44

### Re: Sum of All Positive Numbers is -1/12?

11/13/2015 2:42 AM

Exactly! They would never know if / when they were done....on the other hand, if one adds for long enough without sleep, - 1/2 becomes more plausible, and supports perhaps a theory that first light from the universe passed us between 1914 and 1918, and we await this event for nought.

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#22

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 6:58 AM

The whole thing falls apart at the S1 stage. Taking an average value of 0 and 1 as the answer to infinity is nonsense. I really don't believe these fellows believe it themselves either. Next please ........

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#42

### Re: Sum of All Positive Numbers is -1/12?

11/12/2015 10:00 PM
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#43

### Re: Sum of All Positive Numbers is -1/12?

11/12/2015 10:30 PM

Yes. You should read pages 3-4. At the end of page three it states that one gets a nonsensical result. (Just in case you misunderstand the meaning of "nonsensical" I gave you a link to a definition.) What is not pointed out in that paper is the first infinite sum of the divergent section is actually a convergent series not a divergent series. When one applies a convergent analysis to a divergent infinite series, one will get a false result.

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#26

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 9:48 AM

It sounds right from the formula proposed.

But the sum of all positive numbers to me means no negative numbers can be used.

The proof is full of negative numbers - which by careful selection has given -1/2.

Mathematicians out there could no doubt produce 42.

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#27

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 9:56 AM

Come to think of it, this specious analysis is supposedly one of the lynch pins of string theory. That can explain a lot!

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#28

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 10:12 AM

I was talking to a coworker about a month ago and this topic came up. He is much more mathematically inclined than I and shared with me he had an email exchange with some other folks on the proof. Below is the text from his email:

The whole thing is based on S1 which isn't convergent. Using very elementary methods,

….Obviously!!!

They mentioned the Riemann zeta function in a huge hand-wave at the beginning, which when you look it up (click here), says that:

But their S1 doesn't meet that criteria… the exponent in the Reimann zeta sum for S1 is -1, whose real part is not greater than 1... it's -1!

The fact that they use this sum in physics and get a non-infinite result tells me that they haven't modeled the reality correctly, not that the obviously infinite sum converges.

Physicists should only be allowed around mathematics under the supervision of responsible adults.

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#32

### Re: Sum of All Positive Numbers is -1/12?

11/10/2015 4:43 AM

Nice, would this dismiss string theory then?

It's a nobel prize you know.

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#34

### Re: Sum of All Positive Numbers is -1/12?

11/10/2015 8:27 AM

No, only human beings can earn a Nobel Prize for their work. The person and not the work is the prize winner. I know that is a semantics argument but sometimes one must get the semantics correct.

Alexander Polyakov has performed some string theory work and he did earn the Nobel prize. I'm not sure if his prize is for work using string theory or not.

String theory is a problematic subject in Science today. Peter Woit's book "Not even wrong: The failure..." still does a very good job of summing up the hopes and drawbacks of string theory.

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#35

### Re: Sum of All Positive Numbers is -1/12?

11/10/2015 9:20 AM

You are correct.

I am not able to find a reference to Polyakov being awarded a Noble Prize. In 2013 he was awarded the Fundamental Physics Prize for for his many discoveries in field theory and string theory including the conformal bootstrap, magnetic monopoles, instantons, confinement/de-confinement, the quantization of strings in non-critical dimensions, gauge/string duality and many others.

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#36

### Re: Sum of All Positive Numbers is -1/12?

11/10/2015 9:50 AM

Oops, my mistake, that's the wrong prize.

None the less a significant prize. My apologies to any I offended with this error. I still applaud professor Polyakov on this work.

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#37

### Re: Sum of All Positive Numbers is -1/12?

11/10/2015 9:54 AM

No offense here. I went to look for what his Nobel Prize was in and discovered the Fundamental Physics Prize.

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#38

### Re: Sum of All Positive Numbers is -1/12?

11/10/2015 10:01 AM

Allow me a final comment on string theory.

String theory is a complicated, convoluted, unproven (demonstrated) theory. That does not mean that it is a false theory or that it should be dismissed. String theory has yet to predict something we can test for that differs from other less complicated theories.

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#39

### Re: Sum of All Positive Numbers is -1/12?

11/10/2015 11:33 AM

I have a good friend who is a mathematician; he is currently a calculus professor at Kansas State University.

For several consecutive summers, he spent time at MIT working on string theory.

He could be wrong, but he thinks string theory is BS.

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#40

### Re: Sum of All Positive Numbers is -1/12?

11/10/2015 11:45 AM

At the very least, string theory violates Occam's Razor for little to no reward. I agree, it looks like BS to me.

Then again, sometimes one should spread some BS on the ground to produce something more fertile.

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#29

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 10:19 AM

It looks good to me.

So let's say you give me a dollar today; then tomorrow you give me \$12; the following day \$24; \$36, and so on: at the end of time we will be all square.

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#30

### Re: Sum of All Positive Numbers is -1/12?

11/05/2015 11:21 AM

At the end of time, your lender will still owe you money. Is this what caused the housing bubble to collapse?

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#31

### Re: Sum of All Positive Numbers is -1/12?

11/06/2015 4:17 PM

Zeno, the Greek, disproved this milenia ago.

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