In the End, It All Adds Up to -1/12 (1+2+3+4+ ··· ∞ = -1/12)

You might think that if you simply started adding the natural numbers, 1 plus 2 plus 3 and so on all the way to infinity, you would get a pretty big number. At least I always did.

So it came as a shock to a lot of people when, in a recent video, a pair of physicists purported to prove that this infinite series actually adds up to ...minus 1/12.

To date some 1.5 million people have viewed this calculation, which plays a key role in modern physics and quantum theory; the answer, as absurd as it sounds, has been verified to many decimal places in lab experiments. After watching the video myself, I checked to make sure I still had my wallet and my watch.

www.nytimes.com...

This is easily the most bizarre and mind blowing mathematical result I've ever come across in my life. I'm not sure I really believe it myself. It seems absurd that the answer should be anything other than infinity, if there's an answer at all. I'm amazed that I hadn't heard of it before, but the article does note it's one of the "best-kept secrets in math":

“This calculation is one of the best-kept secrets in math,” said Edward Frenkel, a mathematics professor at the University of California, Berkeley, and author of “Love and Math: The Heart of Hidden Reality,” (Basic Books, 2013), who was in town recently promoting his book and acting as an ambassador for better math education. “No one on the outside knows about it.”

The great 18th-century mathematician Leonhard Euler, who was born in Switzerland but did most of his work in Berlin and St. Petersburg, Russia, was the first one down this road. Euler wanted to know if you could find an answer to endless sums of numbers like 1 plus 1/2 plus 1/3 plus 1/4 on up to infinity, or the squares of those fractions.

Here is the video from the article:


And a more detailed video:


And some links for those who wish to read deeper into the subject:

periodicvideos.blogspot.co.uk...
www.nottingham.ac.uk...
en.wikipedia.org...
terrytao.wordpress.com...

reply to post by ChaoticOrder


I agree,
This doesn't sound reasonable at face value. But I am admittedly terrible at math. I'll have to watch the videos when I get home later.

Totally counterintuitive and I had never heard of this before.

Thank you so much for sharing this. I will now be pondering this all day!

Thanks for sharing. I watched the videos and at first was in the mindset that this was just wrong. I thought to myself, they are just manipulating the numbers to get the answer they want! After more contemplation, I began to realize the significance of their application. Certainly, in research one would need a finite number to focus upon or else one could never conclude a theory. Fascinating stuff, and I am in no ways a mathematician, but they certainly do explain it well in the videos.

Edit add: I believe this is an useful formula to grasp an idea mathematically, but think that it still is not concise nor accurate. In their 26 count string theory comparison, they say this formula assists them in knowing there are 24 levels. Personally, I would think that in truth, one would only have a guesstimate being they have applied an assumed differentiation into the equation.

ChaoticOrder
en.wikipedia.org...

I like this quote from the Wikipedia link:

In Ramanujan's second letter to G. H. Hardy, dated 27 February 1913, he wrote:

"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal.

Interesting that he mentions an expectation others might point him to the lunatic asylum.

reply to post by ChaoticOrder


I agree with him it is mathematical hocus pocus

Usedfor string theory

Which until they come up with a verifiable prediction

Is philosophy.

One of them also says string theory is at like 22 dimensions

The number of dimensions in String keep going up

This is just averages and number play because physics hates infinity

Which is why I love it

Does this answer an old question I have?

I don't remember what Greek philosopher said it, but they ( ancient Greek philosophers ) believed that numbers were finite because they have a starting line.
I'm surprised it ends in the minus tho... lol

Now, I'm going to get my mind blown by these videos, I guess.

Another_Nut
reply to post by ChaoticOrder

 

I agree with him it is mathematical hocus pocus

Usedfor string theory

Which until they come up with a verifiable prediction

Is philosophy.

One of them also says string theory is at like 22 dimensions

The number of dimensions in String keep going up

This is just averages and number play because physics hates infinity

Which is why I love it

I agree.

The first sum makes total sense:
1+1-1+1-1... = 1/2

The second sum is where the "magic trick" occurs, as they intentionally off set the second/duplicate set of calculations and since it is using the oscillating (+ then -) this sum results in the illusion of 1/4. If they just did with S2 what they did in S1 and simply used the calculations as is (1+2-3+4-5+6..) it results in (3 0 4 -1 5 -6) which CANNOT be reduced to 1/4.

Then they use the S2 results (1/4) to algebraically arrive at -1/12.

It seems because we can never count to infinity, they are using that limitation and a "magic trick" is S2 to cause us to think the answer is -1/12 because it fits their "theory".

God Bless,

The concepts discussed in this video really seem to be at the heart of it:



A key aspect of this how we get a different answer simply by changing the placement of the brackets. When you think about it, the placement of the brackets can only really be valid if you know how the series ends; does it end with +1 or -1? The answer is that it doesn't end, so it's not logically valid to say the answer is 0 or 1.

This is one of the examples he gives in the video:
S = (1-1)+(1-1)+(1-1)+(1-1) ... = 0

But if we shift the brackets forward by one place:
S = 1(-1+1)(-1+1)(-1+1)(-1+1) ... = 1

But notice one thing: we had to add an extra 1 to the second equation in order to properly fit out last set of brackets in there, which he doesn't mention in the video. The first equation uses eight 1's but the second equation uses nine. So essentially the use of these brackets is just another way of defining how the series should end. If you format the brackets as they are in the first equation then we're essentially saying that it ends with -1, but if we shift the brackets forwards by one place we're saying that it must end with +1.

But of course it doesn't end with a +1 or a -1 because it never stops, it goes on for infinity, therefore the most rational answer would be 0.5 or 1/2 and in the video he shows one method which leads to that conclusion. But my main question would be, is it even valid to attempt to apply a finite value to an infinite series. Infinity is essentially just an abstract concept, so I don't know if it's truly valid to merge that concept into real mathematics. It seems a bit like trying to quantize 1+1+1+...∞ and give it a finite value when obviously the answer is infinity.

reply to post by ElohimJD

The first sum makes total sense:
1+1-1+1-1... = 1/2

The second sum is where the "magic trick" occurs, as they intentionally off set the second/duplicate set of calculations and since it is using the oscillating (+ then -) this sum results in the illusion of 1/4.

Offsetting the numbers in the way he did does nothing to invalidate the process he used. 1+1-1+1-1... = 1/2 is what allows them to reach the final result of 1/4, it's basically just a more intuitive version of the problem but it's based on the same concept. That is where the true trick occurs, without that assumption the rest is impossible.

To him and others, this is just another example of what the eminent physicist Eugene Wigner called the “unreasonable effectiveness of mathematics.” Why should such woolly and abstract concepts as zeta functions or imaginary numbers, the products of a chess game in our minds, have such relevance in describing the world?

Korzybski theorized that mathematics is a language similar in structure to our environments AND our nervous system, which explains why it is so valuable in science. This makes sense when you realize that everything is an interaction between observer and observed. It's important to remember that 'infinity' is not a thing, its a process in our brains. We imagine what would happen if a given series was extended forever, for example.

reply to post by ChaoticOrder


ya but it not just the positive numbers...they do the negatives too.....to calculate S.....so I guess it makes some sense that its not a real big large insane number...I would have guessed it came out to be 0 then...unless I have no idea what im looking at which is possible.

ElohimJD

Another_Nut
reply to post by ChaoticOrder

 

I agree with him it is mathematical hocus pocus

Usedfor string theory

Which until they come up with a verifiable prediction

Is philosophy.

One of them also says string theory is at like 22 dimensions

The number of dimensions in String keep going up

This is just averages and number play because physics hates infinity

Which is why I love it

I agree.

The first sum makes total sense:
1+1-1+1-1... = 1/2

The second sum is where the "magic trick" occurs, as they intentionally off set the second/duplicate set of calculations and since it is using the oscillating (+ then -) this sum results in the illusion of 1/4. If they just did with S2 what they did in S1 and simply used the calculations as is (1+2-3+4-5+6..) it results in (3 0 4 -1 5 -6) which CANNOT be reduced to 1/4.

Then they use the S2 results (1/4) to algebraically arrive at -1/12.

It seems because we can never count to infinity, they are using that limitation and a "magic trick" is S2 to cause us to think the answer is -1/12 because it fits their "theory".

God Bless,

agree!

this is just so stupid, taking an average to calculate something...

If you set your head on fire but you sit in ice water, your average temperature may be good, but how healthy is your state ??

illusionists

In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.

SeeWikipedia

The remarkable thing is that physicists are able to use these manipulations to actually predict measurements to a high degree of accuracy.

saneguy
The remarkable thing is that physicists are able to use these manipulations to actually predict measurements to a high degree of accuracy.

Some guy claiming to be a physicist on the Wiki talk page doesn't seem to be too thrilled with the popular video:

Wiki Talk

it troubles me that some guy does those videos on youtube, and has obviously no knowledge of the most basic mathematics (university level this is week 3 of a 5 years), but expresses those false things with the authority of some postdoc and doesn't scrutinize his own results, when the FIRST line of this article says the series is DIVERGENT. then in the end "yeah this is an odd result, but then i guess many physics results are counterintuitive, so this can, after all, be correct". sorry, this is no justification, this is not a mathematical approach and no physicist will say that this sum is really -1/12 (i am one). the series is divergent and will remain divergent until the end of time, the -1/12 is not totally incorrect but one has to be very careful in the wording, what's certainly false is that "the series of natural numbers converges to -1/12".

CynConcepts
Thanks for sharing. I watched the videos and at first was in the mindset that this was just wrong. I thought to myself, they are just manipulating the numbers to get the answer they want! After more contemplation, I began to realize the significance of their application. Certainly, in research one would need a finite number to focus upon or else one could never conclude a theory. Fascinating stuff, and I am in no ways a mathematician, but they certainly do explain it well in the videos.

Edit add: I believe this is an useful formula to grasp an idea mathematically, but think that it still is not concise nor accurate. In their 26 count string theory comparison, they say this formula assists them in knowing there are 24 levels. Personally, I would think that in truth, one would only have a guesstimate being they have applied an assumed differentiation into the equation.

edit on 2 21 2014 by CynConcepts because: (no reason given)

Religion, 1 Prophet, 12 disciples
Religion 24 elders in heaven

Religion has been telling us a lot more than most realize...

reply to post by ChaoticOrder


This is the point. The definition of the undefinable.

AND, with this definition, we know, that which is unknowable, we can understand, the infinite.. its the place where the intellect of man meets that which is infinite.

reply to post by Arbitrageur

In modern terms, Dr. Frenkel explained, the gist of the calculations can be interpreted as saying that the infinite sum has three separate parts: one of which blows up when you go to infinity, one of which goes to zero, and minus 1/12. The infinite term, he said, just gets thrown away.

And it works. A hundred years later, Riemann used a more advanced and rigorous method, involving imaginary as well as real numbers, to calculate the zeta function and got the same answer: minus 1/12.

“So Euler guessed it right,” Dr. Frenkel said.

Those of us who are not mathematicians probably wouldn’t care so much about infinity except that it crops up again and again in calculations of things, like the energy of the electron, that we know are finite, or in string theory, which physicists would like to hope is finite.

In this case, our current understanding of the very solidity of reality depends on coming up with a consistent way to assign values to infinite sums.

In the process known as regularization, which is a part of many calculations in quantum theory, physicists do something similar to what Euler did, arriving at a real number that corresponds to the quantity they want to know and an infinite term, which they throw away. The process works so well that theoretical predictions in quantum electrodynamics, the fancy version of the familiar force of electromagnetism, agree with experiments to a precision of one part in a trillion.

Which is remarkable given that infinite quantities have been thrown away, or “swept under the rug,” in the words of the California Institute of Technology physicist Richard Feynman, who helped invent a lot of this stuff but thought it was more than faintly scandalous.

Likewise, it is no surprise that the factor 1/12 shows up a lot in string theory equations, Dr. Frenkel said. Why it all works is still a mystery.

“Quantum physics needs its own Riemann to come and give a rigorous explanation of these mysteries,” Dr. Frenkel said.

www.nytimes.com...

reply to post by ChaoticOrder


It's just wrong. Every trick messes up the answer. And as for the actual addition of all POSITIVE numbers what do you think you get.. They already have an answer for it. It's called infinity. By the way S2 is the most wrong. Breaking infinity into a finite number has the similar effect as dividing by 0. It gives you the ability to make up any answer you want. Slide s2 over twice and tell me what you get? Hahaha... A completely different answer. Dun Dun Dun.

Didn't you guys ever prove that 1 = 0 in Geometry?

This is VERY similar to that. you just ignore basic math principles like dividing by zero, or taking fractions of infinite series and pretending they represent the whole.

The error arise when they use alternating series. Its something like to describe a truck you use a car, because it have 4 tyres.

Basic rules dictate , when non negative value summed up, you cannot arrive at negative value.
Srinivasa Ramanujan error is in the first step, he use alternating series to explain linear series.