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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 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.
I like this quote from the Wikipedia link:
ChaoticOrder
en.wikipedia.org...
Interesting that he mentions an expectation others might point him to the lunatic asylum.
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.
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
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.
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?
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,
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.
Some guy claiming to be a physicist on the Wiki talk page doesn't seem to be too thrilled with the popular video:
saneguy
The remarkable thing is that physicists are able to use these manipulations to actually predict measurements to a high degree of accuracy.
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)
www.nytimes.com...
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.