SUMMATION

     

Recently a very strange result has been making the rounds. It saysthat when you địa chỉ up all the natural numbers

1+2+3+4+...

then the answer to lớn this sum is -1/12. The idea featured in a Numberphilevideo (see below), which claims to prove the result và also says that it"s usedall over the place in physics.People found the idea so astounding that it even made it into the NewYork Times. So what does this all mean?

The american-home.com.vn

First of all, the infinite sum of all the naturalnumber is not equal to lớn -1/12. You can easily convince yourself of this by tapping into your calculator the partial sums

*

*

*

*

*

*

and so on. The get larger and larger the larger gets, that is, the more natural numbers you include. In fact, you can make as large as you lượt thích by choosing large enough. For example, for

*
you get

*
" style="width:101px; height:16px" class="math gen" />

and for

*
you get

*
" style="width:148px; height:16px" class="math gen" />

This is why mathematicians say that the sum

*
" style="width:126px; height:13px" class="math gen" />

diverges lớn infinity. Or, khổng lồ put it more loosely, that the sum is equal khổng lồ infinity.


*

Srinivasa Ramanujan


So where does the -1/12 come from? The wrong result actually appearedin the work of the famous Indian mathematician Srinivasa Ramanujan in 1913(see this article for more information). But Ramanujan knew what he was doing và had a reason forwriting it down. He had been working on what is called theEuler zeta function. Lớn understand what that is, first consider theinfinite sum

*

You might recognise this as the sum you get when you take each natural number, square it, & then take the reciprocal:

*

Now this sum does not diverge. If you take the sequence of partial sums as we did above,

*

*

*

*

*

then the results you get get arbitrarily close, without ever exceeding, the number

*
Mathematicians say the sum converges khổng lồ
*
, or more loosely, that it equals
*

Now what happens when instead of raising those natural numbers in the denominator to lớn the nguồn of 2, you raise it to some other power nguồn ? It turns out that the corresponding sum

*
" style="width:236px; height:17px" class="math gen" />

converges lớn a finite value as long as the power nguồn is a number greater than

*
. For every
*
1$" style="vertical-align:-1px; width:38px; height:12px" class="math gen" />, the expression has a well-defined, finite value. is what’s called a function, and it’s called the Euler zeta function after the prolific 18th century mathematician Leonhard Euler.

So far, so good. But what happens when you plug in a value of that is less than 1? For example, what if you plug in

*
? Let’s see.

*
" style="width:276px; height:19px" class="math gen" />
*
" style="width:149px; height:13px" class="math gen" />

So you recover our original sum, which, as we know, diverges. The same is true for any other values of less than or equal to 1: the sum diverges.


Extending the Euler zeta function

As it stands the Euler zeta function S(x) is defined for real numbers x that are greater than 1. The real numbers are part of a larger family of numbers called the complex numbers. & while the real numbers correspond to lớn all the points along an infinitely long line, the complex numbers correspond lớn all the points on a plane, containing the real number line. That plane is called the complex plane. Just as you can define functions that take real numbers as đầu vào you can define functions that take complex numbers as input.

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One amazing thing about functions of complex numbers is that if you know the function sufficiently well for some phối of inputs, then (up to some technical details) you can know the value of the function everywhere else on the complex plane. This method of extending the definition of a function is known as analytic continuation. The Euler zeta function is defined for real numbers greater than 1. Since real numbers are also complex numbers, we can regard it as a complex function và then apply analytic continuation to lớn get a new function, defined on the whole plane but agreeing with the Euler zeta function for real numbers greater than 1. That"s the Riemann zeta function.
But there is also another thing you can do. Using some high-powered mathematics (known as complex analysis, see the box) there is a way of extending the definition of the Euler zeta function lớn numbers

*
less than or equal to 1 in a way that gives you finite values. In other words, there is a way of defining a new function, điện thoại tư vấn it
*
so that for 1$" style="vertical-align:-1px; width:38px; height:12px" class="math gen" />

*

and for

*
the function
*
has well-defined, finite values. This method of extension is called analytic continuation và the new function you get is called the Riemann zeta function, after the 19th cenury mathematician Bernhard Riemann. (Making this new function give you finite values for
*
involves cleverly subtracting another divergent sum, so that the infinity from the first divergent sum minus the infinity from the second divergent sum gives you something finite.)

OK. So now we have a function

*
that agrees with Euler’s zeta function
*
when you plug in values 1$" style="vertical-align:-1px; width:38px; height:12px" class="math gen" />. When you plug in values
*
, the zeta function gives you a finite output. What value vì you get when you plug into the zeta function? You’ve guessed it:

*
" style="width:113px; height:17px" class="math gen" />

If you now make the mistake of believing that

*
for , then you get the (wrong) expression

*
" style="width:330px; height:17px" class="math gen" />

This is one way of making sense of Ramanujan’s mysterious expression.

The trick

So how did the people in the Numberphile đoạn phim "prove" that the natural numbers all địa chỉ cửa hàng up to -1/12? The real answer is that they didn’t. Watching the video is like watching a magician & trying to lớn spot them slipping the rabbit into the hat. Step one of the "proof" tries khổng lồ persuade you of something rather silly, namely that the infinite sum

*
" style="width:160px; height:13px" class="math gen" />

is equal khổng lồ

*

The video doesn’t dwell long on this & seems lớn imply it’s obvious. But let’s look at it a little closer to lớn see if it makes sense at all. Suppose that the sum

*
has a finite value & call it . Adding khổng lồ itself you get the infinite sum

*
" style="width:409px; height:14px" class="math gen" />

But this is just the original sum, implying

*
" style="width:124px; height:14px" class="math gen" />

Since

*
it follows that
*
which is nonsense. So the assertion that the infinite sum
*
can be taken lớn equal to 1/2 is not correct.

In fact, you can derive all sorts of results messing around withinfinite sums that diverge (see here). It"s a trick!

The physics

But how did this curious, wrong result make it into a physics textbook, as shown in the video? Here is where things reallyget interesting. Suppose you take two conducting metallic plates & arrange themin a vacuum so that they are parallel to lớn each other. According to lớn classicalphysics, there shouldn"t be any net force acting between the twoplates.


*

Illustration of the Casimir effect. Image: Emok.


But classical physics doesn"t reckon with the weird effectsyou see when you look at the world at very small scales. To vày that,you need quantum physics, which tells us many very strange things. Oneof them is that the vacuum isn"t empty, but seething withactivity. So-called virtual particles pop in & out of existence allthe time. This activity gives a so called zero point energy: the lowest energy something can have is never zero (see here for more detail).

When you try lớn calculate the total energy mật độ trùng lặp từ khóa between the two plates using themathematics of quantum physics, you get the infinite sum

*
" style="width:148px; height:14px" class="math gen" />

This infinite sum is also what you get when you plug the value

*
into the Euler zeta function:

*

That’s unfortunate, because the sum diverges (it does so even quicker than than

*
), which would imply an infinite energy density. That’s obviously nonsense. But what if you cheekily assume that the infinite sum equals the Riemann zeta function, rather than the Euler zeta function, evaluated at
*
? Well, then you get a finite energy density. That means there should be an attractive force between the metallic plates, which also seems ludicrous, since classical physics suggests there should be no force.

But here’s the surprise. When physicists made the experiment they found that the force did exist — & it corresponded lớn an energy mật độ trùng lặp từ khóa exactly equal khổng lồ

*
!

This surprising physical result is known as the Casimir effect,after the Dutch physicist Hendrik Casimir.

Take a moment to take this in. Quantum physics says the energy mật độ trùng lặp từ khóa should be

*
" style="width:215px; height:17px" class="math gen" />

That’s nonsense, but experiments show that if you (wrongly) regard this sum as the zeta function

*
evaluated at
*
, you get the correct answer. So it seems that nature has followed the ideas we explained above. It extended the Euler zeta function to lớn include values for
*
that are less than 1, by cleverly subtracting infinity, and so came up with a finite value. That’s remarkable!

The reason why we see &

*
in the Numberphile đoạn clip and the physics textbook, rather than and
*
is that when you imagine the Casimir effect as happening in one dimension (along a line rather than in 3D), the energy mật độ trùng lặp từ khóa you calculate is rather than .

So why did the Numberphile people publicise this strange "result"? They certainly know about the analytic continuation that makes the functionwell-defined, but that was something that was a little too technical for their video. Knowing they had the analytic continuation method, that would make the final result OK, hidden in their back pocket, they went ahead with their sleight of hand. In doing so they got over a million hits & had the world talking about zeta functions & mathematics. For this they should be congratulated. The mathematics of zeta functions is fantastic and what we described here is just the start of a long list of amazing mathematical properties. In bringing mathematics and physics khổng lồ the public we always have khổng lồ make choices about what we leave out and what weexplain. Where to lớn draw that line is something we all have khổng lồ leave khổng lồ our consciences.

About the authors


*

David Berman is a Reader in Theoretical Physics at Queen Mary, University of London. He previously spent time at the universities of Manchester, Brussels, Durham, Utrecht, Groningen, Jerusalem and Cambridge as well as a year at CERN in Geneva. His interests outside of physics include football, music and theatre và the arts.

Marianne Freiberger is editor of american-home.com.vn.


I appreciate this piece, and it"s helped me make sense of a lot of what"s going on here. However, I have to lớn part ways in calling for the Numberphile folks to be "congratulated." As a math teacher, và one who struggles every day to counter the deeply-ingrained notion that math makes no sense whatsoever, I can"t stand it when that notion is spread across a wide audience và further ingrained into our culture. Sure, those who are already somewhat mathematically inclined are intrigued and want to know more. But those who are not see a video like this and say lớn themselves, "Further confirmation that math makes absolutely no sense." So I"m not about to congratulate them.

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Matt E.Sharon, MA


Well, I think the Numberphile guys made an entertaining đoạn clip and I wish that my school american-home.com.vn teachers had been even half as good with their explanations as these guys are. For me it was the teachers, with their leaps of faith, blasting through the set books & omitting whole chapters (homework : study chapter so & so & do the exercises) who instilled the fear of american-home.com.vn & a sense of futility into all but the three or four kids, out of a class of 25, who could figure out what was going on. american-home.com.vn teachers take a look in the mirror!

NickBerlin, Germany


Isn"t it more important to recognise that american-home.com.vn is about adventure, discovery, fun? This is simply an example of an important aspect of in mathematics - you take some concept, abstract it and extend it, and see where it leads. Shouldn"t more american-home.com.vn teaching & learning emphasise this?

A nice example that I use in my classroom, far simpler than analytic continuation and one that survives the journey more intact, concerns the index laws. You take the well understood concept that a times a times a... N times is a^n, & uncover the fact that a^m times a^n = a^(m+n). Then generalise the concept of power nguồn so you can consider things that "don"t make sense" like a^(1/2) or a^(-3). These are not meaningful under the initial view that "power means repeated multiplication", but the extension và subsequent exploration lead to important & very meaningful results. Learning the index laws can be an exercise in rote memorisation, or it can be a wonderful journey of discovery, where seeming "nonsense" becomes clarified and empowering!

Numberphile demonstrate this side of american-home.com.vn, và should absolutely be congratulated.


"More important" than what? More important than truth, precision, clarity, correctness, understanding, etc.? Many Numberphile videos are fun & all that in addition to lớn being essentially correct. However, they really blew it with the nonconvergent-series videos.You might consider steering students to Martin Gardner"s books based on his Scientific American column "Mathematical Games"; even though he wasn"t a mathematician, he had lots of contacts in the mathematical community, và his writing was clear, entertaining, and essentially correct.


I have khổng lồ disagree, và am quite bemused by the moral tone taken up by some of the detractors of this work. It"s too easy khổng lồ dismiss these videos as incorrect/untruthful etc. In context, they are a valid exploration of mathematical ideas, & thus important & valuable.

In the history of american-home.com.vn, this happened with irrational numbers, negative numbers, imaginary numbers. Formal manipulations that included "nonsense" ended up enriching mathematics. And in modern american-home.com.vn, look at p-adics where, for example. In the 5-adics, 5+5^2+5^3+... Converges but 01/05 + 1/5^2+1/5^3+... Does not. There"s the Umbral Calculus where the formal basis is still only being constructed. Also, the extended complex plane where infinity is just a point lượt thích any other.

Now you can say they sneakily departed from real numbers, but the whole question naturally goes beyond the reals because it involves infinity. Infinity is not a real number, but more relevant, it is not just a "really big number". That it is qualitatively different is important to lớn learn, & apparent from ideas lượt thích divergent and non-absolutely convergent series. As a 15 year old I was introduced to lớn 1+2+4+8+...=-1 by a american-home.com.vn professor on an excursion, và it made a tremendous and positive impression on me and my fellow students. & I use similar things lớn both communicate my love of american-home.com.vn, và to encourage others lớn look at it differently and find their own sources of wonder.

And it is not so far removed from school american-home.com.vn. We teach that infinity - infinity is undefined arithmetically, but in terms of sets we often show that it can have many values (easiest example is remove all even numbers from 1,2,3,4,5... Và you are left with an infinite set, compared to lớn match up 12, 24, 36 etc & show that none are left behind). Exposure lớn such conflicts is a great and fun way to lớn learn about the limits and context of american-home.com.vn. Và a major source of error is not learning the limits, applying things unthinkingly out of context, because too much exposure is only khổng lồ "nice" examples. There is loads of education research on this. These paradoxical results force students to lớn confront the limits, & thus can be used to lớn enhance their mathematical thinking.

The authors of the article recognise the broader context (although were clearly not entirely happy with the presentation). The commenter above, Matt E, a american-home.com.vn teacher, seems lớn miss this broader context. Of course, approaching và crossing boundaries may well mean things need khổng lồ be redefined, concepts generalised, but that"s mathematics. Use it to lớn generate interest, provide historical và real-world context, và thus enrich teaching.

Thinking about Grandi"s series is lượt thích a first step on a journey. Enjoy it, and let students enjoy it too. Relate it lớn Thomson"s Lamp. Bring mathematics lớn life!

p.s. Martin Gardner is great, certainly, but for a bit less puzzle orientation I recommend "The Heart of Mathematics: An Invitation to Effective Thinking", by Edward Burger & Michael Starbird.


I really just wish they had put some sort of disclaimer in their clip about their math trickery. As a current math major who"s done some work with infinite series, I was skeptical of their claim that (-1)^n converges to một nửa and so just ended up being confused when they said that the sum of all natural numbers converging lớn -1/12 was consistent with modern physics. Now, a while after seeing the đoạn clip I decided to bởi vì my own research & came upon this article which was very helpful in understanding what was actually going on. I suppose I have a different perspective on math than the average person, but I was much more satisfied with this, correct explanation, than their hand-wavy one.

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Think about it. At the start with the 1-1+1-1+.... Being wrong, is simply wrong. Because of the part where they say 1-1+1-1+1-....+1-1+1-1+1-1+.... Is as wrong as putting decimals after a repeating number. Even if this is true, you are assuming that the sum ends with -1. In order khổng lồ have the second sum start with +1 which is absurd because if the sum ends with -1 (which defeats the purpose anyway) then Z=0. Và if Z=0 then 2Z=Z is true. Because 0+0=0 not to mention that it isn"t even the original sum anymore.