A math break

by Vera Spillner
10/3/2017 – Or a 'maths' break if you will. Try this on for size: 1+2+3+3 …. =-1/12 (the sum of all natural numbers equals -1/12). Wait what? Sounds like a joke, yet it is actually mathematical practise. How and why this is possible is explained in a fun video by mathematician Brady Haran.

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Math between two Chess games

Ever been bored between two chess games? Well, how about calculating a few huge sums. One can for example identify every chessboard square with one number, such as a1 with 1, a2 with 2 … b1 with 9 etc. Then let's sum them all up. If we do it correctly, the sum 1+2+3...+64 will yield 2080.

Carl Friedrich GaussThe nine year old math genius Carl Friedrich Gauß, who was to become one of the most famous mathmaticians of all times, had to do a similar problem once in school. The class was asked by the teacher to add up all numbers to 100. Gauß only needed a minute to come up with the correct answer. His trick: pair up 100 with 1, then 99 with 2 and so forth, and you will receive 50 packages of equal size, namely 101. 50*101=5050. Doesn't make me want to switch places with his teacher.

In fact, this is an instance of a general formula. If one adds up n natural numbers, the sum will always yield n*(n+1)/2, such as for n=100: 100*101/2=50*101=5050.

So if anyone is enthusiastic about this and the break between two chess games long enough: give n a go! It sure will seem as if with increasing n, this product should get ever larger. But does it?

Above: Carl Friedrich Gauß portrait published in Astronomische Nachrichten 1828

Not if modern mathematics is right!

In fact, one can show that the sum of infinitely many natural numbers is not infinite, but can be set equal to -1/12! So 1+2+3+4 … = -1/12. This is not a joke, but rather based on the mathematical theory of analytical continuation.

Two mathmaticians of the Numberphile-Youtube-Channel hosted by Brady Haran show why this is true:

The entire theory is based on ideas by Bernhard Riemann (19th century) und Leonhard Euler (18th century). Both were frustrated with the fact that some sums simply seem to yield infinity, and this escapes our understanding and stops us from learning anything about such sums. The sums in some way act "not well behaved", as mathmaticians put it. This needed to be changed.

So Riemann and Euler looked at certain sums, which were well behaved in some areas but not in others. In fact, they then found a way to somehow transfer the good behavior into the badly behaving areas.


Bernhard Riemann in 1863 and Leonhard Euler (date unknown)

One such sum is the so called Riemann Zeta-Function. It is defined as the sum 1 + 1 / 2^s + 1 / 3^s + ..... . s is a so called complex number. 

This sum has well-behaved areas, such as at s=2. If you add all inverse squares 1+1/2^2+1/3^2 … it can be shown that the sum converges towards pi^2/6. This may seem a surprising value, yet it is finite and thus the sum is well-behaved there.

But there is a problem for s=-1. For this, we find
1+1/2^(-1)+1/3^(-2)+.... = 1+2+3+ … which is just the sum of all natural numbers which caused us a head ache due to its seeming divergency towards infinity.

The trick: The Zeta-Function can be written as a very complicated looking mathematical function:

Zeta function

If one now assumes that this function, which is so well behaved for certain s, can be — somehow — also anayltically evaluated in "bad" values of s, then one can simply insert s=-1 into the formula and arrive at: -1/12.

More than a game

This trick is not only mathematically sound and can be explained much better than by this short article. It also is useful in physics! For example, the sum 1 + 2 + 3 + .... appears in vacuum physics when calculating the force between two plates — the so called Casimir force. There, the minus in front of 1/12 even predicts an attraction, which can be observed. So it's much more than just a mathematical game.

We hope, that nobody has a break between two chess games which is long and boring enough to actually reach -1/12. But then, maybe some of you are interested to learn more about number theory. To those brave readers: dare to use the break between games to visit the really great youtube channel "Numberphile" and enjoy.

Correction October 4: The general formula given in the third paragraph is n(n+1)/2 not over s.

Quantum physicist, polyglot and violinist. Earned her PhD at the Institut für Philosophie in Bonn, studied quantum physics (special area of expertise: string theory) at the Institute for Theoretical Physics in Heidelberg.
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dperezcabrera dperezcabrera 10/22/2017 12:21
If S = -1/12
then S3 = 1 + 1 + 1 + 1 + .. = 0

S - S = 1 + 2 + 3 + 4 + 5 + 6 + ..
- (1 + 2 + 3 + 4 + 5 ..)
= (1 + 1 + 1 + ..) = 0

S - 1 = 2 + 3 + 4
S + S3 = 2 + 3 + 4 + 5 + ..

And then
S - 1 = S + S3
S3 = -1 but S3 = 0

-1 = 0?

Troyjs Troyjs 10/21/2017 10:00
The simplest proof that - 1/12 is an absurd result, is that if:
1+2+3+4... = - 1/12, then
2+3+4... = - 1/12 - 1.

This subtraction can be taken anywhere, such as for Eg., ...98+99+101..., subtracting or leaving out the 100. This gives the result -1/12-100. But 1+2+3... is equivalent to 2+3+4... This gives the result that - 1/12= - 13/12, which is absurd.
Troyjs Troyjs 10/21/2017 09:50
A further argument would be that mathematicians are more committed to the claim that there are infinite number of prime numbers (for example, Euclid's proof). The sum of all natural numbers is equal to the sum of all prime numbers, since prime numbers can be reduced, Eg.,..2+3+5 = 2+1+1+1+1+1+1+1+1, which = 2+3+4+1,and 4 can be taken from the following prime number 7 and added to the 4 to make 5, giving 2+3+4+5, and the process continuing ad infinitum. This gives us the result that the sum of all primes is - 1/12, which is a finite number.
Troyjs Troyjs 10/21/2017 09:24
This is used as a heuristic tool, rather than positing a mathematical fact. The mathematician David Hilbert was very clear that when calculating with infinite values, we can get absurd results. For example, infinity + 3 = infinity, therefore infinity - infinity = 3. Also, the sum of all natural numbers 1+2+3..., is equal to the sum of 1+1+1... to infinity. This implies that - 1/12 is equal to infinity.
mikalziane mikalziane 10/6/2017 02:00
Also worth reading this comment on the very same (old) video
mikalziane mikalziane 10/6/2017 01:49
Thanks Vera for this math break and thanks nanopunk for the link to T. Tao's clarification.
Clearly the result -1/12 is wrong IF it is interpreted as with convergent series BUT it must not and this is what should be said at least in the article. What is also missing is a clearer explanation of how this is used in physics. Strings theories are hardly testable, yet, so this is not a good justification. But there are other application it seems in other areas of physics, such as in the Casimir effect.
bert344 bert344 10/5/2017 11:51
Mathematical (non)sense:
0 x 1 = 0 x 2, so (0 x 1) / (0 x 2) = 1 and if we strike away the equal numbers above and under the division line (remove the zeros) then it shows that 1 = 2.
This only goes to show that not every series of steps that individually look logical, will together form an acceptable chain. In Math, you cannot divide by zero, nor can you “calculate” in a traditional way with infinity.

On the other hand, you can look at “behavioral patterns” of infinite number series.
We all know that the series 1/2 + 1/4 + 1/8 + 1/16 + … converges to 1, but we have difficulty to “see” a similar behavior if the row goes up and down, like 1 – 1 + 1 – 1 + ….
If you write it as (1 – 1) + (1 – 1) + … then the result would be 0, but if you take the first 1 aside, then the rest of the series equally seems to add to 0, and the sum would then be 1.

With an infinite series that seems to jump between 0 and 1, the average “behavior” is 0,5 and when going into infinity the series looks like a “line” that trembles around 0,5.
That is why one could say that practically the “limit value” of the series is 0,5.

You could compare this with electrons: when they move as electricity in a copper wire, they behave like individual parts, but in an atom they swirl around the kernel so fast that they behave as a shell.
rgorn rgorn 10/5/2017 11:50
It's quite simple: you cannot effectively add an infinite number of quantities. Adding never comes to an end and so you never get to the sum. Only conclusion: an infinite number of quantities has no sum. Period. Even 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2 is nonsense per se.
SambalOelek SambalOelek 10/5/2017 08:42
math is just a way to avoid thinking a bit more out of the box , right?

chess is also a box

go is also a box
what isn't a box?
SambalOelek SambalOelek 10/5/2017 08:12
maths sucks in explaining reality

my two cents

it's just a fools hobby
esrefeskinat esrefeskinat 10/5/2017 02:22
This is totally wrong. The series 1-1+1-1+1 ... is not convergent, so the sum S1 can not be claimed to be 1/2. Everything that follows depends on this result and is thus wrong.
jeroen jeroen 10/5/2017 01:09
nice article, but it reminds of an act of crime
KingZor KingZor 10/5/2017 12:46
This reminds me of a great mathematician named Costello who, using the same reasoning as these guys, proved that 7x13 = 28

fons fons 10/5/2017 03:00
I always find it funny when people in the comments go against some of the biggest authorities of the planet on a certain subject.

I don't claim to understand this, but I am also not going to claim all of this is just nonsense.
Mr TambourineMan Mr TambourineMan 10/4/2017 06:32
A Go player, and a math student I met last summer showed me this so-called evidence. As I also had studied maths earlier, but not so much, I asked him if not the sum of all natural numbers, which I thought was infinite, not - 1/12, yet anyway twice as big as the sum of all odd numbers, or the sum of all even numbers. But I received the answer that both of them was infinity large. Question. Should not the sum of all odd numbers or the sum of all even numers then be -1/24? Hope someone can give me an answer and a tip on how I can overthrow his own logic next time!
KWRegan KWRegan 10/4/2017 05:09
We used a pic of this equation in a real physics book (on String Theory no less) in our St. Patrick's Day post in 2014: https://rjlipton.wordpress.com/2014/03/17/happy-st-patricks-day-2014/
hansj hansj 10/4/2017 03:34
Nonsense, sheer nonsense.
nanopunk nanopunk 10/4/2017 02:04
Here is a rigorous explanation about this series by Terence Tao:

Basically, the article conclusion -as stated- is pure nonsense; -1/12 is certainly a valid LOWER ORDER term but there is then another (positive and very large) term.
Frederic Frederic 10/4/2017 01:04
I don't accept this result, so I am checking it manually. I have got to 16,231 + 16,232... Might take a while for me to get to infinity. But seriously: the article is genuine and valid. Vera is one of the three smartest people I have met in my life. I basically believe _everything_ she tells me (or plays on her violin).
KingZor KingZor 10/4/2017 12:30
As a complete non-mathematician, I'm glad to see bert344's comment. The shifting of the second row in 2*S2 struck me as arbitrary. Likewise, it's not made clear why S2 is added to S at the end.
kenard kenard 10/4/2017 12:00
I thought the answer was 42 :-D
brian8871 brian8871 10/4/2017 02:24
Here's another demonstration of some funky math. The sum of the series 1 + 2 + 4 +8 + 16 +... is infinite, right? Okay, let x = 1 + 2 + 4 + 8 + 16 +.... In that case, 2x would be 2 + 4 + 8 + 16 +.... That means 2x - x = -1 since all other terms cancel out. So then x = -1, meaning the sum of the series is -1. Of course, the fallacy is that infinity minus infinity cannot be defined, so it's wrong to say 2x - x equals anything.
bert344 bert344 10/4/2017 12:50
As aMathematician my first impression was: "this is BS". Especially when 2*S2 was calculated by shifting the second row, I had to object, because you cannot do this - it destroys the rythm of the row!
But there is an easier way: put the first 200 numbers of this row into Excel and then you see that the row of the SUMs alternates like +1, -1, +2, -2, +3,-3 etc.
Then the AVERAGE of these SUMs variates between 0 and a row that converges to 0,5.
So the AVERAGE of the AVERAGES is a row that converges to ... 0,25 QED.
lnlver lnlver 10/4/2017 12:49
If you think this is crazy, check this out: Ramanujan’s Formula for Pi
jmafoko jmafoko 10/4/2017 12:28
frankly i am more confused, what? 1+2+3+..=-1/2?
SambalOelek SambalOelek 10/3/2017 10:55
juliok i see the same things after six beers
juliok juliok 10/3/2017 10:28
I'm playing a series of longer and longer matches with Stockfish.
Lately it keeps telling me I'm doing really well, and I'm likely to end up 1/12 point ahead.
Anybody else seen this?
SambalOelek SambalOelek 10/3/2017 09:11
Serious Math Problem

what if Ivanchuck is getting fired up for a match with Carlsen . Would it be a mathematical elo's clinch or just a genius Ivanchuck defying all laws of mathematics?
Zvi Mendlowitz Zvi Mendlowitz 10/3/2017 07:25
Here is a problem combining infinite series (as in the above article) and chess.
A white king, white pawn and black king are placed randomly on an n x n chessboard. What is the probability that white wins as n -> infinity?
Zvi Mendlowitz Zvi Mendlowitz 10/3/2017 07:19

The hands are interchangeable at 12k/143 where k is any integer from 0 to 142. If k is a multiple of 13 we get one of the 11 (not 12) trivial solutions. To get then "twin" of any k just multiply k by 12 and take the remainder modulo 143.
For example, if we take k=33 then 12*33/143 = 2.769230... which is 2:46:09 + 3/13 seconds. 12*33 is 396, which mod 143 is 110. And 12*110/143 = 9.230769... which is 9:13:50 + 10/13 seconds. This is the twin of 2:46:09 + 3/13 seconds.
Andrea Mori Andrea Mori 10/3/2017 03:35
Well, in fact viewing the sum 1+2+3+4+... as the value at s=-1 of Riemann's zeta is but one possible option. Other options are possible making up for possible different outcomes. But these are just speculations: fact is that the sum 1+2+3+4+ ... diverges as any first year Mathematics (or for that matters Physics) student knows very well (which is actually the correct answer).