Now this is just mindblowing in every sense of the term.

If you take all the whole numbers, from 1 to infinity and add them together, you get:
-1/12



If you don't like the way they manipulate the various series (which I have some issues with), here is a slightly more rigorous proof that the sum of all integers is, in fact, equal to -1/12.

Riemann-Zeta Function

Apparently, this has real implications for physics (esp. string theory).

I'm definitely going to spend some time wrestling with this the next time I have a lot of time, and a reasonably high dose of some psychedelic.

Blessings
~ND

I've been trying to wrap my head around this since that video came out. The best I can do is analogize it to Euclid's controversial 5th postulate. Our intuition says that two parallel lines never meet, but in the physical universe this becomes more complex when we consider something like the bending of space. With the sum of all integers, I assume it's the same. Like Euclid's 5th, you can either operate on the intuitive assumption that it's "infinity" or use a different set of assumptions like in non-euclidean geometry and collapse the thing into something workable.

That's the best I can do with my eager education.

btw, your link to the other proof is the same as the embedded video. I think you meant to link here.

I'm eager to know what others think about this.

1/12...12... interesting, as the duodecimal (base 12) number system could be described as the optimal base for numbering:

is 12 the optimal number base?

Certainly an idea that's reflected in choices made by many ancient societies.

Blue-Velvet: I changed the link, I actually meant to link to the Wolfram site about the Zeta function. Thanks for catching that.

I'm not sure how much I like the idea of comparing it to Euclid's 5th, simply because the 5th postulate only works when constrained into an arbitrary and unrealistic world (Euclidean space), whereas something like the integers are very fundamental to our perception of reality.

It's the appearance of the negative sign that I find so hard to swallow. Geometrically speaking, adding all the integers together should only move you farther away from the origin, and yet somehow if you add enough of them together, you somehow appear on the other side?

I am just beside myself.

What the bloody hell?

I did their proof for myself, and it fit on one sheet of paper!! Why?! What?!

The ONLY thing I find suspect is the shift when adding 2(1-2+3-4+...). I know that where one begins adding them should not make a difference.

Actually, let me try that. I'll scoot one more to the right:

S=1-2+3-4+5...
2S=1-2+3-4+5...
+[1-2+3-4+5...]
2S=1-2+4-6+8-...
2S=1-2[1-2+3-4+5...]
2S=1-2S
S=1/4

The only one that doesn't seem to give a definitive result is when they are lined up. You simply get 2S=2S, which does no good. Is that a source of error? It really shouldn't matter how the addition is lined up. Infinity doesn't care where you start..... does it?



I am seriously dumbfounded.

What do I do now? How can I go on living my life like normal, now that I have this information? There is no normal! Perhaps there never was? Is that the lesson? Argh!

Two comments.

You can't manipulate infinite series in the way they do in the video, it's not rigorous and you end up with nonsensical answers or any answer you want.

e.g.

1 - 1 + 1 - 1 + ... = (1 - 1) + (1 - 1) + ... = 0 + 0 + ... = 0

1 - 1 + 1 - 1 + ... = 1 + (-1 + 1) + (-1 + 1) + ... = 1 + 0 + 0 + ... = 1

When mathematicians talk about summing the series 1 + 2 + 3 + ... , they don't mean summing in the conventional sense of the word because the series isn't convergent, i.e. it is infinite.

What they do mean, is that you can assign a number (-1/12) to this series which behaves like a sum in a technical sense. This number (-1/12) can be calculated in different ways (zeta function, ramanujan sums, etc) and you get the same answer - that's what makes it interesting.

Creo: since you seem to have a handle on all this stuff, how do you get that zeta(-1)=-1/12?

The short answer is that you simply set s = -1 in the formula for the Riemann zeta function and get the answer.

The long answer explains where the formula comes from. The best way to understand this is by analogy. Consider the following series:

1 + x + x^2 + x^3 + ...

This series is finite if -1 < x < 1 and infinite otherwise. For example:

1 + (1/2) + (1/2)^2 + (1/2)^3 + ... = 2

1 + 2 + 2^2 + 2^3 + ... = ∞

But there exists a function which is identical to the above series for -1 < x < 1 but which gives finite values otherwise:

f(x) = 1/(1 - x)

In particular:

f(1/2) = 2

f(2) = 1/(1 - 2) = -1

So, this function gives finite values when the series is infinite. Note that this doesn't mean that the series and the function are the same mathematical object.

Now, the following series is finite for x > 1 and infinite otherwise:

1 + (1/2)^x + (1/3)^x + (1/4)^x + ...

And the Riemann zeta function is simply the function which is identical to this series for x > 1 and which gives finite values otherwise. Again note that this doesn't mean that this series and the Riemann zeta function are the same mathematical object.

I really like this. It had me thinking all day. Respect to the OP for bringing this one up.

There are many ways to look at equations such as this and I have conceived an alternative approach that exposes a flaw that can easily be introduced to the process.

In my alternative viewpoint the first sequence (sequence1) used in this video 1 -1 +1 -1 to infinity=1/2 stops working unless we assume infinity to be an even number.

My reasoning is this.

Assume we have an infinate being switching a light on and off. At the end of the first on period switch (S1) the light has been on all the time. At the end of the first off sequence (S2) we have the light on for 1/2 the time. S3 (second on period 3rd switch) we have an on period (o) of 2/3. S4 = o2/4 or 1/2. S5=o3/5. S6=o3/6 or yet again you could call it 1/2 We can effictively say that if i is even then that o=S/2. If i is odd then we can say that o=S/2 +1/2. Using this logic we can see that for o to be 1/2 as in the video above then infinaty has to be an even number we also see that there is a problem that is 1/2 of infinity is still infinaty and 1/2 of infinaty add 1/2 is also infinity. Oops the whole thing now is pear shaped as the origional assumption that the answer sequence1=1/2 now does not work as the light is now has both on and off periods lasting to infinaty. The light is never half on as this state does not exist. Upon ending each off period the light has spent half it's time in each state. However during the on period it will always have been on for longer than it has been off.

The point I am trying to make here is that even with maths a problem can be approached from many angles and numbers manuilpulated to give you the answers you desire. The -1/12 answer is a really cleaver solution to a unsloveible problem but only works if you approach it from the right perspective and make assumptions that you can average out an infinate number sequence and in doing so you have to treat infinaty an an even number.

Hope this makes enough sense and gives people something to think about.

Uhhh my head, especially now that Ive watched a bunch of their other videos.


Ultimately, you guys are going about the problem all wrong. Here, let me just help you out a little bit. I've found an easy way to summarize this simply.

Creo's posts above are spot on and the video is rather misleading. This link contains all the nitty gritty details:

http://terrytao.wordpres...e-analytic-continuation/


eH

After reading through that link ^ I am 99% sure I now have HPPD. Way too much maths, my head popped.