I'm gonna give you an astounding result Astounding? An astounding result So I'm just gonna write down a little sum I'm just gonna see what answer it gives. 1 + 2 + 3 + 4 + duh duh duh duh duh And I include all the
natural numbers, so all the way up to infinity So what do you reckon the answer to this is, Brady? -Well I would say it would go to tend towards infinity. Yeah, that makes sense, doesn't
it. The answer to this sum is -- remarkably -- minus a twelfth. It's amazing! I mean, I first saw this result
when I start learning a bit of String Theory And what's even more bizarre is that this
result is used in many areas of physics This is a very well known string theory
textbook by Joe Polchinski. As you can see, sort of quite early on, page 22, we have this
statement here which is that the sum of all this is --- basically saying the sum of
all the integers --- all natural numbers all the way up to infinity, is, minus a twelfth. Alright so we're going to prove now, Without getting our knickers in a twist with Riemann zeta functions, we gonna prove in quite simple way, why the sum of all the natural numbers is indeed minus a twelfth. So, to do that we're gonna do this in a
number of steps. We're gonna to look at a few different sums. The first sum that I want to look at, I'm gonna call it S1. And it's 1 -1 +1 -1 +1 -1 and so on. That's the first sum I wanna look at in principle. The second sum I'm gonna look at is 1-2+3-4 and so on Carry on in that process all the way up. And the third one is of course going to be the one we're really interested in which is 1+2+3+4... and so on. So we're gonna evaluate all these 3 different sums. Now the first one is really easy to evaluate. We need to attach a number. Now clearly what is the answer to this? You stop this at any point. Okay, if you stop it at an odd point you're going to get the answer 1. You stop it at an even point you'll get the answer 0, clearly. That's obvious right? So what number are we gonna attach
to this infinite sum? Do we stop at an odd or an even point? We don't know so we take the average of the two. So the answer is a half. There're other ways to prove that this sum is a half by the way, which we can do if you want... No, no, there'll be a link, there'll be a link there, 'cause we've done it before. ok, good ...
See the link there Ok so this is a half and I think intuitively that's the easiest way to say that you either get 0 or 1, and therefore you just take the average... So this is the natural number to attach to this sum. So once we know this we're laughing, ok? Because from this we can achieve everything we want to achieve. The next step is to find out what this sum is. So what I'm gonna do is... I'm going to take two copies of this S2, ok. So I'm gonna add it to itself. So 2 times S2 is equal to... Let me just write it out. I will write it out twice 1 -2 +3 -4 duh duh duh and so on. And then I'm gonna add to it itself, but I'm gonna shift it along a little bit So that's +1-2+3-4 and so on I've just taken two copies and added them together. But I've done it in a particularly nice way, and I just pushed this one along slightly on the bottom. So now see what you get: you take this 1 and this... This and this and I get 1. So 1 + nothing is 1 -2 +1... -1 3 -2 is 1 -4 +3 is -1 duh duh duh and so on. And I 'm gonna keep getting this pattern. So hang on a minute, I just got back to the sum that I started with. Which I know the answer to it: a half. Ok, so therefore I know this sum. So let's divide through by this 2 and I get a quarter. So now I know that this sum, the second sum with somewhere the signs alternate is actually equal to 1/4. So this is my second remarkable result. Now I have everything I need to prove
this crazy -1/12 thing, right. Ok, so let's do it. I'm going to take this one and I'm going to subtract this one. Ok, so I'm going to substract S2 from S. Let's write them (?) Ok, so I'm going to write our S first which is 1 + 2 +3 +4 +5 and so on. Ok. And I'm going to subtract S2. So that's minus and let me just put a bracket. And now, because I'm gonna "minus" all of this 1 -2 +3 -4 and so on. Ok 1 - 1 what's that? -Zero
-Zero, yeah, exactly. So I get nothing from that bit. 2 - (-2) is... 4 Ok 3 - 3... 0 4 - (-4) is... 8 And so on. And the next one, I'm gonna get it from the 5, so I get nothing. From the 6 here I get 12, and so on duh duh duh... And it proceeds in that way. And now you can see we're almost there now right. Because look what I've got here. I've got 4 + 8 + 12. I'll take a factor of 4 out. It's 4 times 1 + 2 + 3 duh duh duh Ok It's my sum I want to... So now I've got a formula So this is just 4 time S, which is my sum. Now I just solve this equation, right, because I know what S2 is. So I have now the expression S minus... I know what S2 is: S2 is 1/4 is equal to 4S. Ok, let's take S from either sides, so I get -1/4 is equal to 3S. Which implies that S equals -1/12. (?) you believe me. That's amazing, I love it... It's so nice... (?) Tony, if I've got a calculator out, and wrote 1 + 2 + 3 + 4 + 5 and I sat here until the end of everything, and then press = -Will I get -1/12 ?
-What do you mean by the end of everything? You can't do it till the end of everything, can you? So, the point is I know it looks like a
bit of mathematical hocus-pocus or thing, but... I tell you the truth it's not, and I'll tell you why we know it's not: And I know you think I've gone about physics too much. But we know it's not because of physics. Because these kinds of sums appear in physics. And in physics we don't get infinite answers. It's amazing. It's amazing and it's just... You know, I was trying to come up with an intuitive reason for this, and I just couldn't. *laughing*
to be honest You have to do the mathematical
hocus-pocus really to see it. And then you just have to believe that you're not measuring physical infinities in nature and those two facts, I think, give you confidence in this result. But... it is clearly counter-intuitive.
It's counter-intuitive because intuitively, you just want to stop the sequence, and in the minute you stop the sequence... then all your intuition for this result goes out the window So what if I do 1 + 2 + 3 + 4 + 5... and I go up to a googleplex ? You get a big number... you won't
get anything like -1/12. You'll have to get to infinity, Brady It's negative. I've added all these positive numbers together up to infinity and I've got -1/12. But it does play a role in lots of different things that the number 12 and and it's so for
example as I said the calculations of the critical dimension in string theory, the 26 dimensions comes from this calculation.
Found this posted on a separate forum from 2010:
Wendell Wagner
Look at the ways that an infinite sum is different from a finite sum. When we say that 1 + 2 + 3 + 4 = 10, we're saying anybody with a knowledge of ordinary arithmetic can solve the problem in a finite amount of time. Furthermore, because of the laws of arithmetic, the summation can be done in various ways. We can add it like this:
1 + (2 + (3 +4))
or like this:
((1 + 2) + 3) + 4
or several other ways.
We can also add it in other orders like this:
4 + 3 + 2 + 1
or this:
2 + 1 + 3 + 4
In other words, finite addition is both associative and commutative.
But what does it even mean for there to be an answer to the infinite addition?:
1 + 1/2 + 1/4 + 1/8 + 1/16 + . . .
You can't say that it's obvious just from the usual definition of finite addition. By the usual definition of infinite addition, that sum is 2. But what does that mean? It means that if you add the numbers in the given order, the sum will get closer and closer to 2. You have to add them in the given order though. Suppose that you were to say, "Well, I don't want to add in the given order. I want to add in my own order. I want to add it like this:
1 + 1/4 + 1/16 + 1/64 + . . . + 1/2 + 1/8 + 1/32 + 1/128 + . . ."
If you were to add it that way, you could say that the sum doesn't get closer and closer to 2. The sum, as far as you can take it, gets closer and closer to 1 and 1/3. That's because you never get around to adding the second part of the sum. So you have to have a different definition for addition when you do infinite addition.
Notice that we could say that the infinite sum:
1 + 1/2 + 1/4 + 1/8 + 1/16 + . . .
could be split up into the two infinite sums:
1 + 1/4 + 1/16 + 1/64 + 1/128 + . . .
The first sum adds up to 1 and 1/3 and the second sum adds up to 2/3, so the two infinite additions give the same answer when added together as when split apart.
Now look at this sum:
1 + 2 + 3 + 4 + 5 + . . .
You can't use the same rule as before to define this infinite sum. You don't get closer and closer to anything. Now, it's true that according to the standard definition in this case, the sum of this is ∞, since the sum increases without bound. But that's a new definition. Nothing in the standard definition of finite addition or infinite addition (when the sum converges toward a finite number) tells you this.
Now look at this sum:
1 + (-2) + 3 + (-4) + 5 + (-6) + . . .
You can't use the rules of finite addition to come up with this sum. You can't use the rule of what the sums are converging toward, because they aren't converging toward a single number. You can't say that the sum is ∞ or -∞, because the sum isn't growing upward or downward without bound. The sum is bouncing back and forth between 1 and (-1). You have to come up with a new definition for the sum. One way is to say that no sum exists. That's a definition though, not something that's clear from previous definitions.
So when you ask how this can be true:
1 + 2 + 3 + 4 + 5 + . . . = (-1/12)
The answer is that it's a new definition for the meaning of infinite sums. You want to know how this can contradict the sum that you know of:
1 + 2 + 3 + 4 + 5 + . . . = ∞.
It's because it's using a different, older definition for infinite sums. It's possible to use different definitions in different parts of mathematics. It's like asking what the answer to this question is:
11 + 11 = ?
The answer can be 22 or it can be 1001 (or it can be many other things). It depends which number base you're using. In decimal numbers, the answer is 22. In binary numbers, the answer is 1001. If you're going to understand mathematics, you have to understand that the definitions are different in different parts of the field.
Something else occurred to me. You might say "Well it's obvious the sum:
1 + 1/2 + 1/4 + 1/8 + 1/16 + . . .
adds up to 2. Nobody could possibly doubt that."
Well, somebody did doubt that. It's called Zeno's Paradox:
http://en.wikipedia.org/wiki/Zeno's_paradoxes
Until someone figured out what infinite addition meant, this was a really hard problem. In any case, it's necessary to make clear definitions in mathematics. Sometimes these will strike you as arbitrary definitions.
I'd also like an explanation of how this isn't hokus-pokus. I'm just an undergrad, but I've been taught that rearranging non-absolutely-convergent series, let alone divergent series, is a big no-no because you can make it equal whatever you want. Is there an area of study where this kind of thing is okay? In what sense does adding a string of numbers give you their average? I feel like I'm missing a piece of the puzzle.
That's probably the most counter-intuitive thing I've ever seen.
I have a feeling that this video should include a "Do not try this on your AP exam" disclaimer.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders two. The third one three beers.
The bartender stops them, and says, "You all owe me one-twelfth a beer".
More jokes of this nature can be found here.
Can someone explain this to me please ? I mean isn't this series divergent by the divergence test anyway? How is it convergent and how is its sum negative?
I'm so lost.
I really want to know where they find all of this brown paper
And that's numberwang.
Sorry. I'll just get my coat.
Well without what definition of convergence I'm using, I too can find all kind of crazy results