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Sum of Natural Numbers Explained · Acedra
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Sum of Natural Numbers Explained
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0:00
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.
0:32
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.
1:00
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.
1:30
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.
2:06
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
2:31
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...
2:58
Let me just write it out. I will write it out twice 1 -2 +3 -4 duh duh duh and so on.
3:07
And then I'm gonna add to it itself, but I'm gonna shift it along a little bit
3:12
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.
3:51
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
4:23
+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.
4:50
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
5:26
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
5:41
is equal to 4S.
5:43
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.
5:56
That's amazing, I love it... It's so nice...
6:00
(?) 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?
6:15
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...
6:37
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.
6:49
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
7:02
intuitively, you just want to stop the sequence, and in the minute you stop the sequence...
7:07
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.
7:21
You'll have to get to infinity, Brady
7:24
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.
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