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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...