You. Yes, you. You think you're smart? So... here's a sum. F = 1 + 2 + 3 + 4 + 5 + 6 +7 ... --> ∞ You think you know the answer? Yes? Mine did...

I don't get it... why is it right to assume that an average is equal to an infinite sum? They base their entire argument on that....????

Because it's infinite It wouldn't apply to any finite sum and that's why it seems unnatural or illogical to us - we're just too used to things having an end

this is why I am not smart To my mind it just isn't possible to come up with an accurate sum for infinity.

It seems a bit convoluted. I see where the explanation is going; but I get the feeling that there's a bit more maths underneath it that explains some of the choices. For example why are they S - S2 - S3 Why does he move the subtraction of the first one along but doesn't move it along one for the second. They appear to want to relate them together but there's no reason why they should be related - why all S and not one called A and the next B and the next C. Either its a daft thing or there's more underlying the theory that they didn't want to go into because it would take more htan 7 mins and break all out minds

It´s S1, S2 and S ;D Because he can do it in the first one (since he's adding and subtracting so the order doesn't matter) and it makes the sum easier to solve, and he doesn't have to do it in the second one (why would he? it's much easier if you don't). It's S because they are Sums, they could have called them anythig really. Call them A, B, C and you'll get the same result. Instead of S - S2 = 4S it would be C - A = 4C, and since we know that A = 1/4 we get C - 1/4 = 4C. We take one C out from both sides and it's -1/4 = 3C which implies that C = -1/12. And they are indeed related if you look at it backwards. They show you the S1 first because that's what you get in S2 and then you can go straight to the result because they already showed you that it's 1/2. And S2 is explained before S because it helps you get the result for S. Well, the guy said that there are other ways to prove that S1 equals 1/2

Darn it Oddy no more maths - my head is already spinning and getting confused and now its - no no no more thinking of it or I'll end up drawing on the walls .

Hmm but wait a sec the whole thing hinges on the first assumption that we average the results of the impossible to solve first equation. So in theory the -1/12 is only based upon an average not an actual value so surely its not actually the correct answer?

But 1/2 is an actual value of the first equation. Impossible as it may seem, that's how infinite sums work. It reminds me of how you solve Zeno's paradoxes. 1 + 1/2 + 1/4 + 1/8 + ... doesn't give you an infinitely big number but a simple, finite 2. It seems weird but it's proven. That's what's awesome in mathematics ^^

*blinks* Oddy I think you broke my brain. It just doesn't make sense to me that an infinite assembly of numbers results in a finite concept.

I know, OR, it's counter-intuitive But look here: 1+ 1/2 + 1/4 + 1/8 + ... = X If we leave the first number and take the factor of 1/2 out from the rest, we get: 1 + 1/2 (1 + 1/2 + 1/4 + 1/8 + ...) = X The equation in the brackets looks just like our X (1 + 1/2 + 1/4 + 1/8 + ...), so we can write it down as: 1 + 1/2X = X so: X - 1/2X = 1 and: 1/2X = 1 which means that X = 2 ^^

See that makes sense but also seems to me to be trying to fit our mental concept of finality into a concept of infinity that we can't easily deal with.

Do you know the story behind the Zeno equation? It's based on the observation that if you want to get from one place to another, you have to get halfway first. But to get halfway, you have to get a quarter of the way there, and to get there you have to travel one-eighth, and so on and so on... infinite halves. Zeno thought it meant that it would take infinite time to get from point A to point B. But we already busted that paradox and we know that it does not mean so. We know that X = 2 and we also know that we actually CAN get from point A to point B, we do it all the time. So what we consider to be infinte may actually be quite finite, depends on how you look at it - infinite parts can make a finite whole

Hmm but taking an infinite amount of time to go from A to B only works if you move in stages that are based upon taking the smallest step first. Plus the concept that you can rank all numbers on a possible A to B scale concept seems to counter the concept of infinity. So surely trying to apply a theory of limitations upon the concept of infinity is just attempting to make it so that "its infinity, but we only really perceive X amount of it so we'll consider the X amount the amount we'll factor for and put arbitrary A and B points that boarder the middle X *thinks that's enough 3ammaths now *

But the time is not infinite. To think that was Zeno's error. We've already proven that it isn't. (Also the first "step" would be the biggest one - 1, the second one would be 1/2, the next 1/4, etc. And well, covering half a distance and than it's half, etc. doesn't mean it has to take exactly one step each time.) Oh, but we DO perceive all the infinite parts, not just a cerain amount of it. Only we perceive it as finite - that's the twist The Zeno example is not about limiting an infinite distance but showing that a finite distance/time is made out of infinite parts. Oh, and in the equation X is time, not distance. Sleep tight! ^^

You're thinking of infinity as something you can tack down. Infinity is not a number; think of it this way... what does Infinity+1= ? Of course you can't have one more than infinity, nor can you divide infinity in half... but you can divide by infinity and there are different degrees of infinity, and not all infinities behave rationally. Infinity is a fickle mistress. Think of Infinity as more a defined process than an actual outcome. Progression towards the Infinite or within the Infinite. Take our Universe... Infinity is governed by the Natural Laws of the Universe, just like everything else. Infinity must obey thresholds and limits... except in a Singularity Event which cosmologists don't understand so much. But that seems counterintuitive doesn't it, how can the infinite become finite or be bound up and restrained? A finite amount of mass, but infinitely dense... Density = Mass / Volume... when the volume reaches zero, in fact it never actually reaches zero it's just progressing in that direction and you have a Black Hole. Infinity is sort of a cord that never quite gets struck.

This still makes no sense to me. Half of infinity is still infinity because infinity is not finite. To my mind, it's analogous to multiplying something by 0: no matter the size of the number you multiply by "0", the answer is still "0". (Even that isn't an apt analogy but I don't know how else to explain the confusion in my deficient brain). It makes complete sense to me that an infinite continuum of numbers exists, but "A" and "B" are finite points on that continuum that we have arbitrarily labelled with mathematical concepts of our own design. Since we label them arbitrarily with mathematical concepts, it doesn't matter that there are potentially an infinite number of points between them. We can arbitrarily calculate the sum of the "distance" between them in whatever arbitrary units we choose (eg by "1s", "10s" or "1/4s"). The point is we have to select an arbitrary unit to count between them and two arbitrary endpoints to get an answer. Once there are defined endpoints, the continuum is no longer infinite conceptually: there are only the number of arbitrary units we've selected between the two endpoints. We can reselect smaller or larger units to count between our endpoints but there is no sum unless we designate "A" and "B". The problem with adding up infinity in it's entirety is that there are no arbitrary endpoints (ie: no "A" and no "B"). My understanding of the definition of infinity is that is that "A" and "B" don't exist. I don't understand how the initial assumption of taking the average 1/2 for the first sum in the video gets around that. The concept of infinity makes perfect sense to me Sparrow and so does your explanation of it. The fact that mathematicians think they can assign a number to the sum of it, does not....lol.

In Zeno's paradox the halves are not halves of infinity. They are, for example, halves of a 40 meter distance (20m, 10m, 5m, 2,5m, etc.). He reasoned that to get to each half it will take a certain amount of time, and since the halves never end, the time wouldn't end either - but he was WRONG. Here they explain it a bit more: It's like Sparrow said, the sum will never reach 1/2 but it will get closer and closer and closer and closer. If you do the sums for Zeno's paradox by actually adding the numbers, you'll see that they get closer and closer to 2, without ever getting there.