A Harmonic Series Paradox (TANTON Mathematics)

[silvascientist]

Finite area, infinite length, sounds pretty normal to me. I mean, isn't that what fractals are all about?

[EclecticSceptic]

What the hell actually. I'm going to try and figure this out. This is so weird.

[ethandrood]

Furthermore an infinitely thin object (such as a line) would have to be infinitely long in order to manifest a length. This may explain why an infinite number of decreasing steps is needed (1/2+1/3+1/4+...) in order to move between 2 points on a line. (There are different kinds of infinity it seems?). In fact some lines can't exist, like a line of length SQRT(2). Lines are very strange!

[EclecticSceptic]

@ethandrood Interesting. However something you said is incorrect. The harmonic series, if viewed as steps from one point to another, would not be 'infinitely decreasing' as you mean. The harmonic series diverges, therefore you would eventually pass the point you wish to travel to. This is unlike the geometric series of Achilles and the Tortoise, which converges to 2.

[ethandrood]

@EclecticSceptic OK I see your point. I still think that the 'paint paradox' is flawed though because a line segment has no area and so cannot be painted.

[EclecticSceptic]

@ethandrood No I think your point about area and lines is good. The thing I pointed out is immaterial really.

[ethandrood]

I think the problem here is with the definitions of a line. A line has length only and no area so we cannot paint a line, we would need infinitely thin paint! Area and length have different dimensions. We cannot compare them. In reality a line doesn't exist.

[messakg123]

What's the answer? :/

[theCamelCase]

Nice! Here's how I resolve this paradox:

Drawing along the line would require an infinite motion of your pen because, in this situation, we have to assume that your ink is of infinitesimal thickness. When you combine an infinite length (motion of your pen) and an infinitesimal rate of growth (rate of ink usage), you get a total (amount of ink used) that is either finite or infinite. In this case, the total is finite.

[ThePrevenge]

I might be wrong, but doesn't the fact that pi has an infinite number of decimals has anything to do with this? Since the finite number (pi^2)/6 (I think it was) includes an unlimited number of decimals that should mean that the number "can always be bigger" if you reveal the next decimal. Still, adding a new decimal wouldn't change the value of the previous decimals. This would mean that writing the number would be an infinite process, and maybe it would be the same with the paint, in some way

[messakg123]

@ThePrevenge nah. The number having infinite decimal places is just because it doesn't fit neatly into our base 10 number system. We could invent a whole new number system and redefine (pi^2)/6 as a whole number but the paradox would still occur. Also, it's not as if (pi^2)/6 "gets bigger" when you look at the next decimal place; its value is the same whether you look at the next decimal place or not.

[ThePrevenge]

@messakg123 I see what you mean, but what I was trying to say is that when you have a number created by an infinite amount of numbers, each created by a diversion of 2 numbers in our number system (in this case zeta(2)) the number you get has to have an infinite number of decimals (in our number system). If you actually draw shapes to represent this number (like in the video) the total number of squares also have to be infinite (obviously). Still, the total area of the shapes can be finite. ...

[ThePrevenge]

@messakg123 This is because, eventually, the next shape you draw will be so small that it will not change any number already written in the total sum, it will just add another decimal. It's the same way if you try to write pi using our number system. If you start with something as simple as 3,14 you can then "make it bigger" by making it 3,141 and then 3,1415 and so on. Still, it will never become bigger than 3,15.

I believe you're right though, this probably has nothing to do with the paradox.

[erdmat61]

The reason this does hold true is the fact that you are using a pen to draw the line. No matter how thin you draw the line, say the line has a thickness of X, you will still have X(1) + X(1/2) + X(1/3) + X(1/4) + ...Then factoring out we have:

X(1 + 1/2 + 1/3 +1/4 + 1/5 + ...). Then no matter what thickness you choose for X, you will still be multiplying it by an infinite sum. So there will reach a point where the line is actually thicker than the boxes.

[DrJamesTanton]

@erdmat61 Good thinking. But are you saying that you still feel we can paint all the boxes? Us humans trying to do infinite things!

[erdmat61]

@DrJamesTanton Yes, as long as you put the boxes in a finite space like a 2 by 1 rectangle and make paint strokes so that you are painting multiple boxes at once! Of course put newspaper down to avoid getting paint where you don't want paint

[AlexMack85]

@erdmat61 The field axioms don't hold for infinite sums if they do not converge. In particular, the distributive law does not hold. So we can't write x(1 + 1/2 + 1/3 + ...) = x(1) + x(1/2) + x(1/3) + ...

[DrJamesTanton]

No matter what size you decide to make the first box, &call that side-length 1 unit, you will get to boxes smaller than an atom and then smaller & smaller still. The mathematics seems to be saying that the areas of these things will always add up to a finite amount & so the areas can all be painted (though it is humanly impossible to paint infinitely many things - we're playing a mind game) but the side lengths add to an inifinte length and can't be drawn! Weird! Is this addressing your concern?

[Elynwen]

Dr. Tanton, I followed your video up until you got to the paint... what if you made your "1" ENORMOUS, like a football field size, and went on until you needed an atom-sized needle and just a molecule of paint to fill it in? And then went even smaller? Are you saying that it's *impossible* to paint something that small, or just not very likely?

[boobah1067]

I might not have been clear earlier: the issue is, why the paradox in your view, since **you are starting with an item bigger by 1/2?**

[boobah1067]

Plus, why do you say a finite amount of paint? In fact, figuring out exactly how much would be infinite ... but would end at the next "infinity" of the new size: the new square (or circle) at 1 1/2 diag, or 1 1/2 diam. Yes?

[boobah1067]

Hi there. Thank you for the tutorial. Why do you say 1 + 1/2 then 1/2 + 1/2. Shouldn't you start # 2 with 1 + 1/2 +1/2 ...? in other words, you don't repeat your term 1 actually. (I think the paradox occurs because we're dealing with a new start point: the original 1 + ... represents one circle, and the 1/2 + 1/2 ... is a different sized circle. Yes, they're both infinite in possibility each, but also different relatively, so one is "greater" from a single point of view. Do you understand?)

[DrJamesTanton]

@boobah1067 G'Day! I am sorry that my handwriting is hard to understand at times, but the math I am doing is correct. Since 1 is bigger than 1/2 we certainly have that S = 1 + 1/2 + 1/3 + 1/4 + ... must be bigger than 1/2 + 1/2 + 1/3 + 1/4 + ..., and since 1/3 is bigger than 1/4, this in turn must be bigger than 1/2 + 1/2 + 1/4 + 1/4 + .... And so on. That is all I am doing. (This might be confusing b/c I am not doing the standard proof everyone sees comparing it to 1/2 + 1/2 + 1/2 + ....)

[boobah1067]

@DrJamesTanton Hello. No, don't get me wrong. I love that you are explicating the way we deal with fractions. Excellent. What I am questioning is HOW we see them. That is, since they relate to the real world of ideals (as in, squares, circles, etc. -- which are real in one sense), they must be understood geometrically. Anyway, can you put here what the standard proof is (this 1/2 + 1/2, etc.)? Why all the 1/2s? Second, do you see that your 1 which is bigger than 1/2 + 1/2 +1/3 etc. (cont'd)

[boobah1067]

@DrJamesTanton (cont'd) leaves out the biggest 1/2? In other words, since 1 + 1/2 + 1/3 ... is bigger by 1/2 than 1/2 + 1/2 + 1/3 the sets are different. In other words, the edges (sides) of the squares are in fact "1" 1/4 smaller (not 1), since each of them makes a square 1/4 of area 1, not area 1. But we always talk of 1 x 1 x 1 as 1, instead of treating it as 1 x 1 = 1 (a start point of existence of 1), but 1 x 1 x 1 is 2, not 1 as we do our counting. It means 1 exists, then replicates.

[boobah1067]

@DrJamesTanton (cont'd) I know how we notate and what's conventional. However, it requires re-thinking the "method" we talk of "1". Sometimes 1 is really 1/2 to the system we are speaking of, but our system still calls it "1". In the case of a square's sides, area "1" is achieved by 4/4, each of them "1". The arithmetic system works out but creates these "paradoxes". By the way, I love the enthusiasm in your voice. :)

[DrJamesTanton]

I am afraid to say that I don't understand your comment. Each square is "thicker" than the portion of line that it sits on. So filling in an area uses more ink than drawing the think line. So ... how can the areas use a finite amount of ink, but the line an infinite amount?

[tomatoe0000]

However an ideal line has no thickness, and therefore a line has no area. In contrast, ink applied from the stroke of a pen does have thickness and area. That is why an infinite amount of ink is required to trace the line, but only a finite amount of paint is required to fill the bizarre shape.