But Σn-sqrt(n) looks like: 1-sqrt(1)+2-sqrt(2)+3-sqrt(3)+…
You’re no longer dealing with just integers at this point, and you can see the sum diverges because it does not decrease. It doesn’t approach the limit of the n-sqrt(n), the series approaches zero.
“And since every number can be multiplied by itself, every ball is a square root, therefore all balls go into the drawer.”
The question is valuable and useful and productive.
But to make it accessible in an article like this, it has to be posed in layman’s terms. If it is asked in mathematically precise terminology, it’s so impenetrable that the only people who will get it already know the answer and the implications.
The hope of “mind-boggling” in the title is that someone who formerly didn’t grok the notion of infinity might, with the help of this article, actually get it. It’s not an easy thing to get your head around, and the first time it happens, it’s a rush.
[quote=“ActionAbe, post:19, topic:49422”]So it actually takes the form:
Σa=0 + 1 -1 + 2 + 3 + -2 + 5 + 6 + 7+ 8 + 9 -3 + 10 +11 + 12 + 13 + 14 + 15 + 16 -4 + 17 + 18…
Σa=0 and you can see it because if you follow the sum by eye…[/quote]
Please, no, you really can’t do that. Σa diverges to infinity, and you can see it because grouping terms leaves you with only non-negative integers:
Σa=0 + (1-1) + 2 + (3-2) + 5 + 6 + 7 + 8 + (9-3) + 10 + 11 + 12 + 13 + 14 + 15 + (16-4) + 17 + 18…
Σa=0 + 0 + 2 + 1 + 5 + 6 + 7 + 8 + 6 + 10 + 11 + 12 + 13 + 14 + 15 + 4 + 17 + 18…
Something important to understand about infinite series is that you can’t arbitrarily shuffle the terms without changing the sum. In fact, the Riemann series theorem says that for a conditionally convergent series, you can rearrange it so that it converges to any answer you choose.
That is in some sense what’s happening here: ultimately all the balls are destined to be put in and taken out, but all the finite examples are portions of a rearrangement where each has to wait longer and longer first, so that the count doesn’t converge to that.
The question tries to draw a physical analogy to a mathematical concept, which is where the problem lies. It seems likely that physics does not contain actual infinities. So, no, actually you don’t end up with all the balls in the drawer, because universes never contain an infinite number of balls.
This mathematician puts an infinite number of balls into a drawer at once - what happens next is so infinitely mind-boggling your puny finite-state minds couldn’t handle it all!
I was going to say something snide, but I decided against it.
What’s going on is that every natural number enters the box once and leaves the box once never to return. The counter intuitive part is simply that, as the numbers get larger, they take progressively longer to exit and your limit diverges.
Of course, your limit argument makes sense if, for example, you don’t want to distinguish between the numbers between 1 and 10 and the numbers between 1000 and 1010. (i.e. you think of both of them as a generic segment of length 10.) If, however, 1 is always the number 1 and not something like a segment of length 1 you have to do a point by point analysis and your limit argument is nonsensical.
Now, there are cases where both of these kinds of analysis make sense. There are even hybrid versions that show up pretty naturally. You just have to decide what it is that you are talking about and not use similar-sounding but distinct ideas.
You’ve been Huffing BoingBoing, I can tell 
I think that you have misunderstood the setup. When you “add” number to the box, you do not perform any summation. When you remove a number you do not perform any subtraction. You should be thinking of the numbers as something like objects with the numbers written on them. (i.e. a bell with a 2 and a ball with a 3 isn’t the same as a ball with a 5.)
Mathematical logic can be our only guide in a world that no longer has any link to our physical reality.
That link was broken the moment you said “infinite”.
There is no ‘infinity.’ It’s an imaginary thing.
And like all imaginary things, it behaves according to its own imaginary ruleset.
The fact that some results are “counter-intuitive” should hardly be surprising, let alone “mind-boggling.”
I think you’re right it doesn’t converge. But I’m still right about it equaling zero. Granted, I’m cheating but Riemann says I can, which is the best kind of cheating 
. I think that it was hard for Matt Parker to find an example that actually decreases monotonically using only integers to make his point, but I think that point is still successfully made and rigorously defensible.
I’m confused. How have I misunderstood the setup? I was attempting to translate it to abstraction. I’m aware that if we convert the setup to real objects it fails. Gabriel’s Horn (as an object with infinite surface area but finite volume) fails in real life because there’s no such thing as infinitely small particles with which to fill it. I think the author was using analogy, and all analogies are imperfect.
The numbers should be considered as sets of numbers, not a summation. It goes like this:
{1} \ {1} = emptyset
{2}
{2,3}
{2,3,4} \ {2} = {3,4}
{3,4,5}
{3,4,5,6}
{3,4,5,6,7}
{3,4,5,6,7,8}
{3,4,5,6,7,8,9} \ {3} = {4,5,6,7,8,9}
etc.
Here \ denotes “set minus”; the removal of elements of the latter set from the former. Notice that, for example, {2,3} is the set containing 2 and 3, not the number 5. You never do any addition or subtraction so questions about infinite sums are not really germane.
Also, @chenille’s comment was that Riemann’s theorem tells you that you cannot rearrange the sum to get convergence. Of course, those series aren’t even conditionally convergent so the theorem doesn’t apply directly. I don’t know if the theorem is even true. It is kind of hard to tell - after all, to get a sensible answer to something like Σn you have to go through some fancy techniques like zeta function regularization. When you do that things like
1-1+1-1+1-1+… = 1/2
happen (or maybe -1/2, I forget). So I’m pretty suspicious about that sum vanishing.
There is no ‘freedom.’ It’s an imaginary thing.
And like all imaginary things, it behaves according to its imaginary rules set.
Imaginary numbers existed as a mathematical exercise for quite a few centuries before they found an application in electrical engineering.
Instead of imagining putting the balls into the box and the drawer, you could imagine doing it all in one step. All the balls that are square roots go straight into the drawer, and all the balls that are not go in the box. Not a single ball would be put in the box.
But that would be true whether you did it all at one fell swoop, or one at a time, sequentially. And equally true for both finite sets and (imaginary) infinite sets.
Because, see, what you did there was change the rules in midstream.
The initial rule was “place each ball in the box and then leave it there until its square turns up in numeric sequence.”
When sequentiality is removed (“all at once”), the new rule perforce becomes “place each ball in the box, but immediately remove it if its number can be squared.”
Those are two different rulesets with (unsurprisingly) two different results.
This new rule will result in an empty box for any number of balls, finite or infinite - and that’s true whether they’re processed sequentially or simultaneously.
It’s not the infinity confusing you here, it’s the fact that you changed the ruleset without noticing.
Imaginary things are especially sensitive to that. (-:
Here’s a snippet from the Wikipedia article I linked above.
There was no rules change in the article.
It’s a very common teaching method to introduce a counter-intuitive concept by providing a series of examples that are illustrative of the idea that progress from simple but imperfect towards complex but more accurate.
Feel; free. I certainly did.
I didn’t say they were useless imaginary things.
Lots of imaginary constructs are useful. Shoot, basic Euclidean geometry is jam-packed with imaginary objects (infinitely small points of no physical extension; infinitely long lines with no thickness; infinite, unbounded planes, and on and on).
And it’s incredibly useful.
But it shouldn’t be surprising that it doesn’t always behave as we’d expect based on real-world experience. At certain points (often the result of one infinity or another, and sometimes, questionable assumptions) it diverges.
Models based on imaginary constructs sometimes do.
This is still less than mind-boggling, in my book.
Beautiful and well-written article on the intricacies of infinity! However, I’m going to contend with some of it - mainly, the assertion that “infinity is not a number.” That infinity that you say doesn’t exist at the end of the number line very much does exist if you consider the extended real line from real analysis. Or, my favorite, the unsigned infinity at the top of the real projective line, created by joining the two infinities at either end of the number line.
These infinities work just fine as numbers - you can do number-y things to them like adding, subtracting, multiplying, and dividing. (Unsigned infinity even lets you legitimately divide by zero - no joke!) They’re not badly-named so-called “real” numbers, though - they exist outside of R and can’t be expected to act like other “real” numbers. You can’t do algebra with them, for example.
What’s more, the way that you do use the “∞” symbol to “count all the whole numbers” isn’t really what we use that symbol for. At that point you’re talking about cardinality, and if you’re looking at the cardinality of the natural numbers, now you’re looking at aleph-null, which is also a number - a cardinal number, greater than any finite number, which can also have all sorts of number-y things done to it.
Let me clarify by the way - I don’t really take so much issue with not using the most specific and correct mathematical vocabulary. I’ll take hand-waviness and conceptual understanding over “rigor” any day. But characterizing infinity as “not a number” does it a disservice - infinity is one of the most beautiful and interesting numbers of all (provided, of course, you’ve said which infinity you’re talking about!) We just have to treat infinity with care and caution, since it’s not quite like any other numbers we usually play with.
There are no numbers, they are imaginary things.
Especially zero. There is literally nothing like zero.
