Of course Iâm wrong, because the author is a mathematician and Iâm just a software guy. I learned this lesson years ago when I KNEW that âAsk Marilynâ (remember her?) was wrong when she wrote about the Monty Hall problem, and I only unknew it when I sat down to write a software simulation to prove she was wrong. But Iâll say it anyway:

The premise of the question makes no sense: Nothing âends upâ when youâre going to infinity. The number of balls in the box does keep increasing, forever and ever. No?

I think thatâs the point of the question. If you think of infinity as a number along a number line and think you can imagine infinity balls being placed either in a box or a drawer, it shows that you are thinking of infinity as a really big number - ball no. â would stay in the box, because thereâs no â^2. If you let go of that idea, larger and smaller infinite numbers make more sense.

The way to solve the conundrum is to tackle the problem from a different angle.

As stated;

âInstead 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 drawerâ.

Im waiting for someone to invent the infinitely small number.

Put a mathematician in a room forever and I bet theyâll come up with it.

Infinities come in negative numbers also.

The number â0â can behave as an infinitely small number too.

Thereâs simply no end to the fun you can have with infinity!

Yep:

Be very careful when playing with infinity â it tends to drive people insane (seriously).

I read a book a while ago about two groups of mathematicians working on trans-finite set theory back in the day; one (the Moscow school) were very much into apophatic mysticism in the Orthodox tradition, and they fared fairly well. The other â French Freethinkers â tended to go insane after too long (as did Georg Cantor, who invented the field).

That sounds like a balance issue: if you delve too deep into the Yang, only the Yin can bring you out again.

Which is one reason why Iâm very comfortable with religious believers, IFF (if and only if) they also recognize reality. As long as you donât hate science â or other human beings â mystical thinking can be a good counterweight in the modern world.

One of my favorite youtube videos deals with a similar topic, the sum of all of the natural numbers.

If you have ever done or are currently doing calculus then you are already using the smallest number. Congratulations!

Put *all* the mathematicians in a room forever. `#FTFY`

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They still wonât come up with it though, because theyâll be arguing over how to split the lunch delivery bill.

Were they made insane by thinking about infinity (so romantic!), or was their insanity inherent and combined with their creativity led them to explore radical new ideas (also so romantic!).

Hmm, religion as an immunization against pissing ones pants and losing it in the face of vast cold infinity.

It was pretty clear (to me) that the answer was going to be that the box was empty, because every number is the square-root of some other number, but I donât think the answer is right.

The question is nonsensical at itâs premise, and so canât be answered. The best you can do is un-ask the question.

The question is: if you simply keep doing this, which balls will end up in the box?

Thatâs not a valid question, because it contains the word âendâ in it. âWhich balls will end up in the box?â simply doesnât make sense, because the answer *relies* on infinity having no end, and **not** being like a regular number.

You canât say âwhat happens when you reach infinity?â and then answer it by saying âsince infinity is not a number, and you canât treat it like a number, and you canât ever reach it, *this* is what happens when you reach infinity.â

An absurd question can have any absurd answer you like. Or, better, no answer.

Except that when you do higher-level physics, these âabsurdâ questions are immensely useful and hold to be true in our tangible universe (see: Bessel functions, Fourier analysis, etc.) Your calculator or computer or phone uses these principles, not just to exist as a functional device, but to actually calculate things like sine functions. Take the number pi, for example. Itâs a number, somewhere between 3 and 4. How do you calculate pi to umpteen million digits? You use the fact that there are infinite sums that âend upâ at *exactly* pi.

Exactly. The supposed paradox is only because this mathematian appears not to understand limit theory, which is pre-Calc, I think.

Lim n-> oo (n-sqrt(n)) =/= 0.

Somewhat relatedâŚ

Thatâs is not the correct formulation. Itâs a summation problem, whatâs known as a telescoping sum.

So it actually takes the form:

ÎŁa=0 + 1 **-1** + 2 + 3 + **-2** + 5 + 6 + 7+ 8 + 9 **-3** + 10 +11 + 12 + 13 + 14 + 15 + 16 **-4** + 17 + 18âŚ

ÎŁa=0 and you can see it because if you follow the sum by eye, itâs obvious that for every number, youâll eventually be able to subtract it as a square root.

The equation n-sqrt(n) describes a simple sum, not an infinite series.

If people really want to understand this, you can pick up an old edition of Stewartâs Calculus for a few bucks. Itâs near the end of the book, but if youâve already done a little precal then you can work your way to it. It makes more sense than you think it does, and actually my main criticism of the article is that heâs right, at first:

Where I think he goes very slightly awry is that you can actually develop intuition for this stuff. Most mathematicians have intuition about infinite series, but usually it comes after wrangling them long enough to develop it.

Yes but the infinite series approaches n-sqrt(n). I see I wrote it wrong.