How to have mind-boggling fun with infinity



Of course I’m wrong, because the author is a mathematician and I’m just a software guy. I learned this lesson years ago when I KNEW that “Ask Marilyn” (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 I’ll say it anyway:

The premise of the question makes no sense: Nothing “ends up” when you’re going to infinity. The number of balls in the box does keep increasing, forever and ever. No?


I think that’s 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. ∞ would stay in the box, because there’s no ∞^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;
“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”.


Im waiting for someone to invent the infinitely small number.

Put a mathematician in a room forever and I bet they’ll come up with it.


Infinities come in negative numbers also.
The number ‘0’ can behave as an infinitely small number too.


There’s simply no end to the fun you can have with infinity!




Be very careful when playing with infinity – 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 – French Freethinkers – 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 I’m very comfortable with religious believers, IFF (if and only if) they also recognize reality. As long as you don’t hate science – or other human beings – 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.

ASTOUNDING: 1 + 2 + 3 + 4 + 5 + … = -1/12


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


They still won’t come up with it though, because they’ll 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 don’t think the answer is right.

The question is nonsensical at it’s premise, and so can’t 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?

That’s not a valid question, because it contains the word “end” in it. “Which balls will end up in the box?” simply doesn’t make sense, because the answer relies on infinity having no end, and not being like a regular number.

You can’t say “what happens when you reach infinity?” and then answer it by saying “since infinity is not a number, and you can’t treat it like a number, and you can’t ever reach it, this is what happens when you reach infinity.”

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


Except that when you do higher-level physics, these “absurd” 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. It’s 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 “end up” 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.

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Somewhat related…

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That’s is not the correct formulation. It’s a summation problem, what’s known as a telescoping sum.

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, it’s obvious that for every number, you’ll 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 Stewart’s Calculus for a few bucks. It’s near the end of the book, but if you’ve 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 he’s 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.

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

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