How many sides does a circle have?

[Ravingstarfish]

Just that really. My sons tutor has taught him this and several people have said she’s wrong.... so how many sides does a circle have?

Or, in other words, 10 coins are being added for every 1 coin that is removed. How does that tend towards 0 coins?

i think it is infinite
or equal to the circumference value.

i mean the the number of sides of circle.

I have 2

Complete nonsense. If A puts in 10 coins a day and B removed 1 coin a day the total numbers of coins in the chest at day 9 will be 9n. As we tend n to infinity the number of coins in the chest also tends to infinity. That's actual fairly simple to understand without hidden logical twists.

Or we could say, after n days, all the coins labelled 1 to n are no longer in the chest. So as n tends to infinity, that means all numbers are no longer in the chest.

Part of the confusion is over whether one infinite set is bigger than another. If we take the set N of all positive integers (1, 2, 3, 4, 5, 6...) and the set E of even numbers (2, 4, 6, ...) it might appear that E is smaller than N but in fact they are exactly the same size. Their infinities are equal.

I think however we are meant to envisage appraising the whole situation after an infinite amount of time. With an infinite amount of time, every coin put in will have been removed.

With an infinite amount of time, angel A will still have added 10x more coins than angel B has removed?

But the size of the set of coins A has added is equal to the size of the set of coins that B has removed. The set of counting numbers (1, 2, 3, 4, ... ) is no bigger than the set of multiples of 10 (10, 20, 30, ... ).

@DadDadDad

Sorry I'm not trying to be difficult and genuinely interested in understanding this (is it a known maths problem/thought experiment? Can you link me to something with a basic explanation?) BUT my head is saying:

That doesn't work because we're not talking about the set of multiples of 10, i.e. we're not mapping 1:1 like in your examples of all numbers vs even numbers, we're mapping 1:10. The set of multiples of 10 would suggest we're ignoring coins 1-9, 11-19, 21-29, and so on.

With your earlier example of:
1 2
2 4
3 6
4 8

How would that work for this angel-coin thing? You can't say
1 10
2 20
3 30
4 40
because angel A isn't just putting in "coin 10" as angel B is taking "coin 1", she's putting in 10 coins. It doesn't map across equally!

I'm addressing purely your reasoning that Angel A has put in 10x as many coins as B has taken out. For finite steps, that would justify that there are still coins there, but I'm just showing it's not a strong enough argument for the infinite case, because the two "infinities" are equal. (There are larger infinite sets, but that's a discussion for another day ).

I would suggest focussing on my simplified case first. Forget Angel A, and imagine you have a chest with an infinite set of coins labelled 1, 2, 3, 4,.... . Angel B comes along on day 1 and removes coin 1, on day 2 she removes coin 2, and so on. Will the chest be empty after infinite days? (if you can have infinite days). I'd say yes, because every coin gets taken out.

Now imagine there's a second chest which is exactly the same at the start. Angel C comes along on day 1 and removes coin 2, on day 2 she removes coin 4, on day 3 coin 6, etc. Although C is removing coins at the same rate as B, after infinite days it definitely won't be empty (because it will contain all the odd numbers).

Perhaps I can help.

I think you are confusing infinity with Aleph-Null.

To coin a phrase

There are more infinities in heaven and Earth, Horatio, Than are dreamt of in your philosophy.

@DadDadDad I understand/agree with both your scenarios there! But it feels as though the original scenario is more like the latter one (with Angel C removing even coins only).

Let's take the original scenario, except say that instead of Angel A putting in 10 coins, she's only putting in 2 (so she's at a disadvantage to begin with! ). Angel B comes along and still removes one coin: the even one. All the odd ones will still remain after infinite days, just like in your last paragraph there!

out of interest, there was long running saga on Digital Spy, regarding whether 1 was different to 0.999 recurring.

(PS if it's too exhausting to try and explain this to me, I understand! I'm just curious and perplexed in equal measures)

OK, let's not get diverted to aleph-null and 0.9 recurring yet.

So, I think, @goodwinter, you agree in what we shall call Scenario 1 (chest starts full with 1, 2, 3, ... and B takes out a coin a day in the order 1, 2, 3...) that the chest should "end up" empty. (I've put it in inverted commas because there is no end).

Now imagine we study the set-up in a bit more detail. We've told B that the chest is infinitely full, and the way it works is it has a little drawer at the front, and when she opens it she can see the next few coins in the sequence and can take the one she wants. She can't see into the back of the chest, but that's fine, she's an angel and believes us that they are all there.

But what if in practice A is a devil and is being economical with the truth. In fact, the chest is empty, but every night A sneaks along with his infinite bag of coins and puts the next ten coins into the chest. Why does this make a difference - you could just think of the bag as extra storage for the chest? From B's point of view (and ours) every coin that exists appears in the chest and gets removed eventually.

The process may be infinite but each number involved is finite and that means whatever number you care to name will eventually be removed from the chest.

I think you are confusing infinity with Aleph-Null.

Aleph-Null is just the cardinality of a countable infinite set, so it describes a certain type of infinity, so no confusion in my mind.

There are more infinities in heaven and Earth, Horatio, Than are dreamt of in your philosophy.

Well, isn't that an open question? We can construct a hierarchy of cardinalities but I thought it was unproven whether there are any gaps. For example, it may be that the cardinality of the reals is aleph-one, it which case they have all been dreamt of, it's just we can't prove it.

every coin that exists appears in the chest and gets removed eventually.’

Isn’t that the flaw in this logic, there is no ‘eventually’. Infinite means it is not finite, there is no end or eventually. It’s not a definite number, it’s never ending. To say there is a infinite amount of coins means this is a never ending scenario. If it ends, it is not infinite, there is a stop point and the number of coins (however large) becomes countable and then you could potentially name that number. All you’re saying is the number I do big we don’t have a name, nor do we know the point it will stop.

OhForkIt - I think you are confirming the wider point I'm trying to make.
The reason for setting out these scenarios was to show that we think we can talk about an "infinite number" or what happens when we get to infinity, but that it's sloppy language that can lead us into contradictions.

If we go back to the circle that started this thread, some people have said if you keep increasing the number of sides of a polygon it gets closer and closer to a circle, so at infinity you arrive at a shape with infinite sides. But, as you say, we never get to infinite sides - the polygons never actually reach the shape of a circle. So the argument is not watertight.

@DadDadDad Still not 100% convinced, but your last reply did really help! It's like that spinning dancer illusion: it seems to be completely different depending on the way you look at it (still not sure why it's not 9, 18, 27, 36, 45, 54.... coins all the way to infinity but the "A is a cheeky devil" scenario was a great counterpoint)

But seriously thanks for your patience and perseverance in trying to explain it to me!

Thanks, goodwinter - there's probably places out there on the web that express it better than I can. But I'm enjoying myself (ex-Maths teacher, good to get back in the saddle now and then).

I've just noticed that this thread is on AIBU. Surely, not appropriate for a maths problem - we really need a CLURT board (Come, Let Us Reason Together ).

OK - found it - here's the rabbit-hole if you want more: en.wikipedia.org/wiki/Ross%E2%80%93Littlewood_paradox

I must have come across this a long time ago, and forgotten it's origins, although interestingly I'd remembered that it uses a ratio of 10 to 1. The other element that's given here is that the time between each step halves which means that an infinite set of steps can be performed in a finite time - another brain-bender.

@DadDadDad Ha, CLURT would be great! I did wonder if you were a maths teacher - I was always the kid at school who the teachers got sick of from all the questions ;) I just wanted to understand stuff!

Hang on a sec - it's a paradox?! So it's not as straightforwardly "zero coins" as you would have me believe

DadDadDad: We can construct a hierarchy of cardinalities but I thought it was unproven whether there are any gaps. For example, it may be that the cardinality of the reals is aleph-one, it which case they have all been dreamt of, it's just we can't prove it.

Actually, as DadDadDad probably knows (or at least once knew), it is a bit more interesting even than that. The 'continuum hypothesis' the hypothesis that the cardinality of the reals is aleph-one is undecidable, not just unproven.

That is, the question 'Are there any gaps?' as DadDadDad has it, has no answer (at least in standard mathematics). This does not mean we do not know the answer -- that would not be very interesting; rather, we know there is no answer and nor will there ever be an answer.

Strange, hein?

[More? See, e.g., Mathworld, or for a little more discussion, Stanford Encyclopedia ... both authoritative sources for this kind of stuff.]

I was arguing for zero coins because others had argued for infinite coins. The scenario demonstrates that you can come to different conclusions based on exactly how you define which coins are added or removed. If someone had argued for zero, I would have played devil's (!) advocate on the other side to draw out the contradiction. (or paradox).

No no, I get you - I just feel slightly vindicated that I had a logical position and wasn't just being totally dense :)