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Old 03-30-2007, 02:46 PM   #1
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Default School Help - Infinity

OK, I have a math quiz Monday, and there is going to be a bonus problem about the concept of infinity. It goes something like this:

There exists a town that has an infinite number of hotels, and each hotel has an infinite number of rooms. One day, a group of cannibals (yes, my math teacher is very odd ) destroy all of the hotels and set up their own, which has an infinite number of rooms as well. The first question is: Is it possible to put all the people who lost their hotel room into this new hotel? If so, how would you organize everybody to fit in it?

To explain in concrete terms, say we had 2 hotels of infinite rooms. We would organize these people into the new hotel by having everyone from Hotel #1 go into an odd room while everyone from Hotel #2 goes to an even room. Hence, we have a procedure to fit everyone inside. In equation terms, the room number for someone from Hotel #1 is 2x-1 and for Hotel #2 2x. If there were 3 hotels involved, the equations would be 3x-2, 3x-1, and 3x, and so on.

So how would I explain this procedure in terms of infinite hotels? I thought of using summations or modulo, but both of these are concepts we have not covered in class yet, and our math teacher told us that it does not require advanced math, but rather logically reasoning. Thanks for your help.
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Old 03-30-2007, 03:00 PM   #2
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Default Re: School Help - Infinity

Yes, you could.

"Infinity" can be thought of as just a representation of "REALLY big". So, if you have a REALLY big number of hotels, with a REALLY big number of rooms, and all the occupants of those rooms had to move into another hotel, they would, combined, make a REALLY big number of required rooms.

But since the cannibals' hotel has a REALLY big number of rooms, it could fit that REALLY big number of people.

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Old 03-30-2007, 03:13 PM   #3
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Default Re: School Help - Infinity

He's talking about sets of infinity which may be greater than or less than other sets of infinity. I read something about this before. Can't recall what it was exactly... I think it was a report about an old proof that some mathematician did to prove that one set of infinity can be greater than another set of infinity.

I scanned over this: http://www.c3.lanl.gov/mega-math/wor...ty/inbkgd.html

Looks like it might help you.
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Old 03-30-2007, 03:20 PM   #4
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Default Re: School Help - Infinity

Accommodate all of them by taking all the guests in room N and moving them to room 2 × N, then putting each of the new guests in room P^S, where P is the (H+1)-th prime number, H is the hotel number, and S is the room number.

That should work. As long as you can count the infinities you should be able to fit them in. I had originally thought you couldn't fit them in because you couldn't count the infinities, but alas I think that you can. There are an unlimited number of rooms, and if you keep moving people in this fashion you can fit everyone into the one hotel.
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Old 03-30-2007, 09:39 PM   #5
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Default Re: School Help - Infinity

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Originally Posted by Reach View Post
Accommodate all of them by taking all the guests in room N and moving them to room 2 × N, then putting each of the new guests in room P^S, where P is the (H+1)-th prime number, H is the hotel number, and S is the room number.

That should work. As long as you can count the infinities you should be able to fit them in. I had originally thought you couldn't fit them in because you couldn't count the infinities, but alas I think that you can. That should work; there are an unlimited number of rooms, and if you keep moving people in this fashion you can fit everyone into the one hotel.
So...

what you're saying is:

∞+∞=∞

???
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Old 03-30-2007, 09:47 PM   #6
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Default Re: School Help - Infinity

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Originally Posted by Afrobean View Post
So...

what you're saying is:

∞+∞=∞

???
It's not that simple.

It's easy to say ∞+∞=∞, but the question clearly asks how you are going to go about fitting everyone into the one hotel. Simply saying this doesn't address the true problem. Unless of course you conveniently remove all of the rooms and decide this hotel is a giant lobby you can just throw everyone into ;lD

Essentially imagine the hotel, which has an infinite number of rooms. Suddenly, an infinite number of carriers/busses/whatever line up infront of the hotel, each with an infinite number of seats and passangers in it.

The hotel is already full from the infinite number of cannibals, but paradoxically the hotel can still hold more individuals. How it can do this is what I addressed.
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Old 03-30-2007, 09:54 PM   #7
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Default Re: School Help - Infinity

OH GOD NOT ANOTHER INFINITY TOPIC

Okay, first of all, number of rooms does not matter. Number of rooms * number of hotels in town = ∞*∞ = ∞. Likewise, number of people times number of rooms = ∞*∞ = ∞.

So the number of people in town is ∞. If the new hotel has ∞ rooms, then the equation to find how many people go into a room is ∞/∞, which is indeterminate.

therefore infinity is retarded bye
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Old 03-30-2007, 10:00 PM   #8
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Default Re: School Help - Infinity

"There exists a town that has an infinite number of hotels, and each hotel has an infinite number of rooms. One day, a group of cannibals (yes, my math teacher is very odd ) destroy all of the hotels and set up their own, which has an infinite number of rooms as well. The first question is: Is it possible to put all the people who lost their hotel room into this new hotel? If so, how would you organize everybody to fit in it?"

You can't destroy an infinite amount of hotels to begin with.
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Old 03-30-2007, 10:01 PM   #9
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Default Re: School Help - Infinity

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Originally Posted by inflames07 View Post
You can't destroy an infinite amount of hotels to begin with.
I was just about to edit that in.

Then again, you can't have an infinite number of hotels to begin with.
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Old 03-30-2007, 10:05 PM   #10
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Default Re: School Help - Infinity

Yeah, I don't like the idea of infinity being used in place of a number. It ruins logic.

The way I think of infinity is a lot simpler I think. Imagine a line in 2 dimensions. It has an obvious beginning and end, and thus has a finitely defined length. Now, imagine that this "line" is really a circle turned on it's side within the third dimension. Projected to the 2nd dimension, this circle would be a line, but in actuality, it is really a circle with no beginning and no end. I like to think of the concept of infinite within the 3rd dimension in a similar manner. Additionally, thinking of it in this way explains the concept of sets of infinity which are of different values of other sets of infinity, because like the circle in the 2nd dimension, the size of the circle would affect the size of its infinity. I don't know if I verbalized my thoughts well enough for what I said to make sense, but at least it makes sense in my own mind ^_~
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Old 03-30-2007, 10:07 PM   #11
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Default Re: School Help - Infinity

Sure you could destroy an infinite number of hotels. You would have to destroy them at an infinite rate, though.

It's purely hypothetical anyway.

Tokzic, is again oversimplifying the problem >__>

I can imagine him being the manager and having an infinite number of people mad at him because they don't have a room 8)
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Old 03-30-2007, 10:09 PM   #12
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Default Re: School Help - Infinity

I just think of "infinity" as a variable instead of an object. It shows that the numbers can increase in value forever. Although I think forever is a bad word. I see "forever = infinity"
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Sex kills time and it's free.

Instead of taking her out to a movie and buying her popcorn, bend her over the arm of a couch. It's very economical. Just make sure you are using the proper protection, because then it can become VERY, VERY GOD DAMN UNECONOMICAL if she pops a baby 9 months down the road.
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Old 03-30-2007, 10:14 PM   #13
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Default Re: School Help - Infinity

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Originally Posted by Reach View Post
Tokzic, is again oversimplifying the problem >__>
Point out a flaw in my reasoning.

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I can imagine him being the manager and having an infinite number of people mad at him because they don't have a room 8)
rofl
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Old 03-30-2007, 10:22 PM   #14
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Default Re: School Help - Infinity

Or maybe you should try really hard at the actual questions and not wory about it.



But that's just my opinion.
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Old 03-30-2007, 10:29 PM   #15
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Default Re: School Help - Infinity

Quote:
Originally Posted by Tokzic View Post
Point out a flaw in my reasoning.



rofl

Fiend, edited >_>

Too bad there is still a reasoning flaw.

The problem is here:

Quote:
So the number of people in town is ∞. If the new hotel has ∞ rooms, then the equation to find how many people go into a room is ∞/∞, which is indeterminate
First sentence, correct. However, then you go wrong. It's not just ∞/∞ = indeterminate.

We're going to have to go back to my post quoting afro. You see, there are already an infinite number of people in the hotel, so it is full. Likewise, once you put one of those carriers full of an infinite number of people into the hotel, the hotel will be once again, full. And yet you still have an infinite number more carriers to place into the hotel Uh oh.

Your method does not work because it doesn't account for the fact that just 'sticking them in' anywhere will fill up the hotel, whether you would like to think so or not You'll have an infinite amount of people wandering around that can't find a room because an infinite number of them are already occupied.


And yes, I do realize this is very very hard to understand. I had to edit my post because I was initially wrong, and wasted a good deal of my time contemplating the answer >_> Interesting that your teacher ended up asking you this. I'm sure many people will realize you can fit them in by counting the infinities, but the method of doing so isn't so easy.
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Old 03-30-2007, 10:43 PM   #16
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Default Re: School Help - Infinity

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Originally Posted by Reach View Post
infinite amount
Please... don't say this. Infinite is not a quantity. If it was a quantity, it would simply be a very large number (possibly an infinitely large one? haha), and as such would not actually be infinite.

Actually, I think this is the underlying problem with this in my mind. Infinite is not a quantity, and as such, you cannot "fill" a hotel with capacity of infinite, even if it were possible to have an "infinite amount" of guests.

One more thing I think someone else already touched on: it would be impossible unless the move were done at a rate of infinite people per 0 seconds. Now, not only does that not make any sense at all, but it is also not only physically impossible, but also logically impossible as far as I can figure.
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Old 03-30-2007, 10:49 PM   #17
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Default Re: School Help - Infinity

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Originally Posted by Afrobean View Post
Please... don't say this. Infinite is not a quantity. If it was a quantity, it would simply be a very large number (possibly an infinitely large one? haha), and as such would not actually be infinite.

Actually, I think this is the underlying problem with this in my mind. Infinite is not a quantity, and as such, you cannot "fill" a hotel with capacity of infinite, even if it were possible to have an "infinite amount" of guests.

One more thing I think someone else already touched on: it would be impossible unless the move were done at a rate of infinite people per 0 seconds. Now, not only does that not make any sense at all, but it is also not only physically impossible, but also logically impossible as far as I can figure.
There's nothing wrong with saying an infinite amount. An amount is how much of something. You have infinite people in the hotel. I am well aware of infinity being a concept.

Really, your problem is that you keep trying to visualize the problem XD You can't afro. As much as you, and I, would like to try and visualize it happening, you can't and it only confuses the hell out of you.

Also, you're right, you're never going to get everyone in unless the rate of movement is infinite. But that doesn't matter - You can keep fitting people in and in a hypothetical sense, can fit everyone in if you want to. Remember, it's hypothetical, stop talking about it like it is a real event.


Oh yea, one more thing.

The thread is going to turning into a squabble fest soon. Can anyone refute my method of getting everyone into the hotel? Noone has even addressed it (probably because it works ;p). There's no point in just going off on a tangent about your opinion here; the man asked a question and wanted an answer (which he now has, unless there are alternative answers to the problem at hand, not your own hypothetical problem ;p).
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Old 03-30-2007, 11:22 PM   #18
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Default Re: School Help - Infinity

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Originally Posted by Afrobean View Post
Actually, I think this is the underlying problem with this in my mind. Infinite is not a quantity, and as such, you cannot "fill" a hotel with capacity of infinite, even if it were possible to have an "infinite amount" of guests.
This is true - I ignored it to follow through on the problem as much as I could, but again, we have ∞/∞ as the equation for number of people per room in the old town.

If you ask me there are so many logic crossroads that this question is hardly worth answering. He's probably just seeing what thought processes everyone goes through and throws them marks if they look at all well-thought out.

Reach, you don't even verify that you could fit that many people in the new hotel. Establish that how you can before you even touch on the "if so, ..."
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Old 03-30-2007, 11:27 PM   #19
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Default Re: School Help - Infinity

TECHNICALLY, infinity is a concept of running on forever with no end (was that redundant?). But, if you take infinity and turn it into a ridiculously large number, as Afrobean said, it will destroy the concept of infinity as a whole. Besides, if cannibals (I agree your math teacher is weird...) came and tried to destroy the hotels, they have a problem: All of the hotels can never be destroyed.
Honestly, I think that this is some kind of trick question involving an infinity paradox...but don't take my word for it. I'm not in Precalculus yet.
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Old 03-31-2007, 12:23 AM   #20
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Default Re: School Help - Infinity

Quote:
Reach, you don't even verify that you could fit that many people in the new hotel. Establish that how you can before you even touch on the "if so, ..."
You already agree ∞+∞=∞. It comes down to more specficially, how you place the infinities into the hotel. This has nothing to do with the hotel being able to hold that many people or not, because the hotel has an unlimited number of rooms.

It is a paradox. Which is probably why so many people are confused. Here's how it would go (you are the manager 8) )

Tok: Yo, wats up?
dude: yo I got infinity men right here that need a room. Iz u hotel full?
Tok: We're full yo
dude: Wat? We can't get in?
Tok: Oh, yes we have room for your men
dude: wat???
Tok: We have rooms ready for you
dude: yo u said u was full
Tok: We are
dude: you ain't makin no sense
Tok: I know

Ad nauseum. It's a fun scenario ^^

But this is so fruitless...

Put it this way, if you can't count the infinities (i.e. you can't put them into groups of infinite sets) then you, for all practical purposes would win because we wouldn't be able to fit everyone in because of what I've already said before. However, as long as you can count the infinities then you can place the counted infinities into the hotel properly with the correct method.

If it's any consolation for it all, a similar question puzzled philosophers and mathematicians for a very long time. I think Galileo had it figured out.

Quote:
All of the hotels can never be destroyed.
They can, and I already said how.

Don't turn this into the other infinity thread we have going...or did it die yet finally?
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