School Help - Infinity

[EternalWrath]

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.

[GuidoHunter]

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.

--Guido

http://andy.mikee385.com

[Afrobean]

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.

[Reach]

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.

[Afrobean]

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. 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:

∞+∞=∞

???

[Reach]

Re: School Help - Infinity

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.

[Tokzic]

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

[inflames07]

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.

[Tokzic]

Re: School Help - Infinity

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.

[Afrobean]

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 ^_~

[Reach]

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)

[inflames07]

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"

[Tokzic]

Re: School Help - Infinity

Tokzic, is again oversimplifying the problem >__>

Point out a flaw in my reasoning.

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

[Ice wolf]

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.

[Reach]

Re: School Help - Infinity

Point out a flaw in my reasoning.



rofl

Fiend, edited >_>

Too bad there is still a reasoning flaw.

The problem is here:

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.