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The math buffs out there have probably read some variation of this story before. It's a story that somewhat explains the nature of infinity as it is defined by mathematics.
So there exists this place called the Hotel Infinity. It's a hotel that has an infinite number of rooms. However, currently all the rooms are full. A man comes in and asks the desk manager, "Do you have any rooms available for me?" The desk manager replies, “All our rooms are full, but I'll see what I can do."
The manager thinks for a while, and then grabs the intercom and makes the announcement to all the guests, "Attention, if you are currently in room N, please move to room N+1." So the guest in room 1 moved to room 2, the guest in room 2 moved to room 3, etc. After the shift, the manager moved the man into the newly vacant room 1.
The next day, another man comes in and asks the desk manager, "Do you have any room for me and my infinite friends outside?" The desk manager once again replies, "All our rooms are full, but I'll see what I can do."
So the manager again thinks for a while and then makes an announcement to the guests, "Attention, if you are currently in room N, please move to room 2*N." So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, etc. After the shift, all the odd numbered rooms became vacant, and the manager was able to accommodate all of the man's friends.
On the third day, another man comes in and tells the desk manager, "Outside, I have an infinite number of buses, each with an infinite number of people. Do you have any rooms for us?" The desk manager once again replies, "All our rooms are full, but I'll see what I can do."
So once again after some thought, the manager announces to the hotel, "Attention, if you are currently in room N, please move to room 2^N." So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, etc. The manager then proceeds to label each of the infinite buses with a different prime number, starting from 3 and going up. Because there are infinite prime numbers, every bus can get a number. The people inside each bus are also counted off starting from 1 and going up. The manager then tells all the people to go to room B^N, where B is their bus's number, and N is their personal number. Once again, the manager was able to accommodate all of the guests.
So the moral of the story is, infinity is pretty awesome.
There was another variation of the third example I heard that, instead of using prime numbers, uses the fact that the 1-dimensional integer plane and the 2-dimensional integer plane are of the same cardinality.
Basically, you can just enumerate all the buses using natural numbers, and enumerate the people in each bus using natural numbers as well. The bus number is the x-coordinate and the person number is the y-coordinate. You then map the corresponding (x,y) pair to some room number and you're done. The interesting part of course is, how you map 2-d to 1-d?
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I consider myself a smart man, or at least above average, and I am in a major that involves a lot of math, and still to this day cannot wrap my head around the concept of infinity.
Edit: Cool story though, thanks for sharing.
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Cool story, that deals with countable infinities, like the natural or even rational numbers.
What gets really cool is the uncountable sets like the real numbers.
Stuff like...there are as many real numbers in the interval [0,1] as there are in the whole real line.
Math ftw ^^
If someone showed up with uncountable many people the hotel would be in trouble xD
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infinity and chemistry are the only things that make my brain hurt 
infinity is infinitely cooler than chemistry though.
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Oh yeah Zortch I totally agree that the uncountable sets are very cool. Things like Cantor's DIagonal Argument are just so simple, yet so genius, I'm just left in awe at it's beauty.
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I've heard this story before, although in a slightly different form. I still think it's cool though.
I don't get why you have to number the busses with prime numbers, though. Why can't you just use integers, since those are infinite too.
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If integers were used, several (read: infinite) rooms would have more than one person residing in them.
For instance person 4 on bus 2, and person 2 on bus 4, or 2^4=16 and 4^2=16, would both be in room 16. Not to mention they would be in a room already occupied by someone that was already in the hotel to begin with. Using the prime numbers, however, avoids this, due to what makes a prime number a prime number, namely, indivisibility. (One of you math whizzes please confirm or deny this answer, as I am not absolutely sure that I am right.)
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Prime numbers were used in this case because their properties are very useful for what we're trying to do. Lemonwalrus is correct that we cannot just start taking integers to the power, because we will have overlapping values. However, there is a way using just integers, which I'll explain after describing the reasoning for using primes.
The idea behind using prime numbers is that any two prime numbers taken to some positive power will never be equal, because of the fact that they're prime numbers. so 3^x and 5^y will never be equal for any x and y. Now this is very useful, because you know that no people from two separate buses will have the same room. And since there are infinite prime numbers, and each bus is given a different prime number, we are assured that there is no overlap.
Now, to explain the method that uses integers. First, we can think of the room numbers as a 1-dimensional coordinate system or a line, that just has all the positive integers from 1 to infinity. Then, we think of each bus having a positive integer value, and each person in the bus having a positive integer value. We can imagine this as being a 2-dimensional coordinate system or a plane.
Now we don't take the bus number to the power of the person's number like in the prime solution, because that will definitely have overlap. We use a different mechanism that kind of looks like a spiral. Hopefully this next part makes sense.
![[image loading]](http://img167.imageshack.us/img167/24/infinityyd1.jpg)
If you look at this picture, you can see that we have a 2-d coordinate system, and I've labeled the points of the coordinate system with numbers in a spiral type pattern. Point (0,0) has value of 0, point (1,0) has a value of 1, point (1,1) has a value of 2, point (0,1) has a value of 3, etc. Now this construction shows that we can map the infinite 2d plane to an infinite 1d line, and vice versa. What this means for the hotel manager is that he takes the bus number as the x-value, the person number as the y-value, and puts him in the room that the spiral indicates. So for example, Bus number 3, person number 1 corresponds to the coordinate (3,1), which means that his room number is 11. Now there is the issue of the current guests of the hotel. This is dealt with by pretending that they are all part of a Bus with a Bus number of 0.
This construction also shows that the set of rational numbers is also a countable set, and is equivalent to the set of natural numbers. By definition, any rational number can be represented as a fraction, with an integer numerator and denominator. Thus, if we just use the x-axis as the numerator and the y-axis as the denominator, we can map rational numbers to natural numbers.
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I ran across a similar problem in computer science class. The goal was to construct a stream to display all elements of an infinite matrix. I can't exactly remember the solution though =(... something about recursively interweaving the rows of the matrix.
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Slithe, you are awesome. More!
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EPT
