2009-03-11, 06:31 AM
The infinite hotel problem.
So just imagine that there's this hotel with an infinite number of rooms. One day a person walks into the hotel and asks for a room. The attendant looks to him and responds, "My dearest apologies, but we're completely booked."
The person is disappointed at first but gets a brilliant idea. "Um... what if you move the person from room 1 to room 2 and then move the person from room 2 to room 3 ... and so forth. And I'll take room 1."
The attendant looks at him and tells him, "Here's your key."
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Well thankfully the attendant was able to squeeze in one more person. Good thing this hotel has an infinite number of rooms right?
BUT! Little did he expect this to happen:
A countably infinite number of guests show up!
The attendant freaks out a bit, "Oh no, what are we going to do. I might be able to fit one of you, maybe two, but an infinite number of you?!" Luckily for that man, one of the guests had studied abstract algebra and found a great solution to his dilemma.
"What if you do this: Take the guest in room 1 and place him in room 2. Take the guest in room 2 and put him in room 4. Take the guest in room 3 and put him in room 6. And do that for all of the guests. All of us out here will take the odd rooms."
The attendant breathed a sigh of relief, "Enjoy your stay."
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I both love and hate being math major. The above is a not so intuitive proof for the reason why there are just as many even numbers as there are natural numbers. Most people would intuitively think, "Oh there are obviously twice as many counting numbers as there are even numbers." But that's not the case ;3
I both
So just imagine that there's this hotel with an infinite number of rooms. One day a person walks into the hotel and asks for a room. The attendant looks to him and responds, "My dearest apologies, but we're completely booked."
The person is disappointed at first but gets a brilliant idea. "Um... what if you move the person from room 1 to room 2 and then move the person from room 2 to room 3 ... and so forth. And I'll take room 1."
The attendant looks at him and tells him, "Here's your key."
------------
Well thankfully the attendant was able to squeeze in one more person. Good thing this hotel has an infinite number of rooms right?
BUT! Little did he expect this to happen:
A countably infinite number of guests show up!
The attendant freaks out a bit, "Oh no, what are we going to do. I might be able to fit one of you, maybe two, but an infinite number of you?!" Luckily for that man, one of the guests had studied abstract algebra and found a great solution to his dilemma.
"What if you do this: Take the guest in room 1 and place him in room 2. Take the guest in room 2 and put him in room 4. Take the guest in room 3 and put him in room 6. And do that for all of the guests. All of us out here will take the odd rooms."
The attendant breathed a sigh of relief, "Enjoy your stay."
===============
I both love and hate being math major. The above is a not so intuitive proof for the reason why there are just as many even numbers as there are natural numbers. Most people would intuitively think, "Oh there are obviously twice as many counting numbers as there are even numbers." But that's not the case ;3
I both