HOTEL HILBERT has infinite number of rooms. On ar particular night all the rooms were occupied, but a VACANCY sign hung outside. A potential guest arrived, and the desk clerk gave him the key to room 1, after asking the occupant of room 2 to move to room 3, the occupant of 4oom 3 to move to room 4, and so on. The next night an infinite number of new guests arrived and none of the old guests had checked out. ’No problem,’ the desk clerk announced. He moved the occupant of room 1 to room 2, the occupant of room 2 to room 4, the occupant of room 3 to room 6, the occupant of room 4 to room 8, the occupant of room 5 to room 10, in general the occupant of room n to room 2n. That freed up infinitely many odd-numbered rooms for the infinitely many new arrivals.
(From The Man Who Loved Only Numbers, by Paul Hoffman)
http://learn.tsinghua.edu.cn:8080/2002315664/smallstories/hotel-hilbert.pdf
7 Kommentare:
et c'est quoi, la version pour les matheuses?
Quelle question. Etant donnée N, l'ensemble des nombres naturels. Montrer que N est en bijection avec 2N.
:)
Allrightallright...a hotel with an infinite number of rooms, huh?
Please my poor beloved mathematicians...let us be practical for ONCE in your terribly exciting lives:
If there really was an infinity of rooms, the hotel would be infinitely high and it would require an infinity of time to take an infinitely fast elevator to reach the nth room...which means, in MY world, that the night would be long over before even expecting to reach the 1/infinitieth part of the way... and this without taking into account the return trip and the perturbations caused by the infinity of customers trying to get in there infinitieth room at the same time in an infinitely chaotic way....
...And they call that INFINITY FOR ENGINEERS.... P-L-E-A-S-E!!!!!!!!!!
--An infinitely geeky civil engineer wannabe that is infinitely in love with the (infinitely) best mathematician!! Keep it up, cherie: thanks to you mathematics definitively is on the path to redemption!
Je t'aime!
un peu plus simple, la version mathématique =).
PS: @Marc: Since every guest only needs to go from room n to room 2n, the path for each of them is finite and so you will get some sleep ;). But you're right, the question of how to build such an infinite hotel is not answered yet.... It's up to the civil engineers to find a solution ;)
Mwouais, maybe... but the question of how to manage the infinity of customers mowing simultaneously from room n to room 2n remains gloriously unanswered...
et toc!
Non, franchement, ils ne doivent pas tous déménager simultanément.. mon amour. Here's how you do it:
Forget about elevators. Forget about problems to build it. Some things have just been since the beginning of time – nobody is able to explain why the universe is infinite either.
So let's say there is Joe living on the first floor, and James living on the second. And Jack on the third. First, Joe gets the order to move up one floor, he does that (1 person on the stairs... no problem.) The first guest moves into the room. When Joe arrives, he finds James in his room, so he sends him up two stairs. (1 person on the stairs.. no problem.) The second person moves in. ...
Of course you could make it even faster if you allow them to move together (Joe and James). There will still be 1 person on each section of the stairs, and we'll be fine.
You see.. I have to mention here that the way I originally heard this story went further.. because the (infinite) set of real numbers arrived and the hotel director didn't have room for THEM. However, he was not being racist. Dang.
Infinity is amazing... I like this description by Douglas Adams:
"Space is big, really big. You just won't believe how vastly, hugely, mind-boggingly big it is! I mean, you may think it's a long way down the street to the chemist's, but that's just peanuts to space, listen..." and so on.
It's challenging to think about ∞, and mathematicians still have many problems to solve there, as you said mentioning real numbers... Thanks for the story; I'm sure you got talent for being a maths teacher (-;
And last but not least an interesting link: http://www.unendliches.net/
Have fun!
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