Could we get a Math section?
Reply #154 - February 23, 2015, 07:31 AM
For those who are interested.
Hilbert's Hotel
(1) Suppose that we have a hotel with an infinite number of rooms...
And suppose that one day, an infinite number of people come in to stay at the hotel, such that every room is occupied.
Q: What happens if somebody else comes along and wants to take a room, can the hotel accommodate him?
Answer: Yes. All we do is take the person in room 1 and move them to room 2. The person in room 2 moves to room 3, and so on for every room number. This will open up room 1 for our new guest to stay in.
More formally:
For every room n, the person occupying n moves to n+1.
Even though the hotel was full, there will always be a room n+1 for the person in room n to move into.
The first result of Hilbert's thought experiment highlights that : ∞ + 1 = ∞
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(2) Now suppose that an infinite amount of people turn up at the hotel, looking for a room.
Q: Can we accommodate them?
Answer: Yes, we have another very simple solution. What we do is take the person in room 1 and move them to room 2. Then we take the person in room 2 and move them to room 4 and so on.
Generalising:
For every room n , the person occupying n moves to 2n.
What this means is that all the even numbered rooms will be occupied, but all the odd numbered rooms will be made available to accommodate the new guests, as there are an infinite amount of odd numbered rooms.
The second result of Hilbert's thought experiment is that ∞ + ∞ = ∞
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(3) This final example is a little more tricky. We still have our hotel, and it is still full.
Suppose that the hotel is visited by an infinite amount of buses, and each bus contains an infinite amount of passengers.
Q: Can the hotel accommodate them?
Answer: Once again, yes it can.
Although there are various ways to approach (3), here is one of the easier methods, which is known as the interleaving method:
We assign a number to every bus, and assign a number to every seat on every bus. We imagine that the hotel is a bus and that its rooms are seats.
You take the bus number, and take the seat number. If either number is shorter, add 0's to it until they're the same length.
E.g. bus 12 , seat 4892 = 0012 and 4892.
Next, we interleave the two numbers : bus,seat, bus, seat etc...
Such that 0012 and 4892 becomes : 04081922
We then remove the first 0 from the interleaved number, and arrive at the unique number 4081922.
We send the passenger on the bus to that room number. As long as we use this method, we can assign each passenger a unique number.
This is because no person will have the same bus number and the same seat number. Some will have the same bus number, some will have the same seat number. However, no two people will have the same bus & seat number.
The final result of Hilbert's thought experiment is that ∞ × ∞ = ∞
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My mind runs, I can never catch it even if I get a head start.
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Topic: Could we get a Math section?