Imagine that there is a hotel with infinite rooms like 1, 2, 3 and so on. You are the manager of the hotel. Let's say that only one person can stay in a room and all the rooms are full. There are an infinite number of people in an infinite number of rooms. A guest just shows up and he wants a room but all the rooms are full. How will you arrange a room for that person? Supposing that you've arranged a room for that person with your knowledge of infinity. Now, a bus with a hundred people shows up and now you have to arrange a hundred rooms. How will you do that? You've arranged a hundred more rooms but now another bus shows up which is infinitely long and it's carrying infinite number of people. You knew what to do with a finite number of guests but what can you do with infinite number of guests? Let us say that you've somehow managed to arrange rooms for those infinite number of people. Now, there are infinite number of buses with infinite number of people. How will you arrange a room for them?
Kindly answer all the questions :)
Answers
A bit introduction:
Well, this is the 'Hilbert's Theory Of Infinity' (also called Hilbert's Paradox Of Grand Hotel ), Where through the example of hotel with infinite number of rooms with infinite numbers of guests David Hilbert had tried to explain the concept of infinity. Infinity is not a quantity, its just a concept on which we can't apply mathematical rules of calculations such as ∞ times ∞, or 0 times ∞, or ∞/∞.
Let's try to answer the given questions!
(i) Given that in the Hotel of infinity, all rooms are filled with infinite number of guests and each guest stays in separate room. We have to arrange 1 guest who walks into the hotel to get a room. Answer is simple, we can ask all the guests to move next of their room number such as guest in room number 1 moves to 2, 2 moves to 3, 3 to 4 and so on upto n who moves to n+1. Since there are infinite number of rooms, there is a new room for each existing guest, this provides a room for the new guest.
(ii) Here's also the same concept is used. We. We have to open room for 100 guests unloaded from a bus. We can move each existing guest staying in room number n to n+100 thus opened 100 new rooms for new guests.
(iii) Now that seems to be somewhat tricky question if you don't know the actuall concept of infinity. The bus which is infinitely long arrived with infinite number of guests. We have to open infinite number of rooms for those infinite number of guests. In order to provide all those infinite numbers of guests the infinite numbers of rooms, we may use a basic rule of odd and evens. We know that there are infinite numbers of even and odd numbers, we can ask the guest in room number 1 to move in room number 2, 2 moves to room number 4, 3 moves to room number 6 basically guest staying in room number n to move in room number 2n, thus by doing this we have emptied all the rooms with odd numbers and since there are infinite number of odd numbers, these all guests can be assigned rooms with odd numbers. Hence problem solved !
(iv) Now this last question is too much complex but yes that's possible by the concept of infinite prime numbers! This would be possible if we ask the guest in room number n to move in room number 2^n. Say guest in room number 1 moves to 2^1= 2, guest in room number 2 moves to room number 2^2, guest in room number 3 moves to room number 2^3 and so on upto ∞ since there are infinite numbers of rooms. Now all the rooms with 2^n (2 is the smallest prime number) are filled. Now it's time to arrange the passengers. Let's talk about the first bus. We can assign each passenger the room number 3^n where n is their bus number (3 is the next prime after 2). Let's say passenger with bus's seat number 1 moves to 3^1=3, passenger with seat number 2 moves to 3^2=9, passenger with seat number 3 moves to room number 3^3=27 and so on upto ∞. That's about first bus but we have to arrange infinite number of passengers. We can assign the passengers of 2nd bus, the room number of next prime raised to the power of their seat number. 5^1, 5^2, 5^3 upto 5^n . . . and we can do this repeatedly to assign the passenger's to the seat number raised to the power of next prime. Since multiples of two primes can't coincide and prime numbers are infinite - 2, 3, 5, 7, 11, 13, 17, 19, 23 . . . we can assign all the passengers a separate room in the infinite hotel.
Conclusion :
At last the conclusion is that, since in the infinite hotel which was completed filled with infinite number of guests still accomodate infinite number of guests, infinity is an endless concept and we can't ever think about it's end !!!
Answer:
Imagine that there is a hotel with infinite rooms like 1, 2, 3 and so on. You are the manager of the hotel. Let's say that only one person can stay in a room and all the rooms are full. There are an infinite number of people in an infinite number of rooms. A guest just shows up and he wants a room but all the rooms are full. How will you arrange a room for that person? Supposing that you've arranged a room for that person with your knowledge of infinity. Now, a bus with a hundred people shows up and now you have to arrange a hundred rooms. How will you do that? You've arranged a hundred more rooms but now another bus shows up which is infinitely long and it's carrying infinite number of people. You knew what to do with a finite number of guests but what can you do with infinite number of guests? Let us say that you've somehow managed to arrange rooms for those infinite number of people. Now, there are infinite number of buses with infinite number of people. How will you arrange a room for them