Yes, you read it correctly. In this blog entry, You and I are going to IMAGINE a hotel which has infinite rooms. Each room would be one of those standard, regular-size rooms, made to accommodate just one person, but there shall be infinitely many such rooms!
Imagine a night when this Infinite Hotel is fully booked i.e. each room has a guest staying in. A new guest arrives. The proprietor informs this guest that all the rooms in his hotel are occupied. Does it mean that this hotel of his has no room for more?
If you were to answer Yes to this question, then the proprietor would have to decline the guest’s request and that will be the end of this story, making this blog perhaps the dullest piece ever written.
As it turns out, the answer to the above question is no. In this story, the proprietor shall, after a long dramatic pause and fittingly so, inform the new guest that he can indeed accommodate him!
But, how? In any hotel that we have come across, the proprietor would have had no other option but to decline the guest’s request for accommodation. So how can the proprietor of the infinite rooms’ hotel claim to be able to accommodate that guest?
The keyword here is ‘Infinite’ and the story invokes some of the most intriguing and counterintuitive properties of Infinity. This puzzle was first introduced by a mathematician named David Hilbert in his famous lectures on Infinity in 1924. He explained via this Example, his contemporary George Cantor’s contribution in understanding Infinite numbers.
How does the proprietor accommodate the new guest? He can apply a neat trick of moving the occupant of room 1 to room 2, that of room 2 to room 3, that of room 3 to room 4, and so on. Other than the obvious discomfort of shifting guests in and out of the rooms, these transpositions will result in room 1 being made available for the new guest to stay in.
What if there were a finite number of guests, say 10 that needed to be accommodated in the Infinite hotel? The proprietor would still be able to accommodate all ten visitors! The occupant of room 1 will be asked to move to room 11, that of room 2 will be asked to move to room 12, and so on up until the occupant in room 10 is moved to room 20. This will make the first ten rooms of the hotel vacant and the proprietor can easily accommodate the newcomers in those rooms.
These were the circumstances where the proprietor had to vacate only finite number of rooms. What about infinitely many new guests? You may be thinking that no way can the proprietor accommodate an infinite number of new guests in the hotel. Well, get ready to be surprised again!
See, the proprietor of the hotel knows that when it comes to Infinity, a part may be equal to the whole! He will free all the odd numbered rooms in the hotel by moving the occupant of room 1 into room 2, that of room 2 into room 4, and so on.
Now, its little counterintuitive to understand how all the occupants of rooms 1, 2, 3, 4, … can be fully accommodated into even numbered rooms 2, 4, 6, 8, … But, Cantor told us that the infinity of evens is equinumerous to the infinity of all natural numbers. That is, for each natural number n, we can find its even companion, 2n. So, the occupant of, let’s say, 79000th room can be quite easily accommodated in 158000th room. Likewise, the proprietor can shift all its old guests, to the even-numbered rooms, making it possible for the infinitely many new arrivals to be accommodated in the now vacated odd-numbered rooms.
Fascinating isn’t it? Well, allow me to conclude this article by asking you this: Do you think this infinite hotel will be able to accommodate an infinite number of buses where each bus contains infinitely many number of visitors? The answer lies in the infinity of prime numbers!