Written by: Sashwat Paudel - 2022019, Grade X
Posted on: 09 December, 2020
Infinity is the state of being infinite. Infinity is something that has no limit. It is told to be endless or boundless. It is denoted by the symbol ∞. This symbol is known as the infinity symbol. You may have encountered this symbol at least once, whether on the internet or in maths. But what really is infinity? Infinity is perceived as a simple thing, however, it is not.
The infinite hotel paradox is a great proof of how complex and unique infinity really is. The infinite hotel paradox is a thought experiment which was created by the german mathematician David Hilbert, about a grand hotel that has an infinite amount of rooms. Let's say that the hotel is fully booked. What happens when a guest comes to the hotel? Even if the hotel is full, there is no limitation because we have an infinite amount of rooms. To assign the new guest a room, the guest in room 1 is placed in room 2. Likewise, the guest in room 2 is placed in room 3. Every guest in room n moves to room n+1 .This way there is an open room for the new guest to live in. What if there are 60 new guests in the hotel? Won't they be turned down because the hotel is already booked by an infinite amount of people? Fortunately for the hotel, the guests won't have to be turned down at all. This time the guest in room n should be placed in room n+60. As there are an infinite number of rooms, there is space for the new 60 guests without discarding the existing guests. Now let's say that there are an infinite number of guests coming to the hotel. When there are countably infinite numbers of guests, the guest in room 2 moves to room 4. The guest in room 3 moves to room 6, and so on. The guest in room n moves to the room 2n. This way all of the infinitely odd numbered rooms are emptied and there is room for the countably infinite number of guests.
The infinite hotel paradox is a complex paradox thanks to infinity. This paradox proves that infinity is not what we perceive it to be and it is really complex when we get into the fancy calculations and theories. The most fascinating part of all this is that this paradox includes the simplest form of infinity. If we had to consider other forms of infinity, there would be negative numbered rooms, fractional rooms, radical rooms and other complex stuff. This would make the paradox more complex than it is now, because we cannot fill all the rooms using the algorithms and strategies used in the previous part.
