*By abelchen777, This post originally appeared on Function of a Rubber Duck.*

Infinity is a slippery idea. And did you know that some infinities are bigger than others? For example, you have and infinite number of whole numbers. Exactly

*half*of those numbers are odd. So it is fair to say that there are twice as many whole numbers than odd numbers, right? Actually, there are also an infinite number of odd numbers. So if we try to put this into an equation we get:
∞= 1/2 ∞

…which is obviously incorrect. The only logical conclusion is that not all infinities are created equal.

Consider another example. Imagine that you are the manager of a hotel called Hotel Infinity. This hotel has an infinite number of rooms. when the hotel is full, and a guest comes into the lobby asking for a room what do you do? No problem. Just bump everyone up a room, and put the guest in room 1. Every guest goes to room

*n+*1, where*n*is their current room number. Now in a finite hotel this would be a problem because the person staying in the room with the biggest number will be out of luck, but at Hotel Infinity, there*is*no biggest room number.
The next day, an Infinity Tour Bus, containing an infinite number of people, arrives at the hotel. Fitting in one person is one thing, but how do you accommodate for an infinite number of people? First, tell all the people who have rooms to go to room 2

*n, freeing up all the odd number**rooms*. Then, each person on the bus will go to room 2*n*-1, and everyone will be happy. Problem solved.
A little later, and infinite number of Infinity Tour Buses arrive. Now what are we going to do? Reduce this problem to a problem you already know how to solve. Line the buses up beside one another. Number the bus seats diagonally, starting on the top left. It should look something like this:

1 | 2 | 4 | 7 | 11 | 16 |

3 | 5 | 8 | 12 | 17 | … |

6 | 9 | 13 | 18 | … | |

10 | 14 | 19 | … | ||

15 | 20 | … | |||

21 | … |

Now tell everyone to go to the corresponding seat in Bus 1. Now you only have one busload of people, and we already know how to deal with that.

Due to your amazing brain power and a little nifty mathematics, you have been able to solve several accommodation problems here at Hotel Infinity . Is there any situation you cannot accommodate for?

There is, in fact, at least one situation where Hotel Infinity while have to hang a “No Vacancy” sign. You know there are more whole numbers than odd numbers. There are also more real numbers than whole numbers. Suppose guests bus seats were numbered with real numbers rather than whole numbers. Since there are more real numbers than whole numbers, you cannot fit all the guests into Hotel Infinity, even though there are an infinite number of rooms. Has your brain gone numb yet?

*About this contributor: I'm a grade 11 student with a special passion for mathematics. I also enjoy playing badminton, soccer, and reading both fiction and non-fiction books. My favourite field of science is quantum physics.*

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