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Paradox of Hilbertís Hotel

Hotel is full, some guests arrive

Hotel is full, infinite number of guests arrive

Hotel is full, some guests arrive

Hilbertís Hotel is a hotel with an infinite number of rooms and infinite number of guests. Every room is occupied. A new visitor arrives. Can he be accommodated?
Yes, arithmetic with infinite quantities allows to do it.

At first it seems that he cannot, but then the hotel clerk moves the guest in Room 1 to Room 2, and the guest in Room 2 to Room 3, and so on. Every guest is moved to the next room along. This leaves Room 1 vacant for the new visitor. Although the hotel is full, the new guest can always be accommodated in Room 1.

If another visitor arrives, the hotel clerk moves the guests again, and the new guest can be accommodated in Room 1.

Hotel is full, infinite number of guests arrive

What will be if hotel is full and infinite number of guests arrive? Can they be accommodated?

Yes, they will be accommodated thanks to property infinity + infinity = infinity.

For example, the manager asks each guest to take note of their current room number and move to the room whose number is twice that of his own. After everyone has done this, only the even-numbered rooms will be occupied and the odd-numbered rooms will be empty.

So Hilbertís Hotel is full, and yet it has an infinite number of vacancies.

by Tetyana Butler

Complex functions Tutorial
Complex analysis is studying the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of algebra and trigonometry of real numbers may break down when applied to complex numbers.
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