One of the most difficult questions a curious student may ask a math teacher is whether 1/0 = ∞ or not. And then of course comes another thoughtful extension: Is 1/∞ = 0?

Why this is a difficult question? It is difficult because it is usually asked by those who feel that *infinity* is a natural concept, like a number is, and everyone, especially someone who loves mathematics, should not have difficulty answering the question.

In early twentieh century German mathematician Hilbert proposed a thought experiment to show us just how hard it is to wrap our minds around the concept of infinity.

Imagine the Grand Budapest Hotel has infinite number of rooms with a very hardworking manager in Mr.Gustave and his lobby boy Zero. So one night the hotel houseful, with infinite people occupying infinite number of rooms. Just as Mr.Gustave is about to go to sleep one man appears and asks for accommodation. Mr.Gustave who had been told by the owners that no one should be returned is in great trouble, but Zero who had a great love for numbers suggests that if they move guest of room number n to room number (n+1), like person in room 1 to room 2, room 2 to room 3 and so on. The new man can be accommodated easily. Mr.Gustave was a happy man again. Later that night some 40 more people came and all were accommodated using the same trick. So person in room 1 moved to room 41, 2 to 42 and so on.

No body left the next day. At night however a bus of infinite number of seats arrived at the hotel. Mr.Gustave is helpless, but then again zero comes up with a solution. He tells his boss to move the person in room n to room 2n i.e. the guy in room 1 goes to room 2, 2 to 4, 3 to 6 and so on. So Mr.Gustave has rooms 2,4,6,8,10… occupied and 1,3,5,7…. Empty. He asks the people is the bus to unload and allots a room to everyone. He is a happy man again.

The popularity of the Grand Budapest Hotel keeps increasing. One night before retiring, Mr.Gustave hears some chaos outside. He looks out through his window and faints. There were an infinite number of buses each having infinite passengers. On regaining his senses he calls for Zero straightaway. But even Zero seems clueless this time. However after thinking his brains out, he comes up with a brilliant idea. His idea is based on the fact that the number of prime numbers is inifinite. He tells Mr.Gustave “Put the people occupying nth room to room number 2^{n}. Then we can put the people in the 1^{st} bus in 3^{n}-th room, i.e. 1^{st} person in 2^{nd} bus to room 3^{1}, 2^{nd} person to room 3^{2 }and so on. Likewise put 2^{nd} bus passengers to 5^{n}-th room number and so on.” So Gustave had all people accommodated and still some rooms like 6, 12, 15 unoccupied. His job was saved!

However it was possible for Gustave and Zero to get the job done as they had only natural numbered rooms. Had it been real numbered rooms, like negative rooms, fractional rooms, irrational number rooms… then Gustave might have had to quit his job.

Also it would be important to remember that if the charge per room is 100 dollars per night, then the net income doesn’t increase. It remains infinite!

Gustave and Zero shows just how hard it is to grasp the concept of infinity by our relatively finite minds. You can obviously help yourself with this after a good night’s sleep. But you might need to change your rooms in the middle of the night! 😉

One of the most difficult questions a curious student may ask a math teacher is whether 1/0 = ∞ or not. And then of course comes another thoughtful extension: Is 1/∞ = 0?

Why this is a difficult question? It is difficult because it is usually asked by those who feel that *infinity* is a natural concept, like a number is, and everyone, especially someone who loves mathematics, should not have difficulty answering the question.

In early twentieh century German mathematician Hilbert proposed a thought experiment to show us just how hard it is to wrap our minds around the concept of infinity.

Imagine the Grand Budapest Hotel has infinite number of rooms with a very hardworking manager in Mr.Gustave and his lobby boy Zero. So one night the hotel houseful, with infinite people occupying infinite number of rooms. Just as Mr.Gustave is about to go to sleep one man appears and asks for accommodation. Mr.Gustave who had been told by the owners that no one should be returned is in great trouble, but Zero who had a great love for numbers suggests that if they move guest of room number n to room number (n+1), like person in room 1 to room 2, room 2 to room 3 and so on. The new man can be accommodated easily. Mr.Gustave was a happy man again. Later that night some 40 more people came and all were accommodated using the same trick. So person in room 1 moved to room 41, 2 to 42 and so on.

No body left the next day. At night however a bus of infinite number of seats arrived at the hotel. Mr.Gustave is helpless, but then again zero comes up with a solution. He tells his boss to move the person in room n to room 2n i.e. the guy in room 1 goes to room 2, 2 to 4, 3 to 6 and so on. So Mr.Gustave has rooms 2,4,6,8,10… occupied and 1,3,5,7…. Empty. He asks the people is the bus to unload and allots a room to everyone. He is a happy man again.

The popularity of the Grand Budapest Hotel keeps increasing. One night before retiring, Mr.Gustave hears some chaos outside. He looks out through his window and faints. There were an infinite number of buses each having infinite passengers. On regaining his senses he calls for Zero straightaway. But even Zero seems clueless this time. However after thinking his brains out, he comes up with a brilliant idea. His idea is based on the fact that the number of prime numbers is inifinite. He tells Mr.Gustave “Put the people occupying nth room to room number 2^{n}. Then we can put the people in the 1^{st} bus in 3^{n}-th room, i.e. 1^{st} person in 2^{nd} bus to room 3^{1}, 2^{nd} person to room 3^{2 }and so on. Likewise put 2^{nd} bus passengers to 5^{n}-th room number and so on.” So Gustave had all people accommodated and still some rooms like 6, 12, 15 unoccupied. His job was saved!

However it was possible for Gustave and Zero to get the job done as they had only natural numbered rooms. Had it been real numbered rooms, like negative rooms, fractional rooms, irrational number rooms… then Gustave might have had to quit his job.

Also it would be important to remember that if the charge per room is 100 dollars per night, then the net income doesn’t increase. It remains infinite!

Gustave and Zero shows just how hard it is to grasp the concept of infinity by our relatively finite minds. You can obviously help yourself with this after a good night’s sleep. But you might need to change your rooms in the middle of the night! 😉