In "The Joy of X" Steven Strogatz discuss in a chapter on the Hilbert Hotel,a hotel with an infinite number of rooms, the problem of assigning rooms when an infinite number of buses arrive, and each bus has an infinite number of passengers. "There is always room at the Hilbert Hotel" says the manager and Steven Strogatz describes how it is done. Every guest has a room. For exampe bus#45 passenger #77 goes in a certain room. The bus number and passenger number (p, q) represent a rational fraction p/q.There is a one to one correspondence between p/q and hotel room number,the set of integers. Mr. Strogatz says that we can make an exhaustive list of all positive fractions(rational numbers) even though there is no smallest one. Arranging all the integers from one on up I can understand, but I have a hard time in the opposite direction. Even though there is no smallest rational number, he says you can make an exhaustive list of all positive fractions. How do you do this ? All integers I can understand, but going in the other direction toward Zero I can't see how an exhaustive list can be made.