Los Angeles Math Circle

5/13 -- LAMC High School: Understanding Infinity -- Prof. David Weisbart

Suppose that there are two hotels for numbers. The first hotel has rooms labeled 1 through n and each room can take at most one guest. If m numbers get rooms at the hotel and fill the hotel to capacity, then m must equal n. This seems quite natural, but how do we prove it? The second hotel is very strange. Last night, all the natural numbers were guests at the hotel and the hotel was filled to capacity. Tonight, all the fractions have decided to visit the hotel but the natural numbers won't leave. The hotel manager was in a panic because the hotel appeared to be overbooked. Fortunately, the hotel manager's sister is a mathematician and she told her brother not to panic. After rearranging the guests, she was able to fit in not only all the natural numbers, but all the fractions as well. How did she do it and what does this tell us about the infinite?