I am often asked why I chose to study mathematics. Especially from strangers and new acquaintances. Of course, I don't have the time to sit them down and tell them why mathematics is so marvelous and why their view of it is likely skewed by the way they learned it in middle/high school. In these instances I often pull out a simple and 'cool' fact that most people do not know, most will find interesting or make them think, and which does not take twenty minutes to explain the intuition behind it. Some of them are:
- There are 'more' real numbers than natural numbers
- There are the same amount of natural numbers as rationals
- There are non-integer valued dimensions
- The No-Retraction Theorem
I was wondering if anyone else had a similar tool-kit and what is in it? I would like to add some more easy-to-explain-but-interesting facts.