Interesting Math Facts for Layman

Question

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:

• $.9\overline{9}=1$
• $1+2+\cdots+n=n(n+1)/2$
• 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.

Answer 1

I really like the following facts:

1. The hairy ball theorem, which can be interpreted as "there always is a point on Earth where there is no wind."
2. The Borsuk-Ulam theorem, which has, as a special case, "there are always two antipodal points on earth with exactly the same pressure and temperature."

Also, a question I think is really cool is: "can you hear the shape of a drum?"

Note: the answer is no, there are drums with different shapes which produce exactly the same tones.

Answer 2

There are lots of books with the kind of things you are looking for, I particularly like: Professor Stewart's Cabinet of Mathematical Curiosities Professor Stewart's Hoard of Mathematical Treasures 1089 and All That: A Journey into Mathematics And I'm sure you can find many more depending on the specific mind of thing you have in mind!