Yesterday I came a across a new (new to me, that is) proof of the irrationality of . I found it in the paper “Irrationality From The Book,” by Steven J. Miller, David Montague, which was recently posted to arXiv.org.
Apparently the proof was discovered by Stanley Tennenbaum in the 1950’s but was made widely known by John Conway around 1990. The proof appeared in Conway’s chapter “The Power of Mathematics” of the book Power, which was edited by Alan F. Blackwell, David MacKay (2005).
It is a proof by contradiction. Suppose for some positive integers and . Then . Geometrically this means that there is an integer-by-integer square (the pink square below) whose area is twice the area of another integer-by-integer square (the blue squares).
Assume that our square is the smallest such integer-by-integer square.
Now put the two blue squares inside the pink square as shown below. They overlap in a dark blue square.
By assumption, the sum of the areas of the two blue squares is the area of the large pink square. That means that in the picture above, the dark blue square in the center must have the same area as the two uncovered pink squares. But the dark blue square and the small pink squares have integer sides. This contradicts our assumption that our original pink square was the smallest such square. It must be the case that is irrational.
[Note: the squares in the pictures almost work. They are and . As Conway points out, . Indeed .]
If you want to see more examples, look at Miller and Montague’s paper “Irrationality From The Book.” They extend this idea to give geometric proofs that , , , and are irrational.
Also, Cut-the-Knot has 19 proofs of the irrationality of (including this one).