Given any real number, we can find rational numbers that approximate it as closely as we might like. For example, if we want to approximate π to within 1/10000 we might take the rational number 3.1415 = 31415/10000, which we get from the decimal expansion. This uses a fairly large numerator and denominator. Interestingly, there are better approximations to π with much smaller denominators: 333/106 ≈ 3.1415904... approximates π to better than 1/12000 and 355/113 ≈ 3.1415929... approximates π to better than 1/350000.

We'll talk about how to find such unexpectedly good approximations (using continued fractions), and about how well we can hope to do this (theorems of Dirichlet, Thue, Siegel, and Roth).