Hirzebruch noticed "an amusing connection between continued fractions and class numbers" while computing the signature sign(X) of a Hilbert modular surface X attached to a real quadratic number field K. We will discuss how continued fractions show up in the resolution of cusp singularities in a compactification of X. The number of cusps is the class number of K, and, when K has no units of negative norm, sign(X) can be given in terms of class numbers of imaginary quadratic number fields.