The Zolotarev numbers arise as infima attained in a classical rational optimization problem posed by Y. Zolotarev in the late 1800s.  Despite their pre-computational roots, they remain relevant in modern computing, where they appear in several fundamental problems in numerical linear algebra.  This talk will introduce the Zolotarev numbers and the rational functions they are linked to, and then describe how they can be used to create a fast hierarchical method for solving  the inverse nonuniform discrete Fourier transform problem.