A Number System with Base −32

n the present article we explore a way to represent numbers with respect to the base −32 using the set of digits {0, 1, 2}. Although this number system shares several properties with the classical decimal system, it shows some remarkable differences and reveals interesting new features. For instance, it is related to the field of 2-adic numbers, and to a self-affine “fractal” set that gives rise to a tiling of a non-Euclidean space. Moreover, it has relations to Mahler’s 32-problem and to the Josephus problem.