Numbers are everywhere in our daily lives, and positional numeral systems are arguably the most important and common representation of numbers. In this work, we will investigate interesting properties and applications of some variations of positional numeral systems and construct a generalized positional numeral system to model and formalize them in Agda. We will not only examine properties of our representation and its relationship with the classical unary representation of natural numbers but also demonstrate how to translate propositions and proofs between them.