A Combinatorial Model for Rational Base Representations of Natural Numbers

Everyone is familiar with the standard base-10 representation of numbers; others may also be familiar with the binary representation of natural numbers useful for computing. In this talk, we discuss our summer research, which involves representing natural numbers using a rational number as the base instead. In an attempt to generalize results of previous PLU summer researchers, we built an entire combinatorial theory around these representations. In this talk, we define rational base representations and introduce a family of trees that describe the important features of this method of representing numbers. With the combinatorial model in place we describe certain aspects of these rational base representations and answer some relevant combinatorial questions.