In elementary mathematics classes, students are often overwhelmed by different representations of numbers and corresponding operations: usual fractions, decimal representations, binary numbers, etc. What often helps is when students learn the history of these representations, see the limitations of seemingly reasonable representations like Roman numerals, and how other representations overcame these limitations. Still, history was developed somewhat randomly, so the historical sequence is still somewhat chaotic. We believe that providing a unified approach for all these representations would help describe their sequence in a more logical way and thus, help the students even more.
In our analysis, we explore the relation between the foundations of arithmetic, especially related definitions of numbers, and the resulting notations. For example, widely used Peano axioms describe natural numbers as containing 0 and containing, for each x, the next number x + 1. The corresponding definition naturally leads to the historically first representation of natural numbers as I, II, III, IIII, etc. Allowing addition and multiplication by 2 (and powers of 2) leads to binary numbers, allowing general multiplication to decimal numbers, etc. It turns out that a similar foundational description can be found for most historical representations of natural and fractions. For example, allowing the division of 1 by a natural number leads to Egyptian fractions, allowing generic division of integers leads to common fractions, etc.
We believe that exposing students (and teachers) to at least some of these results will help them better understand the relation between different number representations, and thus, will make it easier for them to master the corresponding techniques.