There are many ways to represent a number, commonly known as base expansions. The most frequently used base is ten, which is the basis for our decimal number system. However a more uncommon way to represent a number is the so called Cantor expansion of the number. This system uses factorials rather than numbers to powers as the basis for the system, and it can be shown that this produces a unique expansion for every natural number. However, if you view factorials as products, then it becomes natural to ask what happens if you use other types of products as bases. This paper explores that question and shows there are an uncountably infinite number of bases which can be used to represent the natural, and real numbers uniquely. By using these new and interesting types of bases, it becomes possible to formulate bases in which all rational numbers have a terminating expansion.

Author Bio

I am junior mathematics major at the University of Rochester. I initially learned of the Cantor Base Expansion from Kenneth Rosen's Discrete Mathematics textbook. During my sophomore year, I mentioned to Prof. Ken McMurdy that there are probably more types of strange expansions and he encouraged me to explore this further. It ultimately resulted in the paper, completed during my free time sophomore year. At college, I am the VP of the Society for Undergraduate Math Majors (SUMS), and I am the webmaster for my Tae Kwon Do club. In the future I plan to pursue mathematics with hopes of graduate school.