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.