Rose-Hulman Undergraduate Mathematics Journal Volume 5 Issue 1 Article 4 Generalized Cantor Expansions Joseph Galante University of Rochester,
[email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Galante, Joseph (2004) "Generalized Cantor Expansions," Rose-Hulman Undergraduate Mathematics Journal: Vol. 5 : Iss. 1 , Article 4. Available at: https://scholar.rose-hulman.edu/rhumj/vol5/iss1/4 Generalized Cantor Expansions Joseph Galante University of Rochester 1 Introduction In this paper, we will examine the various types of representations for the real and natural numbers. The simplest and most familiar is base 10, which is used in everyday life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than exponentials as the basis for the representation. It can be shown that the expansion is unique for every natural number. However, if one views factorials as special type of product, then it becomes natural to ask what happens if one uses other types of products as bases. It can be shown that there are an uncountable number of representations for the natural numbers. Additionally this paper will show that it is possible to extend the concept of mixed radix base systems to the real numbers. A striking conclusion is that when the proper base is used, all rational numbers in that base have terminating expansions. In such base systems it is then possible to tell whether a number is rational or irrational just by looking at its digits.