We introduce and study series expansions of real numbers with an arbitrary Cantor real
base β=(β _n) _ n ∈ N β=(β n) n∈ N, which we call β β-representations. In doing so, we …

Similarly to Parry's characterization of β-expansions of real numbers in real bases β> 1, Ito
and Sadahiro characterized digital expansions in negative bases, by the expansions of the …

This paper investigates representations of real numbers with an arbitrary negative base–β<–
1, which we call the (–β)-expansions. They arise from the orbits of the (–β)-transformation …

This article is about the base complement in 𝐵3. The complement method consists of the
course of operations around the centroid. The centroid is played an important role in the …

This paper brings natural numbers from 11112 to 20000 written in ascending and
descending orders of 1 to 9. Most of the numbers are obtained by using basic operations …

We introduce a numeration system, called the beta Cantor series expansion, that
generalizes the classical positive and negative beta expansions by allowing non-integer …

This work brings patterned representations of natural numbers from 1 to 1000 in terms of
single digits from 1 to 9. To bring these results, only basic operations, such as addition …

In this note, we introduce a novel way of doing complex arithmetic that does not involve
separating the complex numbers into their real and imaginary part~. This method uses the …

In the base phi representation any natural number is written uniquely as a sum powers of the
golden mean with digits 0 and 1, where one requires that the product of two consecutive …

Very little is known about the expansions of a real number in several integer bases. We
establish various results showing that the expansions of a real number in two multiplicatively …