Related papers: A number system with base $-\frac32$

In this paper, we study representations of real numbers in the positional numeration system with negative basis, as introduced by Ito and Sadahiro. We focus on the set $\Z_{-\beta}$ of numbers whose representation uses only non-negative…

In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that…

In this paper we consider integers in base 10 like $abc$, where $a$, $b$, $c$ are digits of the integer, such that $abc^2 - (abc \cdot cba) \; = \; \pm n^2$, where $n$ is a positive integer, as well as equations $abc^2 - (abc \cdot cba) \;…

Natural numbers which are nontrivial multiples of some permutation of their base-$b$ digit representations are called permutiples. Specific cases include numbers which are multiples of cyclic permutations (cyclic numbers) and reversals of…

Analysis on fractals is a growing field, with hints of potential for widespread applicability across all of STEM. One of the most heavily researched type of fractals are the nested fractals, fractal shapes defined by virtue of being made of…

There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…

Let $\beta>1$ be fixed. We consider the $(\frak{b, d})$ numeration system, where the base ${\frak b}=(b_k)_{k\geq 0}$ is a sequence of positive real numbers satisfying $\lim_{k\rightarrow \infty}b_{k+1}/b_k=\beta$, and the set of digits…

The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$. In this article, first, another similar…

Dismal arithmetic is just like the arithmetic you learned in school, only simpler: there are no carries, when you add digits you just take the largest, and when you multiply digits you take the smallest. This paper studies basic number…

We study topological, metric and fractal properties of set of numbers $[0;1]$ with given asymptotic mean of digits in their ternary representation. We investigate connection of these numbers and numbers with a given frequency of digits.

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new…

In the paper, I consider appearance of unit's digits in minor totals of a few integer sequences. The sequences include the sequence of even integers, sequence of odd integers and Faulhaber polynomial at $p = 2$. Application of difference…

We study two positional numeration systems which are known for allowing very efficient addition and multiplication of complex numbers. The first one uses the base $\beta = \imath - 1$ and the digit set $\mathcal{D} = \{ 0, \pm 1, \pm \imath…

Let $b$ be a numeration base. A $b$-Niven number is one that is divisible by the sum of its base $b$ digits. We introduce high degree $b$-Niven numbers. These are $b$-Niven numbers that have a power greater than $1$ that is $b$-Niven…

Despite the fact that almost all real numbers are absolutely normal---that is, the digits in their expansions to any base occur in all possible configurations with the expected frequency---not one specific example of an absolutely normal…

The present article is devoted to functions from a certain subclass of non-differentiable functions. The arguments and values of considered functions represented by the s-adic representation or the nega-s-adic representation of real…

With disks and networks providing gigabytes per second, parsing decimal numbers from strings becomes a bottleneck. We consider the problem of parsing decimal numbers to the nearest binary floating-point value. The general problem requires…

In this article, we try to explain and unify standard divisibility tests found in various books. We then look at recurring decimals, and list a few of their properties. We show how to compute the number of digits in the recurring part of…

Simple continued fractions, base-b expansions, Dedekind cuts and Cauchy sequences are common notations for number systems. In this note, first, it is proven that both simple continued fractions and base-b expansions fail to denote real…

We consider positional numeration system with negative base $-\beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $\beta$ is a quadratic Pisot number. We study a class of roots…