Counting the Positive Rationals: A Brief Survey

We discuss the optimal presentations of mathematical objects under well defined symbol libraries. We shall examine what light our chosen symbol libraries and syntax shed upon the objects they represent. A major part of this work will focus on discrete sets, particularly the natural numbers, with results that describe the presentation of the natural numbers under specific symbol libraries and what those presentations may reveal about the properties of the natural numbers thems...

An increasing sequence $(x_i)_{i=1}^n$ of positive integers is an $n$-term Egyptian underapproximation of $\theta \in (0,1]$ if $\sum_{i=1}^n \frac{1}{x_i} < \theta$. A greedy algorithm constructs an $n$-term underapproximation of $\theta$. For some but not all numbers $\theta$, the greedy algorithm gives a unique best $n$-term underapproximation for all $n$. An infinite set of rational numbers is constructed for which the greedy underapproximations are best, and numbers for ...

It is shown that there is an absolute constant $C$ such that any rational $\frac bq\in]0, 1[, (b, q)=1$, admits a representation as a finite sum $\frac bq=\sum_\alpha\frac {b_\alpha}{q_\alpha}$ where $\sum_\alpha\sum_ia_i(\frac {b_\alpha}{q_\alpha})<C\log q$ and $\{a_i(x)\}$ denotes the sequence of partial quotients of $x$.

The independence of the continuum hypothesis is a result of broad impact: it settles a basic question regarding the nature of N and R, two of the most familiar mathematical structures; it introduces the method of forcing that has become the main workhorse of set theory; and it has broad implications on mathematical foundations and on the role of syntax versus semantics. Despite its broad impact, it is not broadly taught. A main reason is the lack of accessible expositions for...

The authors show how to determine the group generated by the ratios $(an+b)/(An+B)$ through its dual.

The concept of nearest integer is used to derive theorems and algorithms for the best approximations of an irrational by rational numbers, which are improved with the pigeonhole principle and used to offer an informed presentation of the theory of continued fractions.

The concept of ``countable set'' is attributed to Georg Cantor, who set the boundary between countable and uncountable sets in 1874. The concept of ``computable set'' arose in the study of computing models in the 1930s by the founders of computer science, including \Godel, Church, and Turing. However, the connection between countability and computability has not been thoroughly studied in the past eight decades. A {\em counting bijection} of a set $S$ is a bijection from the ...

Our number system is a magnificent tool. But it is far from perfect. Can it be improved? In this paper some possibilities are discussed, including the use of a different base or directed (negative as well as positive) numerals. We also put forward some suggestions for further research.

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers.

Let $b>1$ be an odd positive integer and $k, l \in \mathbb{N}$. In this paper, we show that every positive rational number can be written as $\varphi(m^{2})/(\varphi(n^{2}))^{b}$ and $\varphi(k(m^{2}-1))/\varphi(ln^{2})$, where $m, n\in \mathbb{N}$ and $\varphi$ is the Euler's totient function. At the end, some further results are discussed.