Related papers: Countability of the Real Numbers
Let $b \geq 3$ be a positive integer. A natural number is said to be a base-$b$ Zuckerman number if it is divisible by the product of its base-$b$ digits. Let $\mathcal{Z}_b(x)$ be the set of base-$b$ Zuckerman numbers that do not exceed…
We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…
A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by…
We introduce a first-order theory of finite full binary trees and then identify decidable and undecidable fragments of this theory. We show that the analogue of Hilbert`s 10th Problem is undecidable by constructing a many-to-one reduction…
How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This…
We consider compositions of natural numbers when there are different types of each natural number. Several recursions as well as some closed formulas for the number of compositions is derived. We also find its relationships with some known…
We present a short proof of Cantor's Theorem (circa 1870s): if $a_n \cos nx + b_n \sin nx \to 0$ for each $x$ in some (nonempty) open interval, where $a_n, b_n$ are sequences of complex numbers, then $a_n$ and $b_n$ converge to 0.
In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
Four constructions result from a desire to create enhancements to Cantor's infinite real set cardinality. Each continues to keep Cantor's cardinality formulation in place while providing new comparisons of arbitrary infinite sets. To…
We describe an inequality of finite or infinite sequences of real numbers and their quotients. More precisely, we compare the quotient of H\"older functionals of two sequences of numbers with the sum of their quotients. In the last section…
Probably we have observed a new simple phenomena dealing with approximations to two real numbers.
This paper presents a new representation of natural numbers and discusses its consequences for computability and computational complexity. The paper argues that the introduction of the first Peano axiom in the traditional definition of…
We present the basic theory of calculus on dual real numbers, and prove the counterpart of the ordinary fundamental theorem of calculus in the context of dual real numbers.
We formally prove the existence of an enduring incongruence pervading a widespread interpretation of the Bell inequality and explain how to rationally avoid it with a natural assumption justified by explicit reference to a mathematical…
All the already known results on self descriptive numbers, together with the demonstration of the uniqueness for bases greater than 6, are here obtained through a systematic scheme of proof and not trial and error. The proof is also…
It is proved that given any three conditionally convergent series of real numbers, there is a single sequence of natural numbers such that each of the corresponding three subseries sums to either $\infty$ or $-\infty$. An example is…
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
Natural numbers satisfying an unusual property are mentioned by the author in [5], in which their infinitude is also proved. In this paper, we start with an arbitrary natural number which is not a multiple of 10 and non-palindromic, form…
In connection with recent work on gaps in the asymptotic subranks of complex tensors the question arose whether the number of nonnegative real numbers that arise as the asymptotic subrank of some complex tensor is countable. In this short…
The present article is devoted to representations of rational numbers in terms sign-variable Cantor expansions. The main attention is given to one of the discussions given by J. Galambos in [4].