SYSTEMATIC CONSTRUCTION OF NUMBERS

These are my personal notes on Number Systems. I consider Natural Numbers, Integers, and Rational Numbers only in this article. Real Numbers have not been discussed. It requires a thorough knowledge of basic Set Theory to completely understand it.

Theoretical Computer Science, 2002

We are proving in this note a new criterion for the pair {P (x), N } to be a canonical number system. This enables us to prove that if p 2 , . . . ,

The ultimate numbers and the 3/2 ratio , 2020

According to a new mathematical definition, whole numbers are divided into two sets, one of which is the merger of the sequence of prime numbers and numbers zero and one. Three other definitions, deduced from this first, subdivide the set of whole numbers into four classes of numbers with own and unique arithmetic properties. The geometric distribution of these different types of whole numbers, in various closed matrices, is organized into exact value ratios to 3/2 or 1/1. Selon une nouvelle définition mathématique, les nombres entiers naturels se divisent en deux ensembles dont l'un est la fusion de la suite des nombres premiers et des nombres zéro et un. Trois autres définitions, déduites de cette première, subdivisent l'ensemble des nombres entiers naturels en quatre classes de nombres aux propriétés arithmétiques propres et uniques. La distribution géométrique de ces différents types d'entiers naturels, dans de diverses matrices fermées, s'organise en ratios exacts de valeur 3/2 ou 1/1.

A New Theory of Numbers, 2020

“How can all of this be true all at the same time?” This will be the question you will be asking yourself once you discover the amazing inner world hiding behind numbers, as they reveal palindromes, two types of dual characteristics, visible and invisible patterns, perfect plus/minus as well as odd/even balances, and much more. Learn that prime numbers can be organized in a perfect 24-based, yet decimally-based system and aren’t randomly distributed. Discover a whole new way to see numbers as one unified, and I dare say, Intelligent and Logical system. An entire new Number Theory is hereby introduced as well. Find out further aspects hiding behind the Fibonacci numbers, and similarly found ratios point to the square roots of whole numbers. A bonus chapter reveals the number present in our solar system.

2015

In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field No of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of No, i.e. a subfield of No that is an initial subtree of No. In this sequel to [15], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of No that are themselves initial. It is further shown that an initial subdomain of No is discrete if and only if it is an initial subdomain of No's canonical integer part Oz of omnifc integers. Finally, extending results of [15], the theories of divisible ordered abelian groups and real-closed ordered fields are shown to be the sole theories of ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgro...

Cognitive, Affective, & Behavioral Neuroscience, 2014

Number systems-such as the natural numbers, integers, rationals, reals, or complex numbers-play a foundational role in mathematics, but these systems can present difficulties for students. In the studies reported here, we probed the boundaries of people's concept of a number system by asking them whether "number lines" of varying shapes qualify as possible number systems. In Experiment 1, participants rated each of a set of number lines as a possible number system, where the number lines differed in their structures (a single straight line, a step-shaped line, a double line, or two branching structures) and in their boundedness (unbounded, bounded below, bounded above, bounded above and below, or circular). Participants also rated each of a group of mathematical properties (e.g., associativity) for its importance to number systems. Relational properties, such as associativity, predicted whether participants believed that particular forms were number systems, as did the forms' ability to support arithmetic operations, such as addition. In Experiment 2, we asked participants to produce properties that were important for number systems. Relational, operation, and use-based properties from this set again predicted ratings of whether the number lines were possible number systems. In Experiment 3, we found similar results when the number lines indicated the positions of the individual numbers. The results suggest that people believe that number systems should be well-behaved with respect to basic arithmetic operations, and that they reject systems for which these operations produce ambiguous answers. People care much less about whether the systems have particular numbers (e.g., 0) or sets of numbers (e.g., the positives).

Journal of Symbolic Logic, 2001

Introduction. In his monograph On Numbers and Games [7], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including ω, ω, /2, 1/ω, and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered “number” fields—be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with Global Choice, henceforth NBG [cf. 21, Ch. 4], it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields. This can be made precise by saying that whereas the ordered field of reals is (up to isomorphism) the unique homogeneous universal Archimedean ordered field, No is (up to isomorphism) the unique homogeneous universal orderedfield [14]; also see [10]...

2017

In the course of many mathematical developments involving 'number systems' like $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb {C}$ etc., it sometimes becomes necessary to abstract away and study certain properties of the number system in question so that we may better understand objects having these properties in a more general setting -- a relevant example for this paper would be Hausdorffs $\eta_\varepsilon$-fields, objects abstracted from the property that there are no two disjoint intervals in $\mathbb{R}$ of cardinality $\alpha \leq \aleph_0$ whose union is $\mathbb{R}$. My goal will be to reverse this process, so to speak -- I will begin in one of the most abstract mathematical settings possible, using only the undefined notions and axioms of MK class theory to define and subsequently add structure to the class of all ordinals, $O_n$. I proceed in this fashion until the construction (or a subclass thereof) is structurally isomorphic to whichever 'number ...

FigShare, 2024

This work explores the idea of expressing numbers as polynomials in their base. It uses this idea to establish the concept of the cardinality of a number, and uses this to establish several important, practical theorems about the cardinality of numbers in any base. The four basic arithmetic operations of addition, subtraction, multiplication and division are given special treatment, and the limits on the cardinality of the result of applying any of them to two numbers — pure or pure fractional, succinctly established.

Lecture Notes in Computer Science, 2010

We introduce a new number system that supports increments with a constant number of digit changes. We also give a simple method that extends any number system supporting increments to support decrements using the same number of digit changes. In the new number system the weight of the ith digit is 2 i −1, and hence we can implement a priority queue as a forest of heap-ordered complete binary trees. The resulting data structure guarantees O(1) worst-case cost per insert and O(lg n) worst-case cost per delete, where n is the number of elements stored.

Α Generalized Numeration Base is defined in this paper, and then particular cases are presented, such as Prime Base, Square Base, m-Power Base, Factorial Base, and operations in these bases.

IOSR Journals , 2019

Recurrence sequence of bases can be used to construct number systems representations where algorithm of addition can be easily derived from definition of sequence.

British Journal of Mathematics & Computer Science, 2015

The main purpose of the present article is to give a brief description of the "golden" number theory and new properties of natural numbers following from it, in particular, Z-property, D-property, Φ-code, Fcode, L-code. These properties are of big theoretical interest for number theory and can be used in computer science. The article is written in popular form and is intended for a wide circle of mathematicians (including mathematics students) and specialists in computer science, who are interested in the histories of mathematics and new ideas in the development of number theory and its applications in computer science.