Understanding Number System

2021

Cover image: Postage stamp commemorating 150th birth anniversary of Richard Dedekind, whose ideas are fundamental to much of the material in this book.

In today's information age, computers are being used in every walk of life. They are being used by people of all age and profession in their work and in their leisure. This new social revolution has changed the basic concept of 'Computing'. Computing in today's information age is no more limited to computer programmers and computer engineers. It has become an activity of a common man. Rather than knowing how to program a computer, most computer users simply need to understand how a computer functions and what all it can do. Even those who need to program a computer can do their job more effectively with a better understanding of how computers function and the capabilities and limitations of computers. As a result, almost all academic institutions have started offering regular courses on foundations of computing at all levels. These courses deal with the fundamental concepts of the organization, functions, and usage of modern computer systems. Hence we realized that a good textbook that can cover these concepts in an orderly manner would certainly be very useful for a very wide category of students and all types of computer users.

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).

ijens.org

Any system that is used for naming or representing numbers is a number system, also known as numeral system. The modern civilization is familiar with decimal number system using ten digits. However digital devices and computers use binary number ...

HAL (Le Centre pour la Communication Scientifique Directe), 2015

Cognition, 1995

In this short paper, the author presents a new interpretation for the theory of number system and its scale of notation. The spectra of this new theory consist up of some definitions, lemmas, models and theorems. A generalized dynamic model is constructed, which is a link between Arithmetic and Algebra.

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.

Journal of Universal Language

This article examines seven of the 54 generalizations on numeral systems presented by Greenberg in 1978 either as absolute universals or as universal tendencies. Five of the generalizations deal with details of the arithmetic structure of numeral systems, with particular reference to the order of elements in the numeral expression and to the use of subtraction and related phenomena in numeral expressions. It is shown that the details of these generalizations need to be revised * Portions of the material included in this article have been presented at various fora. I am grateful to all those who contributed to the ensuing discussions. This article is dedicated to the memory of the late Joseph H. Greenberg, the giant on whose shoulders I stand.