CM Method and Expansion of Numbers

TL;DR

This paper introduces a simple iterative method based on the center of mass concept for expanding numbers in powers of a fraction, extending to real and complex numbers, and generalizing Jacobsthal numbers and other sequences.

Contribution

It presents a novel iterative approach for number expansion and sequence generation, unifying various well-known sequences within a new framework.

Findings

Method effectively expands rational numbers in powers of r/s

Extension to real and complex numbers demonstrated

Generates sequences like Jacobsthal, Fibonacci, and Pell

Abstract

We show that an iterative method for computing the center of mass (CM) of q units of mass, placed on a unit interval [0, 1] along the x- axis, give rise to a simple procedure for expanding rational numbers less than unity in powers of r/s < 1, with r, s, integers larger than 0. The method is then extended to all numbers, real or complex, though the procedure for none rational numbers is more time consuming. We also show how our method provides a natural way to generalize Jacobsthal numbers. Moreover, the method provides a way to generate infinitely many sequences of numbers, of which many play an important rule in mathematical sciences and engineering, to name few, Jacobsthal sequence, Fibonacci sequence, and Pell sequence.