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Abstract
Continued fractions arise naturally in long division and the theory of the approximation to real numbers by rational numbers. This research considered the implementation on the convergent of θ-expansions of real numbers of x∈(0,θ) with 0<θ<1. The convergent of θ-expansions are also called as θ-convergent of continued fraction expansions. This study aimed to establish the properties for a family of θ-convergent in θ-expansions. The idea of discovering the behaviours of θ-convergent evolved from the concept of regular continued fraction (RCF) expansion and sequences involved in θ-expansions. The θ-expansions algorithm was used to compute the values of θ-convergent with the help of Maple software. Consequently, it proved to be an efficient method for fast computer implementation. The growth rate of θ-convergent was investigated to highlight the performance of θ-convergent. The analysis on θ-convergent revealed the convergent that gives a better approximation yielding to fewer convergence errors. This whole paper thoroughly derived the behaviours of θ-convergent, which measure how well a number x is approximated by its convergents for almost all irrational numbers.
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Additional Metadata
Item Type: | Article |
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Divisions: | Faculty of Science Centre for Foundation Studies in Science of Universiti Putra Malaysia |
DOI Number: | https://doi.org/10.23939/mmc2024.04.1141 |
Publisher: | Lviv Polytechnic National University |
Keywords: | Θ-convergent; Θ-expansions; Θ-expansions algorithm; Continued fraction; Con- vergence errors |
Depositing User: | Ms. Zaimah Saiful Yazan |
Date Deposited: | 26 Jun 2025 04:36 |
Last Modified: | 26 Jun 2025 04:36 |
Altmetrics: | http://www.altmetric.com/details.php?domain=psasir.upm.edu.my&doi=10.23939/mmc2024.04.1141 |
URI: | http://psasir.upm.edu.my/id/eprint/118163 |
Statistic Details: | View Download Statistic |
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