Article
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The Archimedean Origin of Modern Positional Number Systems
Version 1
: Received: 23 November 2023 / Approved: 8 December 2023 / Online: 8 December 2023 (10:06:12 CET)
Version 2 : Received: 8 December 2023 / Approved: 12 December 2023 / Online: 12 December 2023 (10:55:21 CET)
Version 2 : Received: 8 December 2023 / Approved: 12 December 2023 / Online: 12 December 2023 (10:55:21 CET)
A peer-reviewed article of this Preprint also exists.
Manca, V. The Archimedean Origin of Modern Positional Number Systems. Algorithms 2024, 17, 11. Manca, V. The Archimedean Origin of Modern Positional Number Systems. Algorithms 2024, 17, 11.
Abstract
A symbolic analysis of Archimedes’ periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci’s Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented,, by remarking the cruciality of this topic in number theory. Periods of ordinal numbers, and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship among numbers and information, which shed new light on a red line passing through the whole history of mathematics
Keywords
Number Representation Systems; Zeroless positional systems; Arithmetic Algorithms; Ordinals; Recursive enumerability
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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