{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:46:01Z","timestamp":1760150761129,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,27]],"date-time":"2023-12-27T00:00:00Z","timestamp":1703635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>A symbolic analysis of Archimedes\u2019s periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci\u2019s 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 on 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 between numbers and information, which shed new light on a red line passing through the whole history of mathematics.<\/jats:p>","DOI":"10.3390\/a17010011","type":"journal-article","created":{"date-parts":[[2023,12,27]],"date-time":"2023-12-27T05:43:15Z","timestamp":1703655795000},"page":"11","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Archimedean Origin of Modern Positional Number Systems"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1304-0277","authenticated-orcid":false,"given":"Vincenzo","family":"Manca","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Verona, 37134 Verona, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Dijksterhuis, E.J. (1987). Archimedes, Princeton University Press.","DOI":"10.1515\/9781400858613"},{"key":"ref_2","unstructured":"Heath, W. (2002). The Works of Archimedes, Dover Publication."},{"key":"ref_3","unstructured":"Manca, V. (2020). Lezioni Archimedee, Biblioteca Alagoniana. Quaderno N. 6."},{"key":"ref_4","unstructured":"Ifrah, G. (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer, Wiley."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1080\/00029890.2000.12005217","article-title":"Zeroless Positional Number Representation and String Ordering","volume":"107","author":"Boute","year":"2000","journal-title":"Am. Math. Mon."},{"key":"ref_6","unstructured":"Manca, V. (2015). On the Lexicographic Representation of Numbers, Cornell University Library. n. 1505.00458."},{"key":"ref_7","unstructured":"Knuth, D. (1997). The Art of Computer Programming, Addison-Wesley. 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Mathematical Thought from Ancient to Modern Times, Oxford University Press. 3 Volumes.","DOI":"10.1093\/oso\/9780195061352.003.0001"},{"key":"ref_13","unstructured":"Witmer, T.R. (2006). The Analytic Art, Dover Publication, Inc."},{"key":"ref_14","unstructured":"Descartes, R. (1954). The Geometry of Ren\u00e9 Descartes, Dover Publications."},{"key":"ref_15","unstructured":"Netz, R., and Noel, W. (2007). The Archimedes Codex, W&N."},{"key":"ref_16","unstructured":"Cantor, G. (1915). Contributions to the Founding of the Theory of Transfinite Numbers, Dover Publications."},{"key":"ref_17","first-page":"230","article-title":"On Computable Numbers, with an Application to the Entscheidungsproblem","volume":"42","author":"Turing","year":"1936","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_18","unstructured":"Dedekind, R. (1963). Essays on the Theory of Numbers, Dover Publications."},{"key":"ref_19","unstructured":"Frege, G. (1960). The Foundation of Arithmetic, Northwestern University Press."},{"key":"ref_20","unstructured":"Peano, G. (Italy 1960). Formulario Mathematico, Edizioni Cremonese."},{"key":"ref_21","unstructured":"Hilbert, D. (1924). David Hilbert\u2019s Lectures on the Foundations of Arithmetic and Logic 1917\u20131933, Springer."},{"key":"ref_22","unstructured":"Kaplan, R., and Kaplan, E. (2003). The Art of the Infinite, Oxford University Press."},{"key":"ref_23","unstructured":"Shannon, C. (1998). The Mathematical Theory of Communication, University of Illinois Press."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/1\/11\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:42:41Z","timestamp":1760132561000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/1\/11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,27]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["a17010011"],"URL":"https:\/\/doi.org\/10.3390\/a17010011","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2023,12,27]]}}}