{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:10:30Z","timestamp":1783901430055,"version":"3.55.0"},"reference-count":8,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2018,10,1]],"date-time":"2018-10-01T00:00:00Z","timestamp":1538352000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,10,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>Binary representation of integers [5], [3] and arithmetic operations on them have already been introduced in Mizar Mathematical Library [8, 7, 6, 4]. However, these articles formalize the notion of integers as mapped into a certain length tuple of boolean values.<\/jats:p>\n                  <jats:p>In this article we formalize, by means of Mizar system [2], [1], the binary representation of natural numbers which maps \u2115 into bitstreams.<\/jats:p>","DOI":"10.2478\/forma-2018-0020","type":"journal-article","created":{"date-parts":[[2019,2,24]],"date-time":"2019-02-24T04:31:16Z","timestamp":1550982676000},"page":"223-229","source":"Crossref","is-referenced-by-count":0,"title":["Binary Representation of Natural Numbers"],"prefix":"10.2478","volume":"26","author":[{"given":"Hiroyuki","family":"Okazaki","sequence":"first","affiliation":[{"name":"Shinshu University , Nagano , Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2019,2,23]]},"reference":[{"key":"2026071215162265034_j_forma-2018-0020_ref_001_w2aab3b7b2b1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261\u2013279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007\/978-3-319-20615-8_17.10.1007\/978-3-319-20615-8_17","DOI":"10.1007\/978-3-319-20615-8_17"},{"key":"2026071215162265034_j_forma-2018-0020_ref_002_w2aab3b7b2b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026071215162265034_j_forma-2018-0020_ref_003_w2aab3b7b2b1b6b1ab1ab3Aa","unstructured":"[3] Donald E. Knuth. The Art of Computer Programming, Volume 1: Fundamental Algorithms, Third Edition. Addison-Wesley, 1997."},{"key":"2026071215162265034_j_forma-2018-0020_ref_004_w2aab3b7b2b1b6b1ab1ab4Aa","unstructured":"[4] Hisayoshi Kunimune and Yatsuka Nakamura. A representation of integers by binary arithmetics and addition of integers. Formalized Mathematics, 11(2):175\u2013178, 2003."},{"key":"2026071215162265034_j_forma-2018-0020_ref_005_w2aab3b7b2b1b6b1ab1ab5Aa","unstructured":"[5] Gottfried Wilhelm Leibniz. Explication de l\u2019Arithm\u00e9tique Binaire, volume 7. C. Gerhardt, Die Mathematische Schriften edition, 223 pages, 1879."},{"key":"2026071215162265034_j_forma-2018-0020_ref_006_w2aab3b7b2b1b6b1ab1ab6Aa","unstructured":"[6] Robert Milewski. Binary arithmetics. Binary sequences. Formalized Mathematics,7(1): 23\u201326, 1998."},{"key":"2026071215162265034_j_forma-2018-0020_ref_007_w2aab3b7b2b1b6b1ab1ab7Aa","unstructured":"[7] Yasuho Mizuhara and Takaya Nishiyama. Binary arithmetics, addition and subtraction of integers. Formalized Mathematics,5(1):27\u201329, 1996."},{"key":"2026071215162265034_j_forma-2018-0020_ref_008_w2aab3b7b2b1b6b1ab1ab8Aa","unstructured":"[8] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics,4 (1):83\u201386, 1993."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/26\/3\/article-p223.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2018-0020","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,12]],"date-time":"2026-07-12T23:54:57Z","timestamp":1783900497000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2018-0020"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,1]]},"references-count":8,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,2,23]]},"published-print":{"date-parts":[[2018,10,1]]}},"alternative-id":["10.2478\/forma-2018-0020"],"URL":"https:\/\/doi.org\/10.2478\/forma-2018-0020","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2018,10,1]]}}}