{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T03:17:23Z","timestamp":1773717443850,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T00:00:00Z","timestamp":1735776000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Classic formulae for entropy and cross-entropy contain operations x0 and log2x that are not defined on all inputs. This can lead to calculations with problematic subexpressions such as 0log20 and uncertainties in large scale calculations; partiality also introduces complications in logical analysis. Instead of adding conventions or splitting formulae into cases, we create a new algebra of real numbers with two symbols \u00b1\u221e for signed infinite values and a symbol named \u22a5 for the undefined. In this resulting arithmetic, entropy, cross-entropy, Kullback\u2013Leibler divergence, and Shannon divergence can be expressed without concerning any further conventions. The algebra may form a basis for probability theory more generally.<\/jats:p>","DOI":"10.3390\/e27010031","type":"journal-article","created":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T07:44:53Z","timestamp":1735803893000},"page":"31","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Their Fracterm Calculus"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2492-506X","authenticated-orcid":false,"given":"Jan A.","family":"Bergstra","sequence":"first","affiliation":[{"name":"Informatics Institute, University of Amsterdam, Science Park 900, 1098 XH Amsterdam, The Netherlands"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4689-8760","authenticated-orcid":false,"given":"John V.","family":"Tucker","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Bay Campus, Fabian Way, Swansea University, Swansea SA1 8EN, UK"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2005). Elements of Information Theory, Wiley.","DOI":"10.1002\/047174882X"},{"key":"ref_2","unstructured":"Ehrich, H.-D., Wolf, M., and Loeckx, J. (1997). Specification of Abstract Data Types, Wiley."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Bergstra, J.A. (2020). Arithmetical datatypes, fracterms, and the fraction definition problem. Transmathematica.","DOI":"10.36285\/tm.33"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Anderson, J.A., and Bergstra, J.A. (2021). Review of Suppes 1957 proposals for division by zero. Transmathematica.","DOI":"10.36285\/tm.53"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1145\/1219092.1219095","article-title":"The rational numbers as an abstract data type","volume":"54","author":"Bergstra","year":"2007","journal-title":"J. ACM"},{"key":"ref_6","first-page":"70","article-title":"Relations of zero and \u221e","volume":"1","author":"Okumura","year":"2017","journal-title":"J. Technol. Soc. 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Transreal Foundation for Floating-Point Arithmetic. Transmathematica.","DOI":"10.36285\/tm.91"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Bergstra, J.A., and Tucker, J.V. (2020). The transrational numbers as an abstract data type. Transmathematica.","DOI":"10.36285\/tm.47"},{"key":"ref_16","first-page":"11","article-title":"Construction of the transreal numbers and algebraic transfields","volume":"46","author":"Gomide","year":"2016","journal-title":"IAENG Int. J. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Dos Reis, T.S. (2019). Transreal integral. Transmathematica.","DOI":"10.36285\/tm.v0i0.13"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Jansen, N., Stoelinga, M., and van den Bos, P. (2022). Symmetric transrationals: The data type and the algorithmic degree of its equational theory. 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