### The rational numbers as an abstract data typeThe rational numbers as an abstract data type

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 Author Bergstra, J. A. ♦ Tucker, J. V. Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2007 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Rational numbers ♦ Abstract data types ♦ Algebraic specification ♦ Computable algebras ♦ Division-by-zero ♦ Equations ♦ Field ♦ Initial algebra ♦ Meadow ♦ Total versus partial functions Abstract We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that $0^{™1}$ = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system? ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2007-04-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 54 Issue Number 2

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Source: ACM Digital Library