On rational decimal numbers for lower secondary mathematics teachers

Haukkanen, Pentti and Tossavainen, Timo (2025) On rational decimal numbers for lower secondary mathematics teachers. Annales Mathematicae et Informaticae, 62. pp. 209-222. ISSN 1787-5021 (Print) 1787-6117 (Online)

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Abstract

For the purposes of mathematics teacher education, we provide an algebraic and number theoretic explanation of how one can determine, from a and b, whether the decimal expansion of a/b, where gcd(a,b) = 1, contains a pre-period, and if so, how long it is. To that end, we observe that explanation cannot be based solely on group theory because the residue classes of remainders do not form a subgroup of the multiplicative group modulo b. However, the collection of residue classes does form a submonoid of the multiplicative monoid modulo b.

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