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Infinitesimals in the real number system4/29/2023 ![]() ![]() Enacting the instructional sequence provides lenses for mathematics teacher educators to notice and eliminate difficulties of their students while developing relationships among multiple representations of real numbers. This method provides infinitesimals as mathematical objects. Keywords: Eudoxus, hyperreals, infinitesimals, limit ultrapower, universal hyperreal. This further prompted their reasoning with decimal representations of rational and irrational numbers as rational number sequences, which leads to authentic construction of real numbers. discuss two logic techniques of constructing nonstandard models of real number system. A construction of the real number system based on almost homo. Although we will not carry out the development of the real number system from these basic properties, it is useful to state them as a starting point for the study of real analysis and also to focus on one property, completeness, that is probably new to you. ![]() In particular, results showed that prospective teachers reasoned about fractions and decimal representations of rational numbers using long division, the division algorithm, and diagrams. Having taken calculus, you know a lot about the real number system however, you probably do not know that all its properties follow from a few basic ones. Using the set of real numbers as a limit space will not do, we would then have to give each real number an infinitesimal weight. We found that the instructional sequence enhanced prospective teachers’ understanding of real numbers by considering them as quantities and explaining them by using rational number sequences. In this article, we propose an instructional sequence that addresses quantitative relationships for the construction of real numbers as rational number sequences. The understandings prospective mathematics teachers develop by focusing on quantities and quantitative relationships within real numbers have the potential for enhancing their future students’ understanding of real numbers. ![]()
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