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The Concept of Number: From Quaternions to Monads and Topological Fields (Mathematics and Its Applications)

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This book consists of lectures that I have given, in Darmstadt since 1974, under various titles, for beginners or students of the third semester.

I have tried to concentrate on those themes concerning numbers, which seem to me to be the most important on essential and historical grounds, and which in my opinion ought to belong to any wide-ranging, general education of every mathematics student. The content of that, so it occurred to me, should be to use the study of the number systems to reveal Mathematics as a visible unity, an aspect that is just what is lost from the vision of third-semester students. In order to open up the subject of interest from all sides, tools of knowledge are used from Analysis, Algebra, Geometry and Topology. Therefore also I have preferred geometrical approaches and renounced generality, if thereby the underlying ideas of a proof became more intuitive (e.g. in Liouville's Theorem on transcendental numbers). After the essential unity of Mathematics, I have emphasised its historical continuity. In that respect, numerous indicators enrich the text, from simple citations of names and dates, to explicit quotations (Hamel, Hamilton, etc.). So also, in order to treat the conjectured earliest Greek concept of real number, as worked out by 0. Becker [4], I have referred without proofs to some facts about Continued Fractions; and I have given a further development in Sections 2E and 3C. I confess gladly, that the working out of this theme, as well as of others, in this text has given me great pleasure.
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