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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. |
