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This monograph is intended as an exposition of some central results on irrational numbers, and is not aimed at providing an exhaustive treatment of the problems with which it deals. The term “irrational numbers,” a usage inherited from ancient Greece which is not too felicitous in view of the everyday meaning of the word “irrational,” is employed in the title in a generic sense to include such related categories as transcendental and normal numbers.
The entire subject of irrational numbers cannot of course be encompassed in a single volume. In the selection of material the main emphasis has been on those aspects of irrational numbers commonly associated with number theory and Diophantine approximations. The topological facets of the subject are not included, although the introductory part of Chapter I has a sketch of some of the simplest set-theoretic properties of the irrationals as a part of the continuum. The axiomatic basis for irrational numbers, proceeding say from the Peano postulates for the natural numbers to the construction of the real numbers, is purposely omitted, because in the first place the aim is not in the direction of the foundations of mathematics, and in the second place there are excellent treatments of this topic readily available.
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