Numbers: Rational and Irrational.

Ivan Niven.

Published by Random House, New York, NY, 1961
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Bibliographic Details

Title: Numbers: Rational and Irrational.

Publisher: Random House, New York, NY

Publication Date: 1961

Binding: Paperback

Edition: 1st Edition


First Edition Thus [1961]; First Printing, so stated. Very Good in Wraps: shows indications of careful use: slight spine lean; light wear to extremities, with several faint creases near corners; mild rubbing and faint soiling to wrapper covers; the heel of the backstrip has been reinforced with cellophane tape; there is a small child's crayon mark at the front endpaper; the binding is slightly cocked, but remains secure; the text is clean. No longer 'As New', but remains clean, sturdy, and quite presentable. NOT a Remainder, Book-Club, or Ex-Library. 8vo. 136pp. New Mathematical Library. Trade Paperback. Ivan Niven's lucidly written text discusses the properties of the natural numbers, integers, rational numbers, irrational numbers, real numbers, algebraic numbers, and transcendental numbers. He defines the complex numbers but does not delve into their properties. The text is not an axiomatic development of the real numbers. For that, the reader can consult Edmund Landau's text Foundations of Analysis. Niven assumes the existence of the numbers and explores their properties. He also addresses methods of proof. Before Niven proves a result, he discusses how he will prove the result or proves a special case of the result in order to help the reader understand the proof. He also illustrates his results with an abundance of examples. The material on natural numbers, integers, and rational numbers in the early chapters will be familiar to most readers. In the chapter on real numbers, he proves the existence of irrational numbers. He then explores the properties of irrational numbers and contrasts them with those of the rational numbers. He introduces algebraic and transcendental numbers in a chapter that discusses why certain trigonometric and logarithmic numbers are irrational. In this chapter, Niven appeals to results that he does not prove in order to explain why three famous geometric construction problems from antiquity that are supposed to be solved using only an unmarked straightedge and compass cannot be solved. The final chapters on approximating irrational numbers by rational numbers and the existence of transcendental numbers make extensive use of inequalities. The inexperienced reader may wish to consult the text an Introduction to Inequalities (New Mathematical Library) by Edwin Beckenbach and Richard Bellman before studying the final chapters of Niven's text. Otherwise, these chapters could pose considerable difficulties. The appendices are well worth reading. In the first appendix, Niven proves there are infinitely many prime numbers; in the second, he proves the Fundamental Theorem of Arithmetic. The third appendix provides an alternate proof of the existence of transcendental numbers to the one given in the last chapter of the text. The proof in the appendix relies heavily on set theory, so the reader unfamiliar with set theory may wish to consult the text "Naive Set Theory" by Paul Halmos before tackling it. The final appendix on the irrationality of certain trigonometric numbers, which is a modification of an appendix added to the Russian translation of the book by I. M. Yaglom, provides an alternate approach to that given in the chapter on trigonometric and logarithmic numbers. The exercises, for which solutions or hints are given at the end of the book, are grounded in Niven's exposition. The reader who has striven to understand his arguments and who has carefully checked their details should find the exercises reasonably accessible. Ivan Morton Niven (19151999) was a Canadian-American mathematician, specializing in number theory. He was born in Vancouver. He did his undergraduate studies at the University of British Columbia and was awarded his doctorate in 1938 from the University of Chicago. He was a member of the University of Oregon faculty from 1947 to his retirement in 1981. He received the University of Oregon's Charles E. Johnson Award in 1981. Niven completed the solution of most of Waring's problem. Bookseller Inventory # 44514

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