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12 of 12 people found the following review helpful:
4.0 out of 5 stars
An interesting study of the properties of numbers., July 5, 2004
This review is from: Numbers: Rational and Irrational (New Mathematical Library) (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. His text is part of a series (Anneli Lax New Mathematical Library Series) of books published by the Mathematical Association of America intended to be accessible to high school students that explore advanced topics not addressed by the high school curriculum. Accordingly, 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 (Undergraduate Texts in Mathematics) 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 reasonable.
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9 of 9 people found the following review helpful:
4.0 out of 5 stars
The history of how new numbers were needed and then defined, October 5, 2002
This review is from: Numbers: Rational and Irrational (New Mathematical Library) (Paperback)
The history of how the concept of number has evolved over the centuries is an amazing example of necessity being the mother of invention. Each new advancement was in response to a need for new numbers to solve or extend the valid solutions of mathematical equations. Sometimes, as in the circumstances for negative integers, there was an immediate practical necessity. In other cases, the need was a theoretical one so that equations could be solved. What makes the evolution of numbers most interesting is that in each case, the additions defined a superset, and all of the operation rules used on the previous set still applied. This is a fact that is much under-appreciated, even among mathematicians. In this book, Niven steps through the development of the expanding supersets of numbers, starting at the positive integers and ending with the complex. His explanations of the "new" sets of numbers are clear, and one can see the logical consistency that is interwoven into the definitions of all types of numbers. Problem sets are given at the end of the sections with solutions to many of the problems placed in an appendix. As an undergraduate, I worked through all of the required problems in my courses, but never really appreciated how clean and unencumbered the consistencies between sets of numbers are. It took many years of teaching before I really understood the intrinsic beauty of how all the numbers are defined. Had I read this book, I would have achieved this level of joy in the time it took me to read it. Which was about two and a half hours.
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7 of 7 people found the following review helpful:
5.0 out of 5 stars
A Compact History of Numbers, October 12, 2002
This review is from: Numbers: Rational and Irrational (New Mathematical Library) (Paperback)
In this concise [only 140 pages] presentation of the number system, Professor Niven parallels its historical development from ancient origins in counting to twentieth-century theorems on transcendental numbers, always with clear explanations, and without belaboring the history. A glance at the book's Table of Contents serves to illustrate: Ch.1 - Natural Numbers and Integers
Ch.2 - Rational Numbers
Ch.3 - Real Numbers
Ch.4 - Irrational Numbers
Ch.5 - Trigonometric and Logarithmic Numbers Ch.6 - The Approximation of Irrationals by Rationals
Ch.7 - The Existence of Transcendental Numbers
Ap.A - Proof That There Are Infinitely Many Prime Numbers
Ap.B - Proof of the Fundamental Theorem of Arithmetic
Ap.C - Cantor's Proof of the Existence of Transcendentals
Ap.D - Trigonometric Numbers Following the four appendices is the chapter, "Answers and Suggestions to Selected Problems", addressing the book's problem sets; and a very useful Index. Proofs are very clear, thorough, and understandable; the proofs and explanations gradually increase in complexity from the beginning chapters to the appendices, as the cover notes state: > Most readers will find the early chapters well within
> their grasp, while ambitious readers will profit by
> the more advanced material to be found in later chapters.
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