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Numbers: Rational and Irrational, by Ivan Niven
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Series: Anneli Lax New Mathematical Library
A superb development that starts with the natural numbers and carries the reader through the rationals and their decimal representations to algebraic numbers and then to the real numbers. Along the way, you will see characterizations of the rationals and of certain special (Liouville) transcendental numbers. This material is basic to all of algebra and analysis.
This book deals with the number system, one of the basic structures in mathematics. It is concerned especially with ways of classifying numbers into various categories; for example, it provides some criteria for deciding if a given number is rational (i.e., representable as a common fraction) or irrational, if it is algebraic or transcendental. In the course of the later chapters, the reader is introduced to some of the more recent developments in mathematics. Professor Niven’s book may be read with profit by interested high school students as well as by college students and others who want to know more about the basic aspects of pure mathematics. 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.
- Sales Rank: #1130199 in Books
- Brand: Brand: Mathematical Assn of America
- Published on: 1961-01
- Original language: English
- Number of items: 1
- Dimensions: .30" h x 5.54" w x 8.50" l, .40 pounds
- Binding: Paperback
- 140 pages
- Used Book in Good Condition
Review
A fine job of presenting the materials at a level which is attainable for the intended reading public. -- The Mathematics Teacher
About the Author
Ivan Niven attended the University of British Columbia and the University of Chicago. He taught at the University of Illinois and Purdue and has lectured for two summers at Stanford.
Most helpful customer reviews
0 of 0 people found the following review helpful.
Great source for proof skills.
By LearningIsAwesome
The reason for my 4 stars instead of 5 is that it has to do more with personal choice than with the quality of the content. For my taste, I rather have more problems to work with than the very few provided here. I would like to give, however, two main reasons as to why I think this is a good book.
1. The content.
It is a good introduction to the properties of numbers. I'm yet to start the last chapter and so far I have learn a great deal on the approximation of irrationals using rational numbers. Also, as explained in chapter 4, its interesting to see how the rational root test can be used in order to see if a given number is rational or not. I remember seeing a trick on my abstract algebra class to justify that cos 20 is not constructible. This very same idea applies to the techniques exhibited here on algebraic numbers.
2. The structure.
Although this book does not offer many problems to work with (in my opinion), it does really well in explaining the theorems. I believe that Niven's approach to explain proofs is excellent for a anyone that is new to proofs. There are also many tricks involved that are worth learning, especially, if you're going into education.
I hope this review helps.
16 of 16 people found the following review helpful.
An interesting study of the properties of numbers.
By N. F. Taussig
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.
8 of 8 people found the following review helpful.
A Compact History of Numbers
By LioNiNoiL
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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