Customer Reviews

Numbers: Rational and Irrational (New Mathematical Library)

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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.

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.

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.

The other reviews give a great idea of the contents of the book. I'll just say that the quality of the exposition is excellent. I'm experienced at reading maths books, but that doesn't mean I like them dense and difficult. This book explains everything very well, leaving you convinced and appreciative of its quality prose.

As a maths teacher, it gives me ideas for how certain ideas can be presented. There are a few parts I find difficult, which helps me to understand my students' point of view when I ask them to digest something difficult.

Much of the material in here I already "knew", but hadn't seen proved or even rigorously treated. This 45 year old book was an excellent way to fill in the gaps in my knowledge.

I bought this item second hand, but am now browsing Amazon hoping to pick up other titles in the series.

I am not a professional mathematician. Nonetheless, I enjoyed and understood most of this book. I learned some new facts, and appreciated the author's gentle writing. (For instance, I learned what a "transcendental irrational number" is.) I think an interested high school student could read this book and gain a lot from it.

My only wish is that the author would have published the solutions to problems in an Appendix. To a professional mathematician or a graduate student, I am sure the problems and proofs would be easy, but I would have gained a lot from being able to see how the problems could be solved.

So, overall, I really enjoyed this and hope that I can find other books like this in math.

There doesn't seem to be a more elementary notion in all of mathematics than that of a number. It could be argued that maybe the set theory is more fundamental, both conceptually and in terms of how we start learning about mathematical objects. However, when we think of mathematics as an independent field of study, it is the idea of numbers that first comes to mind. Learning how to add, subtract, multiply and divide are the first operations that we learn. These operations also serve the purpose of classifying numbers into distinct categories: subtraction only makes sense if we introduce negative numbers, and trying to make sense of division requires the introduction of rational numbers. It is this last category of numbers that this book uses as the stepping stone for the exploration of even more abstract categories, such as algebraic and transcendental numbers.

The book is written in a clear and accessible style. There are no formal prerequisites for understanding of the material in this book beyond the high school algebra. However, the reader should be able to appreciate more abstract proofs and feel comfortable with formal mathematical reasoning. This is particularly true of the latter chapters where some of the proofs that go well beyond the content of this book are only sketched and given in the most general outlines.

Each chapter contains many interesting and educational problems which range from the most straightforward to the more demanding. As is the case with most mathematical knowledge, it is precisely through working these problems out that the reader will master this material. The end of the book contains solutions and hints to select problems, but these are for the most part very brief at best and still require most of the work to be done by the reader. Overall, however, this is an excellent and very educational short mathematical textbook.

This is a good book, very interesting. It has a good way
of explaining mathematical facts and logical reasoning. It
was very easy to read.

Anyone with an interest in math will profit immensely from this gentle yet rigourous introduction to mathematical thinking. The branch of math Ivan Niven was fascinated with the most, number theory, serves as a treasure chest of gems that the reader will comprehend and tinker with. The topics selection is great, as are the exercises that accompany each section. This is not a book on the history of numbers, and no anecdote is found in their pages, instead it is a book on what math is really about: thinking and proof. My recommendation: try and answer every exercise in the book, and if stuck, look at the answers and think them through.

I highly recommend this book for high school students who have any interest in math, it will surely teach and challenge you, as well as pique your curiousity to read more advanced material.

This book was not very helpful. Not detailed enough. Can not be used alone to teach someone the subjects that it contains.