Customer Reviews

Excursions in Number Theory (Dover books explaining science)

5 star:		(3)
4 star:		(3)
3 star:		(1)
2 star:		(1)
1 star:		(0)

 

 

Written for anyone with an interest in numbers, this booktalks about some of the "neat" things about numbers withoutrequiring more than high school mathematical knowledge. Topics covered include prime number, geometric numbers, repeated fractions, and other chestnuts in mathematical recreations. This book is written at precisely the right level to interest a wide range of readers. For college mathematics students, a higher level book, such as "Fundamentals of Number Theory" by Leveque would be recommended as a primer to the theory of numbers.

Excursions in Number Theory (Oxford, publ. 1966; Dover reprint 1988) is a brief pleasure trip across the realm of number theory. C. Stanley Ogilvy's and John T. Anderson's enjoyable text only requires that readers have familiarity with algebra and have a penchant for puzzles. For those interested in more mathematics twenty pages of explanatory notes are found in the appendix.

Using carefully selected examples, the authors present key topics with surprisingly clarity. Although congruences (arithmetic, not geometric), Diophantine equations, and continued fractions may be unfamiliar, the reader rather quickly appreciates the critical roles played by these concepts and tools. For example, congruences prove to be exceedingly helpful in solving a wide range a numeric problems and also reappear in later discussions on irrationals, iterations, and Diophantine equations.

The study of prime numbers is fundamental to number theory, but as yet we have no known formula to produce all primes. Even more disturbing, we have no procedures that are even guaranteed to produce only primes (i.e., not yield an unpredictable mix of primes and composite numbers). There is something fundamental about primes that we seem not to understand. The short chapter, Prime Numbers as Leftover Scrap, offers a fascinating perspective that I have not encountered elsewhere.

Other chapters are more playful, offering curios, puzzles, and oddities. Some examples appear to be little more than amusing numeric coincidences while other oddities prove to have theoretical significance. I am not an avid fan of mathematical puzzles, but I thoroughly enjoyed these diversionary chapters.

As a follow-up to Ogilvy and Anderson, I am now reading:

Number Theory and Its History by Oystein Ore (1948), available as a Dover reprint (1988), is now rather old, even pre-dating computer use in number theory research. The difficulty level is moderate. The historical background is interesting. (3 stars)

Elementary Theory of Numbers (1962) by William J. LeVeque offers detailed proofs underlying number theory and should appeal to readers that enjoy studying mathematics. Topics include congruences, powers of an integer modulo m, continued fractions, Gaussian integers, and Diophantine equations. The Dover reprint suffers from a small font size. (3.5 stars)

Yet another Dover reprint, Number Theory (1971) by George Andrews also targets more serious readers. Andrews uses an interesting combinatorial approach to number theory. Good font size and open page layouts. (4 stars)

Unlike other Dover books, this text does not require an extensive background in math and fluency in the language of proofs. It is, as the title suggests, a delightful excursion through number theory that will ignite your interest in the subject and move you to further study.

I found the author's annotations helpful and I did not mind the occasional use of British vernacular. At many points in the text, Ogilvy & Anderson prompt the reader to pursue a question on their own, rather than walk through a full proof or explanation. This may seem abrupt, but it keeps the text focused and leaves the reader wanting to know more about number theory.

I hope Dover continues to reach out to a general audience with books like this. It condenses a difficult subject into everyday language without condescending to the reader.

Number theory is the most fascinating area of mathematics; I have yet to meet a mathematician that does not enjoy it. Making it even more interesting is the fact that the fundamental principles can be understood by almost everyone. It is the only area of mathematics where many of the most difficult unsolved problems can be understood by people having only a high school mathematics background.
This book is an introduction to number theory that will appeal to everyone with an interest in numbers that understands high school algebra. It begins with a history of the development of numbers and then quite correctly proceeds to a description of some of the most prominent number patterns. Congruence arithmetic, Diophantine equations, prime and lucky numbers, continued fractions and Fibonacci numbers are the remaining primary topics. Secondary topics are some mathematical curiosities and brief histories of some calculating prodigies.
If you enjoy numbers and are interested in learning more about their properties then this is a book that you will enjoy. High school algebra is the only prerequisite to this journey through one of the most fascinating areas of human endeavor.

This is a reprint of one of the books that most inspired
my interest in mathematics as a boy. I highly recommend
it to any high school student interested in mathematics
or perhaps as a gift that might stimulate interest.
It is very short and very readable. I also recommend
the book "Excursions in Geometry" by Ogilvy.

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...but I could not tell how current this reprint edition of a 1966 book is.

This short volume is filled with curiosities about numbers and their theory, but the explanations are sometimes dense, and I was left wondering in places, Exactly why is this topic important to number theory?

Oh, well, good for a summer afternoon...

This is one of the Dover Math Series which is `flawed'. Some of this series are really nice while others are, as this one, abrupt, written [generally by Englishmen] using words not common to a U.S. vocabulary.

This book was originally published in 1966 by the Oxford Press. Brit maths notation is just `off' enough if, as me, you are not `into' it proofs are spoilt until the uncommon words are deciphered.

Maybe you recall in Alice in Wonderland the poem `... little Bat / how I wonder where your at'.--- `Bat' was a math prof at Oxford whose lectures and proofs were unaccustomed. Its not the maths which is Batty here but the nomenclature, the notation and the abruptness.