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Review
'Although this book is not written as a textbook but rather as a work for the general reader, it could certainly be used as a textbook for an undergraduate course in number theory and, in the reviewer's opinion, is far superior for this purpose to any other book in English.' From a review of the first edition in Bulletin of the American Mathematical Society
'… the well-known and charming introduction to number theory … can be recommended both for independent study and as a reference text for a general mathematical audience.' European Maths Society Journal
'Its popularity is based on a very readable style of exposition.' EMS Newsletter
'… the well-known and charming introduction to number theory … can be recommended both for independent study and as a reference text for a general mathematical audience.' European Maths Society Journal
'Its popularity is based on a very readable style of exposition.' EMS Newsletter
Product Description
The theory of numbers is generally considered to be the 'purest' branch of pure mathematics and demands exactness of thought and exposition from its devotees. It is also one of the most highly active and engaging areas of mathematics. Now into its eighth edition The Higher Arithmetic introduces the concepts and theorems of number theory in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical significance. Since earlier editions, additional material written by J. H. Davenport has been added, on topics such as Wiles' proof of Fermat's Last Theorem, computers and number theory, and primality testing. Written to be accessible to the general reader, with only high school mathematics as prerequisite, this classic book is also ideal for undergraduate courses on number theory, and covers all the necessary material clearly and succinctly.
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Most Helpful Customer Reviews
13 of 14 people found the following review helpful
Format:Paperback
This was the first book I read on the theory of numbers. It is a fascinating subject, and this book is the perfect introduction. It is written by Harold Davenport, a famous number theorist of the 20th century. It gives an introduction to several areas of the subject (primitive roots and quadratic residues, sums of squares, continued fractions, quadratic forms, Diophantine equations) which are accessible without much prior mathematical knowledge, and sticks to elementary methods whilst providing hints and pointers to the use of analytic methods (Dirichlet's theorem on primes in arithmetic progressions, Diophantine Approximation) and elliptic curves in the subject.
Also from quite an early point in the book (the first chapter) the author mentions various unsolved problems, some of which are more famous (Goldbach's conjecture) than others (Erdos' covering problem). The reader might get the impression that it is easy to come up with propositions which one can neither readily prove nor dispose of, and this is true on a global and historic scale.
The book is not written in a lemma-theorem-proof style at all. Of course for a more advanced book, the more structured approach is useful. But for a book of this kind, the more continuous approach works brilliantly.
Also from quite an early point in the book (the first chapter) the author mentions various unsolved problems, some of which are more famous (Goldbach's conjecture) than others (Erdos' covering problem). The reader might get the impression that it is easy to come up with propositions which one can neither readily prove nor dispose of, and this is true on a global and historic scale.
The book is not written in a lemma-theorem-proof style at all. Of course for a more advanced book, the more structured approach is useful. But for a book of this kind, the more continuous approach works brilliantly.
7 of 8 people found the following review helpful
Format:Paperback|Amazon Verified Purchase
Introduction
Several years ago, a tutor showed me her copy of this book and highly recommended it as a primer for this topic. I have compared the contents list (seventh edition, 1999) of mine, against this new copy contents list and both are pretty close to each other. Although this edition will be much updated.
Why is this book worth recommending?
This book encourages the reader to return to its pages again-and-again. This book i.m.h.o has a high level of initial readability, rather than featuring many equations, to create a level of understanding that is rewarding. For example, the initial topics gently explain about 'primes'. Then book clarifies this by branching into topics such as 'Congruences' and 'Quadratic residues' in a way that an author of a few complex analysis books would be proud.
Conclusion
This is a book thats not your final destination in mathematics, but a book to help you reach it.
Several years ago, a tutor showed me her copy of this book and highly recommended it as a primer for this topic. I have compared the contents list (seventh edition, 1999) of mine, against this new copy contents list and both are pretty close to each other. Although this edition will be much updated.
Why is this book worth recommending?
This book encourages the reader to return to its pages again-and-again. This book i.m.h.o has a high level of initial readability, rather than featuring many equations, to create a level of understanding that is rewarding. For example, the initial topics gently explain about 'primes'. Then book clarifies this by branching into topics such as 'Congruences' and 'Quadratic residues' in a way that an author of a few complex analysis books would be proud.
Conclusion
This is a book thats not your final destination in mathematics, but a book to help you reach it.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
6 reviews
62 of 65 people found the following review helpful
This is a MUST BUY if you want to learn Number Theory!
8 Jun 2000
Format:Paperback
This book is an AMAZING introduction to the Theory of Numbers. It assumes no previous exposure to the subject, or any technical mathematical knowledge for that matter. Its prose is lucid and the style appealing. Davenport chose NOT to write a lemma-theorem-proof kind of book, and the result is a marvelous, eminently readable introduction to the subject. Its wonderful to read a book where good prose is used to appropiately substitute a massive collection of uninviting symbols. I've also been reading other books on Number Theory, such as Hardy & Wright, but none are as clear as this one.
I found the chapter on quadratic residues (which includes the reciprocity law) to be especially well written. The section on computers and number theory is excelent as well. A concise and coherent discussion of crytography and the RSA system is included here. The organization of the book's chapters is fantastic. Each chapter builds up on results proven in the previous ones, showing well the connections between the different aspects of Number Theory. The exercises of the book range from simple to challenging, but are all accesible to someone willing to put effort into them.
This would be an excelent source for learning number theory for mathematical competition purposes, such as the ASHME, AIME, USAMO, and even for the International Mathematical Olympiad. The book contains much more than what is needed for these competitions, but the olympiad/contest reader will benefit greatly from a study of Davenport's work.
The book can certainly be used for an undergraduate course in Number Theory, though it might need supplementary materials, to cover a semester's worth of work. I know the book has been used in the past in previous editions as the main text for Math 124: Number Theory at Harvard University.
I would also recommend this book to anyone interested in acquanting themselves with Number Theory.
Awesome! There is simply no other word that describes The Higher Arithmetic.
22 of 32 people found the following review helpful
Good book, but if you have the money, there are better
3 July 2004
Format:Paperback
Well, this is definitely a very good introduction to number theory. The author provides clear, readable proofs of all the most basic theorems on topics such as congruences, sums of squares, etc. He explains things quite well. However, despite costing almost 2.5 times as much, I would recommend Hardy and Wright's book An Introduction to the Theory of Numbers more highly than Davenport's book. Seriously, although it may seem good that Davenport doesn't require a knowledge of calculus as a prerequisite for his book (which Hardy DOES require), one probably shouldn't learn number theory until one has a good backrground on topics ranging from improper integrals to infinite series. Because Davenport does not require calculus as a prerequisite, he neglects HUGE aspects of what could actually be considered BASIC number theory: namely, the basic analytic aspects (such as Tchebycheff's results on the Prime Number Theorem) and the additive theory (i.e. partitions and such, as well as the basics of the generalized theory surrounding Waring's problem for high powers of integers). So, my recommendation is, wait until you know integral calculus and the theory of infinite series BEFORE buying a book on number theory, and then buy Hardy and Wright's book rather than this one.
1 of 1 people found the following review helpful
A classic that is still valuable
1 Jan 2012
Format:Paperback
The principal virtue of this text is that it can be taken up by readers with no more than ordinary high school level mathematical maturity yet it can aptly serve as the text for an undergraduate level first course in Number Theory. It is a model of clear and concise mathematical enunciation.
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