- Paperback: 250 pages
- Publisher: Cambridge University Press; 8 edition (23 October 2008)
- Language: English
- ISBN-10: 0521722365
- ISBN-13: 978-0521722360
- Product Dimensions: 15.2 x 1.5 x 22.8 cm
- Average Customer Review: Be the first to review this item
- Amazon Bestsellers Rank: #6,79,841 in Books (See Top 100 in Books)
The Higher Arithmetic: An Introduction to the Theory of Numbers Paperback – 23 Oct 2008
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'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
Now into its eighth edition and with additional material on primality testing written by J. H. Davenport, The Higher Arithmetic introduces concepts and theorems in a way that does not assume an in-depth knowledge of the theory of numbers but touches upon matters of deep mathematical significance.See all Product description
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Most helpful customer reviews on Amazon.com
Now, some comments are in order regarding the use of this text. It is a common trend in math textbooks to present in a theorem-proof style. So naturally students get comfortable with this and it becomes a hindrance when a book does not meet such a format. Davenport's book is not written like this, so if you require the stock format, you will be disappointed. However, the clarity of this text provides an understanding beyond what a stock treatment can if you can go beyond this artificial hurdle.
Andrew Wiles has said that when he wants to review a topic he always picks up Davenport first, then goes to Hardy and Wright. Davenport's book can almost be read like a novel it is so good and clear. But you won't find symbols marking the end of proofs or telling you where they begin. He simply explains as a great teacher would in conversation. Usually, the logical flow is clear enough where you do not need to see "Proof .... QED." Furthermore, this is not an introductory text that a researcher could find nothing of interest in. Despite being elementary of character, it has innovations in development and bears a stamp of Davenport's brilliant mind. Just to provide an example, the section on continued fractions is very original and beautifully done. There is definitely plenty of meat on the bone even for an experienced reader of number theory.
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.