My own first exposure to the book came when I was an undergraduate and one of my friends came back from an REU and gave a talk on p-adic numbers. I was intrigued by one of the counterintuitive results that one gets when working in the p-adic topology — all triangles are isosceles, but none are equilateral! I checked it out of the library and found a book that was mathematically rich while still being clear and lively to read. In the intervening decade and a half, as I have spent much time reading about and working with p-adic numbers in my own research, I have read any number of treatments of the subject, and certainly several other sources come to mind that have various advantages in their different goals and perspectives. But if I had to recommend one book on the subject to a student — or even to a fully grown mathematician who had never played with p-adic numbers before — it would still be this book. It turns out that the p-adic rational numbers are similar to the real numbers in some senses they are locally compact completions of the normal rational numbers, they are not algebraically closed while being very different in other senses the p-adic rationals are totally disconnected and are not ordered, for example.

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My own first exposure to the book came when I was an undergraduate and one of my friends came back from an REU and gave a talk on p-adic numbers. I was intrigued by one of the counterintuitive results that one gets when working in the p-adic topology — all triangles are isosceles, but none are equilateral!

I checked it out of the library and found a book that was mathematically rich while still being clear and lively to read. In the intervening decade and a half, as I have spent much time reading about and working with p-adic numbers in my own research, I have read any number of treatments of the subject, and certainly several other sources come to mind that have various advantages in their different goals and perspectives.

But if I had to recommend one book on the subject to a student — or even to a fully grown mathematician who had never played with p-adic numbers before — it would still be this book.

It turns out that the p-adic rational numbers are similar to the real numbers in some senses they are locally compact completions of the normal rational numbers, they are not algebraically closed while being very different in other senses the p-adic rationals are totally disconnected and are not ordered, for example.

One of the main reasons that we care about p-adic numbers is that, by working in this new topology, one can use the analytic facts about the power series to learn about the algebraic properties of divisibility and answer number theoretic questions. This puts the subject right at the crossroads of algebra and analysis, although it also has its tentacles in number theory, topology, algebraic geometry, dynamical systems, and even theoretical physics, as some view it as a promising approach to studying the non-Archimedean geometry of space-time at small distances.

The first few chapters have very little in the way of prerequisites, and the later chapters require only the basic topics from undergraduate courses in analysis and algebra. The book is filled with exercises, many of which have hints or comments in an appendix, and could easily be used as a textbook for a course or for an independent study. See, for example, the work done by the folks at SAGE. First crack at the new Keith Devlin?

Darren Glass is an Associate Professor of Mathematics at Gettysburg College whose research interests include number theory, algebraic geometry, and cryptography. If you want to confirm that this review is real and under no influence by the editor, feel free to contact him at dglass gettysburg.

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## Fernando Q. Gouvêa

Tthe non-expanded version is now published by Dover. The book is an introduction to the history of mathematics with the needs of mathematics teachers chiefly in mind. The main differences between the expanded second edition and the just-plain second edition are: The expanded edition has a prettier cover. The expanded edition includes problems and projects. Apart from this, the two versions are identical. The Dover version is very affordable; buy copies for all your friends!

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## p-adic Numbers: An Introduction

Set up a giveaway. See all 3 reviews. The completion of this space can therefore be constructed, and the set of -adic numbers is defined to be this completed space. If andthen the expansion is unique.

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## GOUVEA P-ADIC NUMBERS PDF

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## p-adic Numbers

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