Foundations of Analysis (AMS Chelsea Publishing) Reprint Edition

by Edmund Landau (Author)

17 customer reviews

ISBN-13: 978-0821826935

ISBN-10: 082182693X

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Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous ""Grundlagen der Analysis"" - also available from the AMS - answers these important questions.

Foundations of Analysis (AMS Chelsea Publishing)

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Product Details

Series: AMS Chelsea Publishing
Hardcover: 136 pages
Publisher: Chelsea Pub Co; Reprint edition (May 2001)
Language: English
ISBN-10: 082182693X
ISBN-13: 978-0821826935
Product Dimensions: 0.5 x 6 x 9 inches
Shipping Weight: 12 ounces (View shipping rates and policies)
Average Customer Review: 4.4 out of 5 stars See all reviews (17 customer reviews)
Amazon Best Sellers Rank: #563,234 in Books (See Top 100 in Books)
- #156 in Books > Science & Math > Mathematics > Pure Mathematics > Number Theory
- #3886 in Books > Textbooks > Science & Mathematics > Mathematics

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Customer Reviews

4.4 out of 5 stars

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41 of 41 people found the following review helpful

The beggining of it all

By Guilherme on April 17, 2002

Format: Hardcover

Landau's most known book is this little masterpiece. If you want to see everything about numbers proved, from the beggining, assuming just logical and set-theoretical principles and the five Peano axioms, you will find it here. You will see the proof of why 1+1=2, for instance, or why a+b=b+a. Usually people learn analysis with a lot of pictures and assumptions, and every once in a while one asks himself: how does it all begin? Because sometimes you see something which ought to be evident proved, and something which ought to be proved assumed. I recall that when I first met this book I became amazed and read it through with a lot of willing. It is difficult reading, so be prepared. That's because Landau wanted to follow the axiomatic Euclidean style in its most pure way. So the book is in the non-merciful telegram style of presenting everything in terms of "Axioms", "Definitions", "Propositions". Few books before and after strove to reach such pure and clear presentation of arithmetic. Thank God some one had once the patience to write such careful and complete text! In this book the words of Edgar Allan Poe are more than anywhere true: "What I here propound is true:-therefore it cannot die:-or if by any means it be now trodden down so that it die, it will 'rise again to the Life Everlasting'".

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12 of 12 people found the following review helpful

A text, not a textbook

By karen rogers on February 8, 2011

Format: Hardcover

Most "textbooks" really consist of tutorial lectures: motivation, explanation, examples, as well as definitions, theorems, and proofs. But think of a novel: it does not consist (normally) of the author lecturing you about the themes of the novel, explaining what the novel is trying to do, etc. The novel just does it. It is a "text" to be read and discussed, to be considered, to be taught. Landau's Foundations of Analysis is just such a text, a mathematical one. The reader has to figure out much of what is going on and why. (Euclid is also such a text.) In other words you have to dig for yourself. The work of figuring it all out is part of the intended learning. It is not beyond a reader's individual power to do so. A teacher can help with the motivation, explanations, etc.; so can fellow readers. It is a good seminar book. Solitary readers will have to fend for themselves, and individual responses to the text not surprisingly vary. Readers with a mathematical background may find it an easy read; others may or may not find the challenge rewarding. It deserves its classic status and that so many admire it ought at least to make one curious about it.

Many have commented on the content. I will only add my admiration of the fact that from just a few assumptions one can create all of arithmetic and the basis for an easy proof of the intermediate value theorem, something I just couldn't imagine how to do when I first took calculus.

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11 of 11 people found the following review helpful

Very nice book.

By nightowl03d on January 16, 2007

Format: Hardcover

Ever wonder how many of those field axioms in the front of your calculus/analysis book were redundant and what were essential? Are you a skeptic, who is not sure that you can really derive the complex numbers from Peano's axioms? Are you someone who does not know what the hell Peano's axioms are, even though you do know what an Axiom is? Then this book is for YOU!

This book proves the properties of the complex numbers and some of their subsets, (positive integers, integers, rationals, and reals). The proofs are quite rigorous, and provide an excellent foundation for which to study calculus, real or complex analysis. I thought it was a wonderfull read, and surprisingly accessible, given its hieroglyphic initial appearance.

If you decide to read this book in several sittings, I recommend being sure you cover each of the small sections. If you can't finish a full section, restart the section. I think that if you do it that way the results will seem more cohesive.

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7 of 7 people found the following review helpful

Classic and Unsurpassed

By Daniel Kian Mc Kiernan on October 11, 2013

Format: Hardcover

I acquired and worked my way through this book about 25 years ago. I remain very glad to have done so.

Landau begins with the system of the natural numbers as described by the Peano axioms, and from these constructs the system of the integers. Then, from the system of the integers, he constructs the system of the rational numbers. From the system of the rational numbers, he constructs the system of the real numbers. And, from the system of the real numbers, he constructs the system of the complex number.

Landau presented this book as something to be read in a very few days, and then held for possible future reference, by mathematics students who wanted to seee the real and complex number systems rigorously constructed `from scratch'.

Given exactly those ambitions, this book offers virtually nothing in the way of `intuition'. It's axiomata, theoremata, lemmata, and proofs. The spartan exposition will have different effects on different sorts of readers. For those whose intuitions would have closely but imperfectly matched the intuitions of the author, an intuitive discourse might have been a very good thing. But those whose intuitions perfectly matched those of the author would fall asleep were this much material surrounded by prose that offered no surprises. Those whose intuitions were poorly fit by the author would have found such discourse a source of misery. For my part, I doubt that I would have made it through this book had it ever slowed or cut away from the chase.

(I was not a young German mathematics student of the early 20th Century, but a young-ish American economist of the late 20th Century; it took me rather longer to work my way through this book that Landau apparently envisioned.Read more ›

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Most Recent Customer Reviews

Avoid This Antique

Landau's "Foundations of Analysis" is no longer the definitive rigorous development of the real numbers. Read more

Published 3 months ago by Rick Ferraro

This book is fantastic in a sort of "read it once and then ...

This book is fantastic in a sort of "read it once and then just remember the gist of it" way. Read more

Published 9 months ago by Mr. F. Smith

Five Stars

The best and readable!

Published 12 months ago by George H. Mccormack

Modeling irrational and complex numbers

This is a superb book if you want to understand numbers and the laws they satisfy under addition and multiplication. However it's not the most modern treatment of the subject. Read more

Published 15 months ago by Jeremy Roach

but it is one of the best books in mathematics that I have read

I came across this gem of a book many years ago. Reading it was not enough; I wanted to buy a copy. That had to wait, but finally I did it. Read more

Published 15 months ago by Bhaskar Sarkar

Edmund Landau

It is a very interesting book af a great Jewish mathematician from Germany. In fact, a good and relatively cheap edition.

Published 23 months ago by Juan Valverde

What a great book

This book is very great indeed,if you are a mathematician or a mathematics enthusiast or a serious student of mathematics and also want to know why the common algebraic laws as... Read more

Published on May 8, 2011 by Amarnath Bhattacharya

A Complete Rubbish

I got very upset when I finished reading the book. The book contains a complete rubbish not worthy of reading. Read more

Published on October 24, 2010 by Steve

The reals from the integers via Dedekind cuts.

This book dates from 1951 but it's still a satisfactory place to see how the real (and even complex) numbers can be developed from the integers. A quick read.

Published on April 16, 2008 by Mark Flanagan

Numbers, from Peano through Dedekind

This excellent little book starts with Peano's Axioms and finishes with the complex numbers. It is a good source of challenging proofs to support a modern algebra course. Read more

Published on January 11, 2007 by T. H. Rousseau, III

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